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The Ellipse and Minoan Miniature Art:

Analysis of the 5-priestess signet ring from the

Mycenaean Griffin Warrior’s tomb at Pylos

Update 1

Dimitrios S. Dendrinos

Professor Emeritus, University of Kansas, Lawrence, Kansas, US

In residence, at Ormond Beach, Florida, US

Contact: cbf-jf@earthlink.net

December 5, 2017

The Minoan gold signet 5-priestess ring from the Mycenaean “Griffin Warrior”

tomb at Pylos; length to width are at a Golden Ratio. A four-fold magnification.

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Table of Contents

Abstract (updated)

Summary

On Ellipses, their Origins and Geometry.

The origins of the elliptical oval shape: circles, arcs, apses and ellipsoids

The Geometry of an ellipse

An overview of the signet ring

Borchardt’s ellipsoid from Luxor

The likely 3-circle Genesis of an ellipse

A Brief Review of Ellipsoids and Ellipses in Key Archeological Contexts.

Microscale ellipsoids in artifacts

Small scale pseudo-elliptical structures

Large scale ellipses in structures: the roman amphitheaters and forums

The 5-Priestess Ring from the “Griffin Warrior” Tomb at Pylos.

Description of the ring’s iconography and its Astronomy

The ring’s Geometry

Concluding Remarks

Six Notes

Two Maps

Appendix I. Ellipses, Circles, Cones

Appendix II. Ellipse’s directrices and director circles

Appendix III. The Ellipsograph

References

Acknowledgments

Legal Note on Copyrights

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Abstract (updated)

The Mathematics and embedded Astronomy are explored of the almost elliptical in shape

Minoan 5-priestess gold signet ring of the c 1450 BC Mycenaean “Griffin Warrior” tomb at Pylos

found during the 2015 archeological excavation there. It is documented that the shape of the

ring is extremely close, albeit not exactly identical, to the true ellipse of identical major and minor

axes. The likely knowledge of ellipses possessed by the ring’s maker is identified. In the paper, a

detailed description of the ring’s iconography is also offered, which to an extent differs from the

current archeologists’ based description. The iconography’s Astronomy, is found to be associated

with a ceremony dedicated to the fertility of Mother Earth, that quite likely was taking place

around the Winter Solstice. An estimate of the ceremony’s duration, eighteen days, is also

obtained, as having been engraved onto the ring’s iconography.

An evolutionary path on how the true elliptical shape was gradually approximated through the

design of artifacts and structures of various ellipsoidal forms in 2-d, based on a differing number

of circles used to obtain their approximation is put forward. Approximations likely started with

three circles, the Genesis of the ellipse, that took place at an unknown place and time period. It

was followed by the use of four circles, as is the case of the particular Minoan ring under study

in this paper. It likely ended with the use of five, and that was the case of the post-Ramesses II,

Burchardt ellipsoid at the Luxor Temple in Egypt. Prior Neolithic structures that employed

pseudo-elliptical designs are also mentioned in the paper, which identifies a point where a major

phase transition did occur, whereby pseudo or quasi-elliptical oval stone enclosure designs

employing at times one or two semicircles were transformed into ellipsoidal, schemata that

approximated true ellipses. A brief overview is offered of oval (pseudo-elliptical, ellipsoidal and

elliptical) in shape artifacts and built structures from the Epipaleolithic to the Roman Era, offering

a glimpse into, and outlining a theory of the oval shape’s evolution in Art and Architecture, which

is the central theme running through the entire paper.

Update 1. This updated version of the paper contains the exact measurements this author

received on December 3, 2017 from Luci Philips, who obtained them directly from prof. Jack

Davis, from the University of Cincinnati. The author is indebted to both. The measurements

confirmed the author’s finding reported in the original December 2, 2017 version of the paper

that the dimensions of the ring’s top mounted part’s surface (what jewelers refer to as the

“bezel” of the ring) are in fact such that the ratio of length to width is the Golden Ratio. The

precision with which these measurements approach the Golden Ratio leaves little doubt that the

ratio of length to width implanted on the ring was not random but deliberate. The ring thus

represents the first recorded case in the History of archeological artifacts where this ratio is

recorded. This updated version of the paper takes these exact measurements into account. This

particular finding also lends support to the proposition that a designed scaled up drawing was

made by the artist prior to the ring’s actual molding and carving, hence requiring the answer to

the question, how did the maker of the ring approach the shape of a true ellipse – a key question

in the Archeology, Mathematics (Geometry and Algebra), Architecture and Art of ellipses.

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Eye of the Horus, middle 3rd millennium BC, the falcon’s oval eye. Source: [3.18]

Nefertiti’s Bust from a 3-d scan, top view: it contains an ellipsoid, and on its

sides two parabolas. Source of diagram: the author.

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The Flavian Amphitheater (Colosseum) in Rome: a set of five concentric

ellipsoids surrounding a central elliptical in shape arena. Ellipse at a grand scale,

in a fertility depicting Urban Design context at Rome’s Imperial Forum.

Summary.

A Minoan ring from the second quarter of the 2nd millennium BC is found to have been imbued

with state of the art in Mathematics of that Era and some Astronomy. In this paper the story of

how a 3-d ellipsoidal surface was transformed into a shape that in 2-d closely approximates an

ellipse, partly by means of scale, is told. Its miniature structure was in part the agency that

rendered an otherwise in both 2-d and 3-d ellipsoidal form into an ellipse. Against this backdrop,

in a story that extends its tentacles to Egypt and the Temple at Luxor, this paper analyzes the

almost perfect elliptical Geometry of the largest signer ring (among fifteen or so seals and rings)

found in the Mycenaean so-called “Griffin Warrior” tomb from 1450 BC at the Pylos archeological

site at the South-Western corner of the Peloponnese region of Southern Greece, see ref. [2.2].

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The roots of 2-d and 3-d ellipsoids in ancient artifacts and structures are explored at some length,

with special attention paid to the so-called Borchardt 2-d ellipsoid from Luxor, ref. [2.7]. Some

key questions are asked. What was the then current knowledge by the ring’s maker of ellipses.

How was the 3-d ellipsoid’s shape made to approximate and resemble that of a true 2-d ellipse

at its perimeter, by the miniature artist. To address these questions, the paper incorporates some

analysis of both ellipses and 2-d, as well as 3-d, ellipsoids over time. Further, a detail description

of the ring’s composition is included in the analysis. A comprehensive description of the ring’s

ellipsoidal three-dimensional iconography offers a corroborative narrative to the underlying

specific 3-d ellipsoidal ring morphology. Geometric analysis of the ring’s shape and iconography

establishes that the Minoans, in contrast to the artist at Luxor who drew three centuries later a

2-d ellipsoid that obviously isn’t a true ellipse at the scale drawn, were the first to create a 2-d

ellipsoidal oval form that very closely resembles the shape of a true ellipse. Hence, we are forced

to assume that the ring’s artist had acquired some elementary mathematical knowledge of

ellipses, and implanted on the ring a schema that we may refer to as a “proto ellipse”.

It is remarkable that the ring’ bezel has an almost perfect elliptical 2-d surface and length to width

at a Golden Ratio, in spite of 3500 years of wear and tear. Made more than a millennium earlier

than the time the Mathematics of ellipses were discovered by Menaechmus (4th century BC) and

an ellipsograph was invented by Archimedes (3rd century BC), the making of the Minoan

ellipsoidal rings and seals require study by the historian of Mathematics. The specimen is not only

an extraordinary example of Minoan miniature Art, but also a depository of knowledge as to what

exactly the Minoans knew at the second quarter of the 2nd millennium BC about ellipses, three

centuries before the Egyptian artist at Luxor, and a couple of centuries earlier than Amarna, when

and where Thutmose was creating Nefertiti’s Bust implanting an ellipsoid and two parabolas on

it; or the time that the artist who created Tutankhamun’s Mask embedded on it a parabola.

Moreover, the paper advances the thesis that some Astronomy was also incorporated, in a

masterful way, into the theme of the ring’s iconography. The iconography offers us not only a

message regarding the type of ceremony carved on the ring, dedicated to the fertility of Mother

Earth. But it also offers a possible date the ceremony was taking place, during the day of the

Winter Solstice; as well as the possible duration in days (eighteen) of those ceremonies.

Although the central focus of the paper is the Minoan 5-priestess gold ring, a broad review of

related oval artifacts and structures, pseudo-elliptical and ellipsoidal, is also supplied. Even

though the subjects are seemingly unrelated, and only their general morphology links them, the

aim is to stimulate the reader into examining their connections which, albeit subtle, cut deep into

their mathematical foundations. In putting forward a General Theory in the Evolution of Form In

both Art and Architecture, these structures are nodal. Of course, their full treatment in a spatial-

temporal context and under differences in scales is beyond the constraints of a single paper.

Nonetheless, the diverse underlying cultural (socio-economic-religious) and symbolic contexts

that shaped them, guide the reader into drawing not only direct morphological connections

among all these diverse artifacts and monuments, but also derive research suggestions towards

their mathematical treatment, the key scope of this paper.

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1. On Ellipses, their Origins and Geometry.

1.1 The origins of the elliptical oval shape: circles, arcs, apses and ellipsoids

In this section, and for a very brief moment, we shall venture into the fascinating world of pseudo-

ellipses, ellipsoids and ellipses. At the outset, it must be noted that in Geometry the term

“ellipsoid” is used to connote 3-d structures, which in their 2-d cross sections are true ellipses. In

this paper, the term “2-d ellipsoid” is used to identify crude ways oval shapes approximate true

geometrically defined ellipses, or parts of an ellipse, on a plane.

Ellipses have attracted the ancient designers’ attention in the making of both artifacts and built

structures, for a variety of reasons. Such reasons range from the mystical, religious, symbolic and

ceremonial connotations elliptical forms or ellipsoids (in their various approximations to ellipses)

seem to have carried since the beginning of the Neolithic. Qualitative similarities to the female

symbol of fertility, the womb, and in approximating the shape of an egg, or an eye, human or

animal (as for example in the case of the “eye of the Horus”, see ref. [4/17] on its origin), ellipsoids

and ellipses are evolutionary derivatives of the circle and circle-like forms in ancient design

platforms. There has been an extensive literature on the Architecture, Astronomy and Art of

these structures, which has been reviewed, among other sources, by this author in ref. [1.1-18].

Alexander Thom’s analysis on the “egg” related similarities to pseudo/quasi-elliptical (2-d

ellipsoidal) stone enclosures, in reference [2.2] dealing with a Neolithic monument at Carnac, is

notable. That work also covers a number of stone rings and enclosures of the British Isles, offering

some theoretical perspective on how these rings were designed, formed and constructed by their

architects. That work this author extended in references: [1.2] on Gobekli Tepe’s structures

(stone enclosures) C and D; [1.4] on Stonehenge’s Trilithons ensemble; and [1.5] or Brittany’s Le

Grand Menec Western stone enclosure. These references present cases where oval, quasi-

elliptical, or pseudo-elliptical (proto 2-d ellipsoidal) stone structures have appeared in Neolithic

monuments. Very likely, symbolic factors were the underlying major force that propelled the

ancient architect and artist to design structures and artifacts in ellipse-based (arc and apse type,

as well as pseudo-elliptical, proto ellipsoidal, egg-shaped or quasi circular) configurations.

Besides symbolism, purely Architecture function and form as well as Astronomy related factors,

can certainly not be excluded from the calculus that at the end produced quasi-elliptical or partly

elliptical (in general, pseudo-ellipsoidal) floor plans in Neolithic monumental Architecture.

The Minoans, at the boundary between the Bronze and the Iron Ages, elevated the quasi-elliptical

(pseudo-ellipsoidal) designs of the Epipaleolithic and Neolithic Eras to the almost true ellipse

proper shapes we observe on the Minoan rings and seals of the first half of the 2nd millennium

BC. For the first time, we detect the Minoan artist embedding in these artifacts almost exact

elliptical shapes, assisted by the scale of the artifact, as some basic mathematical properties

drawn from the Geometry of ellipses can be detected in the miniature Art of these artifacts.

Since the 12th millennium BC, various small in scale architectonic configurations entailing simple

quasi-circular shapes and a variety of arcs and apses have appeared in Eurasia and Africa. Over

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time, more complex forms developed. They included sections of 2-d (and even 3-d) pseudo-

ellipsoids (the various and numerous stone circles, rings and cairns), mixed with other geometric

shapes (involving straight lines or triangular forms). Thousands of such monuments are dispersed

in Neolithic Western Eurasia. They constitute a potpourri of built structures, diffused at all points

of archeological space-time. Among them certain key monuments have punctuated that complex

ecology of Neolithic Architecture. Closely related to circles, 2-d ellipsoids (in whole or in parts)

have been among the primordial shapes in Archeology’s Monumental Architecture and Art.

Imperfectly drawn and constructed at the beginning of sedentary living, whole or partial in floor

plan design pseudo-ellipsoids are encountered in many notable archeological contexts: from the

11th millennium BC Tell Qaramel (see reference [1.1] for discussion and relevant citations); to the

6th millennium BC (according to this author) Gobekli Tepe structures, especially structures C and

D (see reference [1.2] for discussion and citations); to the Late Neolithic and Bronze/Iron Ages

stone rings and enclosures of Brittany and the British Isles (see reference [1.3] for discussion and

related citations); to middle 3rd millennium BC Stonehenge Phase 3 II Trilithons ensemble (see

reference [1.4] for discussion and associated citations); to Brittany’s Carnac (Le Grand Menec

Western egg-shaped stone enclosure) and the Maltese 4th millennium BC apse limestone

structures (see reference [1.5] for discussion and related citations); all these monuments

represent stellar examples of structures where the architect implanted elements of ellipses (in

effect, designing oval in shape quasi-ellipsoids) into the floor plan of masonry constructions.

The work by A. Thom (and a number of others who have followed in his steps), who (as already

noted) has provided an initial classification of stone circles must be specifically noted, found in

reference [2.2]. The author, in the references cited, has extended the A. Thom classification of

“egg type” structures by incorporating an evolutionary component to it. Thom’s work is an

attempt to add some rationale to a seemingly unrelated variety of floor plans in stone enclosures,

on an attempt to produce floor plan designs consistent with various Astronomy-related

alignments that Thom has claimed to have detected on the placement of individual stones within

these structures. In the pseudo-ellipsoidal design of these stone rings (or enclosures) Thom saw

an effort to derive elliptical forms by their architects who didn’t possess either the Mathematics

of ellipses or the means to draw them (ellipsographs, see Appendix III on the Archimedes one).

All that seemingly unrelated, imperfect nonetheless, or partial in essence drawing of elliptical

shapes (pseudo-ellipsoids) apparently came to an end in the middle of the 2nd millennium BC,

more precisely in the 2700 – 2300 BC time period. The Minoan Civilization with the design and

making of seals and rings in what seem to be perfect elliptical shapes at a microscale, marked a

great leap forward, an unprecedented step in the ladder of evolution in Art. That point in space-

time signals the beginning of an attempt to draw true elliptical designs, the origin of the

perfectly elliptical Geometry in artifacts. How much however of the underlying Mathematics of

true ellipses was mastered by the Minoans is a research question, an answer to which will be

attempted in the paper’s penultimate section’s last sub-section (3.3), when the miniature Art of

a specific Minoan ring from that period will be examined in some detail.

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Although the symbolism part of either 2-d pseudo-ellipsoids (arcs, apses), quasi-circles and circle

-like shapes, or the ellipsoids and ellipses’ morphology is not an issue this paper intends to deeply

delve into, it is noted nonetheless that there is a strong fertility aspect to them. The female womb

has been often associated, by many cultures in different points in space-time, with such forms,

as is the form of an egg, see ref. [1.5]. However, there are numerous other symbolic aspects of

circles and ovals that have been suggested, but all that will not be a topic for analysis here. It

seems nonetheless that the ellipse, as a shape that gradually evolved from the quasi/pseudo

stages of circles to an ellipsoid to a true geometric ellipse, fits this fertility symbolism bill well. In

our brief discussion on the iconography of the ring under analysis here, it will be shown how the

almost true elliptical shape of the ring (standing as a symbol of both, the female womb and the

egg) relates directly to the theme depicted on its top surface, which according to this author is a

ceremony to Mother Earth’s fertility performed at Winter Solstice, possibly for a number of days

immediately prior to and post the precise day of the Winter Solstice. It is shown here that the

ring’s iconography offers a narrative outlining the length of this ceremony in days (eighteen).

1.2 The Geometry of an ellipse

In this brief subsection, the elements of the ellipses’ Geometry and underlying Algebra will be

indicated, since the terminology used in the paper is tied to these elements, see Figure 1.1. As

this paper is not addressed to mathematicians but to the general public, with some interest in

Archeology and some exposure to College-level Mathematics, the formal theorems underlying

the statements made here will not be offered (either formally stated or proved). The interested

reader may access the citations supplied, see ref. [5.1] – [5.6] for the basic elements, theorems,

lemmas, etc. on the beautiful world of ellipses. Origins of the formal Mathematics involved in

ellipses have been addressed by this author in ref. [1.12], on the elliptical ground level floor plan

of the fourth quarter, 4th century BC Tumulus at Amphipolis, the so-called “Kasta Tumulus”, and

the 3-d shape of the Tumulus in the form of an ellipsoid.

Figure 1.1. The basic elements in the Geometry of an ellipse. Sources: ref. [3.1] and [5.3].

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There is a topological and geometric equivalence between ellipses and circles, and a note on this

equivalence is offered in Appendix I. Under perspective one can transform a circle into an ellipse.

A special case (where in this specific perspective the parallel lines meet at infinity) of the above-

mentioned equivalence (or topological transformation) is shown in Figure 1.2. There, the

property of midpoints of parallel chords in circles and ellipses are demonstrated – they remain

invariant, on a diameter inside the circle, which in the case of the ellipse is transformed into a

line joining the points of tangency of the embedding parallelogram, see the diagram at right,

which is nothing but the transformed square of the diagram at left in Figure 1.2, seen in the

special perspective just stated. Notice that the major and minor axes of the resulting ellipse (at

right in Figure 1.2) are nothing but the two diagonals of the square embedding the circle at left.

Figure 1.2. A property of ellipses. Source of diagram: ref. [3.1] and [5.3].

Key terms to be used in this paper are the basic terms of an ellipse, as shown in Figure 1.1. The

center of the ellipse (the origin) is the point (0,0) in an orthogonal Cartesian 2-d space. The major

(longest, to be designated as 2a) and the minor (shortest, to be designated as 2b) axes of the

ellipse are orthogonal (they meet at a 90 angle). Quantities a and b are the necessary and

sufficient elements to know for drawing an ellipse. On the major axis of the ellipse, its two focal

points F1 and F2, the ellipse’s two foci, are found. There are numerous other point, lines and

curves, some of them to be discussed later in the text, of interest. However, the two foci F1 and

F2 are key points for the analysis that follows and the reader must be familiar with them to follow

the narrative. Their distance from the ellipse’s origin O (the center) is designated as length c, and

it is referred to as the linear eccentricity of the ellipse.

Moreover, the ratio c/a is the eccentricity ratio e of the ellipse. To these two focal points, F1 and

F2, the sum of the distances from any point P on the ellipse’s perimeter remains unchanged, and

equal to 2a (the length of the major axis). Put in the form of an algebraic equation {(PF1) + (PF2)

= 2a}. This is the fundamental geometric and algebraic property of an ellipse. It is noted from

the application of the Pythagorean Theorem that:

c = (a^2 – b^2)^1/2 = {(a + b)(a – b)}^1/2.

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Notice that in the case of a circle (where a = b) linear eccentricity is zero; hence, the more “circle

like” an ellipse looks, the smaller its linear eccentricity. Two additional sets of elements

associated with an ellipse are of special interest here also, the two director circles of an ellipse,

and the two directrices of an ellipse, see Appendix II, their role regarding this ring to be further

explore in subsection 3.3. The total area inside an ellipse is given by the simple formula: A = ab,

where is the irrational number 3.141592… (the ratio of a circle’s circumference to its diameter).

It was this particular number, and its approximation, one of the key factors that delayed both

Mycenaeans and Egyptians from discovering the Mathematics of ellipses. The Minoans of the 2nd

millennium BC and the Egyptians of the 14th century BC didn’t possess a good approximation to

number , or had knowledge of irrational numbers. But they mastered to an extent fractions.

However, the fact remains that neither civilizations knew how to solve (or plot on a Cartesian

graph) second degree algebraic equations, notwithstanding that they seem to have had an

intuitive understanding and perception of ellipses, parabolas and hyperbolas. This is evident

from the approximations to these shapes both civilizations attempted and attained in the design

of certain key artifacts at some nodal time periods. This assertion is documented by this paper.

The length of an ellipse’s circumference is a complicate function requiring knowledge of

Trigonometry. In the references supplied the interested reader can find the formula, which will

not be used in this paper. It is noted that ellipses (as do parabolas and hyperbolas) can’t be drawn

by simple means of a ruler and a compass. They require mechanical means, a key factor in this

paper’s narrative. Issues surrounding the Minoans’ means of design of seals and rings (focusing

on the gold signet 5-priestess ring from Pylos), as well as on the design of other artifacts from

Pharaonic Egypt, will be addressed in this paper.

To summarize, of critical importance hence in the Geometry of an ellipse are the following basic

elements: the four vertices (the points where the ellipse’s perimeter intersects the two axes, the

two vertices and two covertices of Figure 1.1); the linear eccentricity c and the eccentricity ratio

e; and the two foci, F1 and F2. It is around these basic elements of an ellipse, plus the issues

associated with its total area A, that the analysis of the section that follows is undertaken. It will

at the same time account for the director circles and directrices, see Appendix II on them, as it

attempts to answer the question: to what extent was the maker of the ring aware of their

presence, when positioning the various elements appearing in the iconography and appropriately

carving their angles and shapes. Moreover, since an ellipse can’t be drawn by employing only a

ruler and a compass, the question of import is how did the ancients derived elliptical designs,

and actually how did they approximate (and to what degree) by drawing ovals the shape of a

true ellipse. Furthermore, and possibly as (if not more) important for the purpose of this paper,

is the question: how did the artist create the elliptical gold top mounted component of the signet

ring. In effect, how did the artist create the elliptical casting mold, in which the molten gold was

poured into, to form the almost precise elliptical shape we now observe on the gold signet ring’s

mounted surface where the complex and elaborate iconography depicting a ceremonial scene of

sorts was laid, carved and shaped by the adding of extremely thin layers of gold sheets.

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It turns out that in this question, profound mathematical issues are involved. More importantly,

for the point of view of Archeology and History, in the manner these elliptical surfaces were made

one finds hints as to the mathematical sophistication and knowledge base of the artist who made

the ring. And this issue is at the heart of the paper. Is this gold 5-priestess signet ring a true

ellipse or a 2-d ellipsoid; and how did it approach the shape of a true ellipse?

1.3 An overview of the signet ring

Figure 1.3. The three publicly available (and at different scales) photos of the five Priestesses

Minoan gold signet ring from the circa 1450 BC Mycenaean “griffin warrior” tomb at Pylos, the

Peloponnese, Greece. At the top, a side view of both the ring’s hoop (band) and the mounted

part (bezel) is given, showing the 3-d structure of the ring’s top surface; at the middle, the top

mounted surface is shown, which shows in 2-d the outline of an ellipsoidal (very closely

resembling an ellipse) shape; the back view of the mounted part of the ring (the bezel) is at

bottom. See subsection 3.3 for exact sizes of all the ring’s parts. Source of photo: ref. [3.2], and

provenance of the University of Cincinnati (Department of Classics), see ref. [2.1] and [2.5].

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The subject of the analysis that follows is the extraordinary (in both miniature Art and Design –

being very close to elliptical in shape) signet ring, shown in Figure 1.3. It is an artifact that was

found in the circa 1450 BC Pylos Mycenaean tomb of the so-called “Griffin Warrior”, see ref. [2.1],

[2.5] and [3.2]. Its complex iconography, at its mounted top side surface, contains a Minoan

ceremonial scene involving, among other components, five human female figures, seemingly five

Minoan priestesses. The five women are placed three to the left and two to the right of a Door

(or a Shrine), which is flanked by two leaning date palm trees.

In Figure 1.3 three sides of the ring are shown. On top, a side view (elevation) is offered indicative

of the shape of a ring’s long side cross section. The mounted part of the ring (the bezel) is a

complex 3-d structure: its top surface (in elevation) consists of two sloping lines flanking a flat

and horizontal line, thus forming an approximate 3-d trapezoid mounted on the hoop (band). The

bottom part of the mounted section of the ring is a concave arc (in cross section, along the length

of the bezel).

However, this short description of the ring’s top surface doesn’t even come close to describing

this magnificent artifact’s structure and iconography. A striking feature of this ring’s top surface

is its shape: a seemingly almost perfect ellipse. Since this is an artifact of the middle of the 2nd

millennium BC, an immediate question an analyst (and historian) must ask is: how much of the

formal Geometry-related properties of ellipses did the miniature Minoan artist who made the

ring did actually possess at the time. Did the maker know the Golden Ratio? An attempt to answer

these core questions, and particularly the maker’s knowledge of the ellipse, is exactly what will

be the subject of this paper. The inquiry can be broken down into three interrelated questions.

First, is the ring’s 2-d perimeter in fact an ellipse, meaning do all points on its perimeter meet the

fundamental property of an ellipse, as stated above; or is it simply an ellipsoid – albeit of

outstanding quality, that makes it look like a true ellipse. Second, did the artist recognize the

existence and importance of at least the two foci of the ellipse; or, put differently, what was

exactly the maker’s knowledge of an ellipse’s formal Geometry fundamentals. And third, is there

any indication on the carving in 3-d of the ceremonial scene to convincingly demonstrate that

particular points and specific properties of geometric interest (in so far as true ellipses go) were

also familiar to the artist, and that the artist acknowledged their existence by appropriately

designing the iconography of the ring, positioning the major figures of this iconography at critical

places, and thus accommodating important geometric points and properties of an ellipse.

The alternative hypothesis to the latter proposition in specific would be that the artist positioned

the various iconographic elements within the seemingly elliptical perimeter (in 2-d) not cognizant

of the underlaying mathematical (geometric and algebraic) factors. Then, one asserts that the

artist acted on the basis of Aesthetics that can’t be directly attributed to mathematical, geometric

or algebraic, factors; or that the maker was simply not cognizant of such geometric linkages

between Geometry and Aesthetics, while crafting the artifact. The first and second questions will

be shown to have a strong answer, reaching the state of almost beyond reasonable doubt. In

fact, the answer to the first question can be considered to be a formal mathematical proof.

14

In the case of the third question, evidence will be presented by this author to indicate that there

are some compelling reasons to argue that the maker was cognizant to some extent of the

Mathematics of ellipses and that the maker of the ring tried to approximate an ellipse’s shape

through the use of four key circles, although further analysis and debate on that third set of

answers, and the question itself, are obviously needed.

Since the maker of the ring in the middle of the 2nd millennium BC didn’t possess either the full

mathematical knowledge on ellipses (as did Menaechmus in the 4th century BC) or the means to

draw them (both being achievements of the last third of the 1st millennium BC), the question is

how did the maker draw the 2-d ellipse-looking ellipsoid and how was the 3-d structure of the

ring’s top mounted part made. The answer to the first of these two last questions is that (as

already mentioned) the ring’s maker most likely employed four circles (shown in Figure 1.4.2 later

in the text). The molding of the ring’s upper section and the carving of its top surface miniature

morphology are addressed in the paper and specifically in Note 3.

As to whether Aesthetics can be totally devoid of mathematical considerations, this is a far

broader issue and certainly not the topic of this paper. It is however the conviction of this author

that Aesthetics and Mathematics are always (and strongly) linked. The relevant questions in this

case of the Minoan gold signet ring are: what level of Mathematics is involved; and to what extent

was the artist aware of the underlying Mathematics while creating the artifact.

1.4 Borchardt’ ellipsoid from Luxor

Ellipses’ mathematical properties (as well as those of the related 2-d shapes of parabolas and

hyperbolas) prevent us from drawing them with the means of Classical Geometry, namely by

using just a compass and a ruler. They require the use of ellipsographs, means not known to the

ancient mathematicians (and artists) until Archimedes, the great mathematician of the 3rd

century BC (c 287 – 212 BC) and the invention of the trammel, see Appendix III. The issue of

course is, how did the Minoan artist of the middle 2nd millennium BC not only draw in 2-d, but

also made in 3-d the ring’s mold, a major topic under scrutiny in this paper, when the

Mathematics of ellipses were to a large extent (but not totally, as this paper contends) unknown,

and certainly the means to either draw or carve them apparently non-existent.

Analysis of the Minoan seals and rings of the Neo-Palatial (1700 – 1400 BC) period, takes us to a

different time period and another spatial context, namely the post Ramesses II era (post c 1213

BC) at the Luxor Temple, and into the core of the 19th Dynasty Egypt. It also brings us to a

personality of some interest in the field of Archeology, architect and Egyptologist Ludwig

Borchardt (10/5/1863 – 8/12/1938), a person that we also come across in the case of Nefertiti’s

Bust, an artifact that will be reviewed in a bit, as its design bears directly on the subject matter

of this paper.

15

Figure 1.4.1. Ludwig Borchardt’s drawing of the Luxor 2-d ellipsoid (a three-century later

drawing than the Minoan ring) with one of his own (among three that he suggested) schemes on

how the Egyptian artist might had approximated a true ellipse superimposed on the original

drawing. The horizontal (approximate major axis) length of the 2-d ellipsoid was reported by

Borchardt to be about 160 centimeters, or three Egyptian cubits; whereas the vertical

(approximate minor axis) length was reported by Borchardt to reach 103.5 centimeters, or about

two cubits. Ratio of major to minor axis is, hence, about 1.55. Source of the diagram: ref. [2.8].

In 1896, see ref. [2.7], and at the age of 33, Borchardt announced to the archeological community

that he had discovered, drawn on limestone blocks at the Temple of Luxor in Egypt, the schema

of an ellipsoid. The not so sensational at the time, but critical as it now seems, announcement

was presented as shown in Figure 1.4.1. He supplied what can be construed as rough directions

on where this schema is to be found at the Temple. Before discussing this drawing, and what it

implies for the analysis involved in this paper, it must be noted that, to the author’s knowledge,

no photographic evidence exists of the schema alleged by Borchardt to have existed back in the

late 1890s at the Temple of Luxor. Hence, the authenticity of the diagram as well as its accuracy

in depicting the original carving can’t be cross-checked and validated. It is also Borchardt who

dated the schema, as having been carved on the masonry blocks at Luxor’s Temple, and placed

it to a period immediately following the reign of Nineteenth Dynasty Pharaoh Ramesses II.

16

No matter the archeological authenticity of the find, its exact dating, and the precise provenance

of the schema alleged by Borchardt to be as in Figure 1.4.1, the mathematical aspects of it, and

their implied worth - as they regard the state of Egyptian Mathematics of the period, as well as

the manner elliptical forms (oval 2-d ellipsoids) were drawn are of extreme interest. In fact, the

Borchardt dating of the Luxor ellipsoid seems to be consistent with the Evolutionary Theory of

how ellipsoids moved toward true ellipses, as suggested by this author in this paper.

Borchardt offered a number of ways that, according to him, the 2-d ellipsoid of Figure 1.4.1 was

drawn back in the 13th century BC (possibly later). The one involving the five circles (three small

ones of equal radii, and two large ones also of equal radii) shown above is one of those schemes

alleged by him. In ref. [2.8], different schemata are shown, on how one could derive both a better

approximation to what the Egyptian artist did draw, as well as to a closer approximation to a real

ellipse and its total area and always using five circles, being still close to what the ancient

mathematician drew on the limestones of the Temple at Luxor, always according to Borchardt.

The reason why the Borchardt schema of an ellipsoid at Luxor, and his suggestions as to how the

artist/mathematician could had attempted to approximate the shape of and construct a real 2-d

ellipse is a subject that directly influences our analysis of the manner in which the Minoan artist

of the middle 2nd millennium BC (a good third of a millennium prior to the mathematician at

Luxor) made the signet ring under investigation here, and how we can derive schemata on what

the Minoan artist attempted to do examining its complex 3-d structure.

Such an attempt is made by this author, and it shown in the diagram of Figure 1.4.2, which shows

the 2-d ellipsoid and its inscribing rectangle. It is suggested that this schema has been the first

approximation and quite likely the basic approximation to the almost elliptical form for the

Minoan 5-priestess gold signet ring from Pylos. It employs just four circles, as opposed to the five

of the Luxor ellipsoid. These circles are of the specific Basic Geometry type: on the x-axis, that is

on the horizontal major axis, the two smaller circles have radii equal to quarter of the major axis’

length, a/2, and of course their centers lie on the major x-axis. On the other hand, the centers of

the two larger circles lie on the minor y-axis, have radii equal to the minor axis, 2b, and their

centers are on the ellipsoid’s covertices.

However, this suggested schema of the Minoan ring must be construed as a successful and basic

although not initial approximation by the artist on 2-d, to the final derivation of the 3-d ellipsoid

on the ring’s top surface. A likely initial approximation to a real ellipse, the quite likely first

ellipse/ellipsoid, will be more closely analyzed in the subsequent subsection of the paper. There,

the proposition will be advanced that, although the Minoan artist did not have at his/her disposal

the means to drawing an ellipse (thus produced a 2-d ellipsoid), chances are that the fundamental

Mathematics of ellipses must had been known to him/her. The fundamental property of ellipses

is that all points on the perimeter have their sums of distances from the ellipse’s two foci being

a constant and equal to the major axis’ length (2a). This fundamental condition seems to hold for

a very large number of points on the 2-d ellipsoid of the Minoan gold signet 5-priestess ring, as it

will be shown in a subsequent section of the paper.

17

Figure 1.4.2. The Pylos 1450 BC “griffin warrior” tomb found Minoan 5-priestess gold signet

ring made sometime in the 1700 – 1500 BC period, in an approximately 2-fold magnification (the

span of the photo is 6”). The ellipsoid’s most likely approximation by four circles (two small in

radius on the major and two large in radius on the minor axes) to a true ellipse. The inscribing

rectangle (in red), the true ellipse’s position of the x (major) axis and y (minor) axis are shown.

Source: the author from a photo of the ring in ref. [3.2], as well as ref. [2.1 and ref. [2.5].

The arcs from the two small circles with centers on the major axis of the 2-d ellipsoid are needed

to smooth out the sharp edges created by the two large circles with centers at the covertices of

the minor axis. Without them, the rough shape of a human eye is depicted, the shape likely to

have fostered the advent of the ellipse, along with its fertility related symbolism of the womb

and the egg, as it has already been alluded in the paper, and will be further argued in section 3.

18

Close-up of a female face (Tyra Banks) with oval eyes from a photo (in the public domain); the

entire photo is at the end of the paper, showing the presence of ovals, ellipses and parabolas

in a human figure’s eyes, arms and shoulders under a special angle.

1.5 The likely 3-circle Genesis of the ellipse

The Borchardt Luxor ellipsoid, from Figure 1.4.1, in combination with the 4-circle Pylos 5-priestess

gold ring ellipsoidal form derived as shown in Figure 1.4.2, bring about a more basic issue, in the

form of a research question. Is it possible that the conceptual geometrically based origin of the

elliptical design was the derivative of a simple 3-circle schema? This subsection, through Figure

1.5.1, tries to address this topic. On the notion of an “origin” behind this ellipsoidal see Note 6.

Figure 1.5.1. The three circles very likely “Genesis” of schemata approximating a true ellipse.

The scheme may be a phase transition between pre- and post-ellipsoids. Source: the author.

19

The simplest, albeit the crudest, approximation to an ellipse – quite possibly the Genesis of the

pure geometric shape – was an attempt to inscribe into it three circles, see Note 6. These three

circles held the property that, the two circles flanking the central circle would have equal radii

and half of the central circle’s radius in length, as in Figure 1.5.1. Four lines, tangent to the three

circles at points E, H, J, L, F, I, K, and M would complement the four arcs formed by the three

circles (FAE), (HCJ), (KBM), and (IDK) in forming the crudest but potentially the original ellipsoid.

In time, and by adding a greater number of circles, some with greater radii than those in Figure

1.5.1 (and at least two circles with radius greater than the minor axis, 2b) would attempt to come

closer to a true ellipse. That is, to come closer to a schema where the total length taken up by

straight lines would be minimized. A maximum for that length is what is shown in Figure 1.5.1.

That ellipsoid would also produce a total area within it which would closely approach the area of

the true ellipse with its two axes being 2a and 2b. It is recalled that the true ellipse’s total area is

A=ab. The verification of this statement, and the exact degree of approximation is left to the

interested reader(s). The place and time where this original schema may have appeared is

unknown at present. An interesting research question would be to search for that Genesis. Eyes

have an Iris; this might had been the key factor for the ancient artist and architect in devising the

primordial, original, Genesis of an ellipse scheme of Figure 1.5.1, where a relatively large central

circle is the basis of the ellipsoid. In this context, one may refer to the gold 5-priestess ring from

Pylos as a “proto ellipse”. It is of some interest to ponder the Geometry of the schema in Figure

1.5.1. The major axis is double in length the minor axis (b=a/2) so that the inscribing the ellipsoid

rectangle (drawn by the red lines) is in a fact a double square. The radii of the two smaller circles

(with centers at O1 and O2) are b/2 (and a/4), whereas the radius of the large circle is equal to b.

The linear eccentricity of the true ellipse corresponding to the (2a,2b) specifications is:

c = b(3^1/2),

falling always outside the range not only of the large central circle but also after the point where

the smaller circles’ center is at (since: c > 1.5b). The eccentricity ratio of the (true) ellipse is:

e = (3^1/2)/2.

To conclude this brief excursion into elementary theory of ellipses, and the quest for the origins

of ovals, arcs, apses, elliptical, quasi-elliptical, partially elliptical, pseudo-elliptical (2-d ellipsoidal)

designs in Neolithic and Bronze Age to Iron Age Architecture and Art, it is noted that, a

progression is established: the Borchardt ellipsoid requires at least five circles; the proposed by

the author as being the basis of the Pylos 5-priestess gold ring requires four circles; whereas the

Genesis schema for ellipses suggested in 1.5.1 requires just three circles. The proposition is hence

advanced that what the ancient designers of artifacts attempted to do is to find a schema that

would place the true ellipse’s foci as close as possible to the two extrema (at right and left) circles

on the x-axis (the true ellipse’s major axis). They were, through their ovals, in effect on a quest

towards discovering the Mathematics of true ellipses. For more on this critical boundary and

phase transition see Note 6 at the end of the paper.

20

2. A Brief Review of Ellipsoids and Ellipses in Key Archeological Contexts.

In this section of the paper, some key pseudo/quasi-elliptical, partially elliptical (2-d ellipsoidal)

and elliptical structures will be presented, at three qualitatively different scales to highlight two

key aspects in a theoretical context of Evolution of Form in Design: first, that there is a dynamical

(one may characterize it as “evolutionary”) view to be taken in so far as circles, arcs (or apses),

ellipse like (2-d ellipsoidal) and finally elliptical structures are concerned. The thesis has been put

forward, see the author’s work in [1.5] among other papers, that Art and Architecture Form in

time and from the upper Paleolithic has evolved from simple and primordial shapes to more

complex ones. Increased complexity, viewed as a fundamental force of evolution in Nature is the

underlying principle here as well. In the case of ellipses, the same theoretical perspective would

hold: that in the case of built structures (and in artifacts) almost or quasielliptical shapes gradually

evolved from simpler and more crude approximations to more complex and closer to perfect

ellipse forms: from oval, ellipse-like shapes, to more accurate and exact schemes replicating

elliptical morphologies; from their more primitive shapes, oval designs advanced to more

mathematically sophisticated ones, that could also be produced by simpler means.

A second point to be made here is that scale was a major factor in this process of evolution in

complexity and transition in form, as smaller scale structures of similar design (but requiring more

complicated means of making) preceded larger scale structures of identical deign, most likely as

the outcome of some form of experimentation. Hence, the presentation of these structures and

artifacts will obey a scale subdivision, whereby the microscale of a few nodal artifacts from the

Late Bronze Age will be succeeded by a presentation of selected small in scale structures from

Early Neolithic; to the Bronze/Iron Age boundary; to finally conclude with certain examples of

key large in scale structures encountered in late antiquity, and specifically in the Roman Era.

Epipaleolithic and Early Neolithic (Pre-Pottery Neolithic A and B) as well as Middle and Late

(Chalcolithic) Neolithic Architecture is characterized by quasi-circular Natufian structures having

been transitioned to arc, apse and quasi-elliptical (ellipsoid forms in 2-d). That evolutionary

process has been addressed in ref. [1.1]. Within this context, some nodal examples are offered

in the subsection on the small scale ellipsoidal structures of this paper’s section. However, before

that subsection, the case of the microscale will be examined, where some key artifacts of the

Bronze Age will be presented, that includes the Minoan rings and seals, the first on record 2-d

ellipsoids, examples of microstructures demonstrating a close resemblance to true ellipses, along

with two later but closely related Egyptian artifacts: Nefertiti’s Bust and Tutankhamun’s Mask.

2.1 Microscale artifacts

In this subsection, some key examples of artifacts are presented, where the subject of 2/3-d

ellipsoids and ellipses come into sight. It is shown and documented that the Minoan Civilization

during the 2nd millennium BC (and specifically, in the 1700 – 1300 BC period) was the first in

mastering the subject of drawing shapes extremely close to true ellipses, and possibly

understanding at least some of its fundamental Geometry. This is primarily documented through

21

an analysis of the gold signet ring in question (the 5-priestess ring from the “Griffin Warrior”

tomb). However, the ring under study here is by no means a unique Minoan 2-d ellipsoidal

specimen. It likely appeared within the framework of a Minoan School of Miniature Art that

produced a number of equal in craftsmanship and dexterity rings and seals that were imbued

with a sense of an “ellipse”. Two more specimens of that type are shown below.

Another context is also discussed in this subsection, where 2-d ellipsoids at a larger scale have

also appeared before the Iron Age was in full swing, and that is the 19th Dynasty Pharaonic Egypt

of the middle 14th century BC. Within this context, two specific artifacts are presented. One is

Nefertiti’s Bust by Thutmose from Amarna, where a 2-d ellipsoid is found, as well as the Geometry

of cones. The other is Tutankhamun’s Mask, an artifact on which an ellipsoid may not have been

embedded but a geometric shape of equal complexity – that of a parabola – was used to make

it. These two examples from Pharaonic Egypt should be considered in conjunction with the

discussion from the previous section regarding the Ludwig Borchardt ellipsoid from Luxor.

Figure 2.1.1. Minoan gold ring from the so-called “Treasure of Aidonia”, a Mycenaean area by

the city of Nemea at the North-Eastern part of the Peloponnese, and by the modern-day city of

Corinth. The about 1” ring is less elongated than the ring from Pylos. Source of photo, ref. [3.10].

22

Minoan Rings. In Figures 2.1.1 and 2.1.2 two key Minoan Art gold rings are shown, representing

pivotal periods in the evolution of Minoan miniature Art as well as evolution in the ellipsoidal

structure of their top surfaces (the upper surface of the ring’s bezel). In Figure 2.1.1 a ring from

the “Treasure of Aidonia” is shown. Aidonia is a site by the ancient Mycenaean city of Nemea, at

the North-Eastern region of the Peloponnese, in Greece. On the archeological adventures of the

ring, and the surrounding legal aspects of its return to Greece, see ref. [4.10].

The ring’s iconography is of interest to the extent that this paper is concerned. It is simpler in

structure than the iconography of the 5-priestess signet ring from the Pylos tomb, which is the

subject of this paper’s analysis. It depicts a ceremony in which three Minoan priestesses approach

and face towards an altar. This altar is capped by the horns of a Taurus, a typical Minoan Bull Cult

motif. For a description (at times inaccurate, however), and for links to other similar Minoan and

Mycenaean signet rings and seals, see ref. [4.11]. Like all other gold Minoan rings, the ring’s

iconography was made by applying successive layers of about half of a millimeter in thickness

gold sheets. It required a metal (out of bronze) needle with a working edge of about a third of a

millimeter, a subject addressed in a bit and further analyzed in subsequent parts of the paper. A

feature of this ring’s top surface, of interest here is the shape of the 2-d ellipsoid the ring sports.

Approaching an elliptical shape to a lesser degree than the 5-priestess signet ring from Pylos, its

eccentricity ratio e is less than that of the ring from Pylos, as it is obviously less “elongated”

(meaning that its foci are closer to the ellipsoid’s center).

Even less elongated than the ring from Nemea is the ring shown in Figure 2.1.2, a Minoan golden

seal/ring also from the Pylos tomb of the Mycenaean “Griffin Warrior”, see ref. [2.1] and [2.5].

The ellipsoidal surface of this ring carries an iconography far more primitive and closer to the

initial stages of the Bull Cult (with bull leaping as a sport) culture of the Minoans than either the

Nemea 3-priestess or the Pylos 5-priestess rings. Moreover, its two ellipsoidal foci (indicative of

the smallest elongation of all the aforementioned rings) are at a smallest distance from the ring’s

center than the other two (from Nemea and Pylos). Hence, it is safe to presume that the Nemea

ring is a progenitor of the Pylos ring, and that the bull leaping ring from Pylos is the oldest of the

three. Chronologically, and with all caveats of Note 2, one would be inclined to place the making

of the Bull seal/ring from Pylos close to the 1700 BC time period; whereas the 3-priestess gold

ring from Nemea would be closer to the 1600 BC, and the 5-priestess signet ring from Pylos close

to the 1500 BC. As already mentioned, dressing of the Minoan rings and seals’ 2-d surface was in

general done by superimposing extremely thin sheets of gold, about half of a millimeter in

thickness. It must be assumed hence, and there is no evidence to the contrary, that a similar

technique was used in the making of the gold signet ring under analysis here. In this particular

case, the trapezoidal (in cross sections, along the major and minor axes, shown on top of Figure

1.3) bed (bezel) of the ring’s surface was formed by pouring liquid gold into a mold. That mold,

in its 3-d ellipsoidal form, must had followed similar processes of making (and in its drawing) as

outlined here for the case of 2-d. The final surface was carved by using the Minoan bronze needle

shown in Figure N.1 in Note 3, at the Notes’ section of this paper.

23

Figure 2.1.2. Minoan gold seal/ring from the Mycenaean tomb of the “Griffin Warrior” at Pylos.

The about 1” in length scene depicts a Minoan preparing to land, hands extended to maintain

equilibrium, having leaped over a charging bull. The author asserts that this may be among the

oldest gold seals/rings of the Minoans, possibly having been created c 1700 BC. It may have been

among the loot obtained by the Mycenaean warrior in battle and conquest, possibly at the island

of Crete during the invasion of the island by the Mycenaeans. The wear and tear showing on the

ring’s surface may be indicative of its suggested historiography. Source of photo, ref. [3.11], and

ref. [2.5] provenance of the University of Cincinnati, Department of Classics.

Both the Nemea ring and the Pylos bull leaping seal/ring are not real ellipses but oval 2-d

ellipsoids. However, what in reality makes them look pretty close to a true ellipse is their scale.

As the artist made adjustments in the manner (s)he drew the ellipse, these adjustments got

swept away in the scale of the artifact. In effect, the scale smoothed out the oval deviations from

a pure elliptical shape for all the Minoan rings under the microscope in this paper. It is apparent

24

that the scale factor didn’t go unnoticed by the maker. Possibly familiar with larger scale built

structures, ellipsoidal stone enclosures where the mechanics of drawing an ellipse were not

known to the architect of the Neolithic (hence the imperfections of all stone ring circles – a

subject to be addressed shortly) was overcome by the Minoans through the means of scale.

Figure 2.1.3.a. Nefertiti’s Bust where an ellipsoid was implanted. Source of photo: ref. [3.12].

25

Figure 2.1.3.b. Nefertiti’s Bust. Computer simulation from a 3-d scan of the conic sculpture,

showing the implanted ellipsoid. Source: the author and his work presented in ref. [1.16].

Nefertiti’s bust. An extremely interesting and relevant artifact for the deliberations of this paper,

is the widely known Nefertiti’s Bust, an artifact extensively studied as to its Geometry by this

author in ref. [1.16]. The, presently at the Neues Museum in Berlin, bust is shown in Figure

2.1.3.a. The simulated bust is shown in Figure 2.1.3.b. The simulation was produced by this author

from a 3-d scanning of the bust available in the public domain. Geometric implications are

discussed by this author in ref. [1.16]. Nefertiti’s Bust, artwork by Thutmose, found in his Amarna

workshop, made c 1345 BC, presents some interest, for the paper’s topic, as it involves cones and

parabolas that are neighboring geometric forms of direct relevance to ellipses and ellipsoids.

It is noted that the bust was found in 1912 by none other than Ludwig Borchardt, see ref. [4.12]

for a biographical note on his life and work, the same person encountered in the discussion of

the ellipsoid allegedly drawn on the limestones of Luxor’s Temple and discussed earlier.

Associated with this discovery is some controversy regarding the Bust’s authenticity, doubts no

longer seriously entertained by the archeological and scientific community. Nefertiti’s Bust

contains a cone-like structure (Nefertiti’s crown), an ellipsoid (the top surface of Nefertiti’s

crown) and a parabola (Nefertiti’s left and right shoulders). The fact that all these mathematically

closely linked geometric shapes are encountered in this sculpture is of interest, and certainly not

26

random, especially since they appear on a sculpture crafted at that historical juncture. One must

assume that Thutmose was aware of the Geometry connecting these three shapes, in both 2-

and 3-d. Not obviously possessing either the Mathematics or the means to construct all these

shapes (two surfaces and a solid) in precise forms, they all appear in approximations, and this is

strong evidence that Thutmose and the Egyptian mathematicians of that era were still not in

command of the ellipses, the parabolas and the cones’ Mathematics. The approximate cone

structure of Nefertiti’s crown is intersected by a plane (at the level of her forehead) resulting in

an approximate circle (detected in Figure 2.1.3.a); and at the top of her crown, the solid is

intersected by a plane at an angle to the cone’s main axis, resulting in a 2-d ellipsoid (seen in

Figure 2.1.3.b). The analysis here must be considered as extending and appropriately amending

the author’s analysis in ref. [1.16].

Tutankhamun Mask. Next, and last in the cases of artifacts at the microscale, another widely

known artifact is briefly analyzed, Tutankhamun’s funerary Mask, a signature artifact of not only

the 19th Dynasty Pharaonic Era, but possibly the entire ancient Egypt. The Mask has been studied

by the author in ref. [1.17]. The analysis of this paper amends and extends the work in ref. [1.17].

Figure 2.1.4. Back side view of the Tutankhamun funerary Mask. Source of photo: ref. [3.13].

Tutankhamun’ Mask is an artifact that was created a bit later than Nefertiti’s Bust. It contains

some evolution in the knowledge base of Mathematics in the lower half of the 14th century BC.

The outline of its frontal and back views contains an approximate parabola. This approximation

must be construed as evidence that the artist had not totally mastered the fundamentals of

parabolas, a geometric form linked to ellipses. But he/she had the conceptual design mastered,

a means to approximating the end shape. They still didn’t have a good enough approximation to

, and they were far from deriving the mechanics that would trace ellipses and parabolas.

27

2.2 Small Scale pseudo-elliptical structures

A very large number of pseudo or quasi-elliptical in shape built architectonic structures can be

presented here, but only those found in four settings will be addressed. They are pivotal in the

course of Evolution in Neolithic Architecture, and they can be used to make the key intended

points. All these nodal structures in Architecture have been studied in some detail by this author.

The first structure is the communal building of a Pre-Pottery Neolithic B (PPNB) site in Northern

Syria, Jerf el-Ahmar (Figures 2.2.1.a, and b); the second set of structures (with an emphasis on

structure D) is the Layer III, structures A, B, C and D at Gobekli Tepe, a 6th millennium BC

(according to this author) construction (Figures 2.2.2.a – d); the third structure is an apse type

complex of the 4th millennium BC Hagar Qim Temple Phase at the island of Malta in the Maltese

Archipelago (Figure 2.2.3); and the fourth structure is the circa 2500 BC Trilithons ensemble at

Phase 3 II at Stonehenge (Figure 2.2.4).

Figure 2.2.1.a. Site plan of the Jerf el-Ahmar (circa 10th to 9th millennium BC) archeological site

at the banks of the Euphrates River in Northern present-day Syria. A number of oval shaped

structures belonging to different settlement phases are shown. Source of diagram: ref. [3.5].

Jerf el-Ahmar. The site plan of Jerf el-Ahmar, no longer a site that can be visited and further

explored under current conditions since the 1991-1999 Tishrin Dam on the Euphrates has

inundated it with water, is a nodal one in the study of ellipsoidal schemes in Neolithic

Architecture. It is also an important site in its capacity (along with Nevali Cori) to offer significant

markers for the dating of another site, that of Gobekli Tepe and its enclosures of Layer III.

28

As all Neolithic sites (and in fact, as is the case with any human settlement) Jerf el-Ahmar was

built in phases. What is of interest to the analysis here, is the architectonic evaluation as to which

of the buildings (shown in Figure 2.2.1.a) are older and which ones are structures of a later phase

in the spatial and temporal evolution of the settlement. The evidence seems to suggest that the

communal building (at the bottom left of the site plan shown in Figure 2.2.1.a and at the center

of the photo in Figure 2.2.1.b) belongs to the last phase of the settlement, c 8500 BC.

Discussion on this site is found in ref. [1.13] by the author, with relevant citations. A major source

of this discussion draws from the analysis in ref. [2.9]. It documents that this is a settlement with

a Natufian type Architecture more primitive than that of Gobekli Tepe’s enclosures, and a site on

the Euphrates River – both strong indicators of it being an older settlement than Gobekli Tepe.

The absence of monolithic dressed bocks of the Gobekli Tepe scale (size) and type (dressing) in

the communal building stone enclosure is further evidence of a prior construction activity.

Figure 2.2.1.b. Jerf el-Ahmar photo prior to inundation with waters from the Tishrin Dam on

the Euphrates. At center is the later phase communal building structure. The dominant

Architecture of the site is an advance form of the Natufian type. Source of photo: ref. [3.5].

In Figure 2.2.1.a quasi-ellipsoidal structures even more primitive than the pseudo-ellipsoidal

scheme of the communal partially in-ground dry ashlar masonry edifice (ceremonial or possibly

the residence of an elite member of the community, hieratic or administrative or both) indicates

a prior phase construction. Rectangular dwellings indicate later construction (possibly of the

Sultanian type, see ref. [1.1]) as well.

29

Gobekli Tepe. The site is a nodal one for Neolithic Architecture, with its well-dressed

sophisticated megalithic monolithic components within the stone enclosures of Layer III, and

specifically as found in structures C and D. These structures’ orthostats and pillars can be

construed as offering markers in the Evolution of Neolithic monumental Architecture. In

numerous respects, as argued in ref. [1.1, structures C and D at Gobekli Tepe ushered the 6th

millennium BC Neolithic Era Monumental Architecture. Studying Gobekli Tepe’s quasi-ellipsoidal

structures’ form offers one the opportunity to set the standards for the design of derivative

enclosures in Western Eurasia. The site’s key structures have been analyzed by this author in a

number of papers, see ref. [1.1], which contains in turn references to prior papers and to work

by the archeologists on the site, with whose views the author strongly disagrees.

Figure 2.2.2.a. Layer III, structure (enclosure) D, the possibly oldest structure of the enclosures

excavated thus far at Gobekli Tepe. The major axis of the complex pseudo-ellipsoidal scheme

depicted in this structure has an (East, North-East) to (West, South-West) orientation. Source of

diagram, the author in ref. [1.18], (p. 41, Figure 5.3).

30

The author’s grounds for disagreement, in so far as dating Gobekli Tepe is concerned, are

founded on Climate, Geology, Demographics, Economics, Human Geography, City Planning,

Urban Design, Architecture, and Art related factors. They are also founded on the archeological

principle that fills do not date structures. This author disputes the C-14 evidence of the

archeological team, as being contaminated by PPNA/B soil from the fill. From the Architecture of

the structures, especially structure D, the ellipsoidal schema of its design will be extracted so that

it will be linked to the subject matter of this paper. That schema is shown in Figure 2.2.2.a. It

consists of two semi-circles (drawn with centers at F and C) that are joined by two straight lines

(shown by the segments AD and BE in the above Figure 2.2.2.a). The schema’s major x-axis is

oriented towards the azimuth of the sunrise point at Summer Solstice and at the back end of it

towards the sunset point of the Winter Solstice.

Figure 2.2.2.b. Gobekli Tepe, Structure D, Layer III, orthostat #43 (the so-called Vulture Stone).

The Art embedded at the very top of this oval shaped enclosure orthostat’s head is a time marker,

as the triple purification symbol (the handbag) is also (and for the first time since) encountered

in the 2nd millennium BC Temples at Uruk. North is up. Source of photo and diagram: ref. [3.8].

31

Here, the focus is the enclosures excavated so far pseudo-ellipsoidal schema. The implications of

the time marker in Art found at the very top of orthostat #43 (the Vulture Stone) in enclosure D

is extensively addressed by the author in [1.1]. By comparing the stone enclosure C, shown in

Figure 2.2.2.c, and enclosure D, shown in Figure 2.2.2.b, one observes that the interior design in

C is less primitive and more elaborate in interior spacing partitioning than that of structure D.

This distinction is, from an Architecture viewpoint, critical in comparatively dating them.

Figure 2.2.2.c. Gobekli Tepe, Structure C Layer III. This structure’s oval shape is more complex

and more pseudo-ellipsoidal than that of structure D. North is up. Source of photo, ref. [3.7].

The floor plan of structure C, Figure 2.2.2.c, requires three circles to construct, as opposed to the

two circles of structure D’s floor plan, see ref. [1.18], (p. 37, Figure 5.1). Hence, one must conclude

that enclosure D was built prior to enclosure C. The entire complex of structures excavated as of

the middle of 2016 is shown in the public domain photo of Figure 2.2.2.d. In it, structures A and

B are shown, along with C and D. From the sizes and compositional complexity of their floor plans,

one could potentially derive a sequence in their construction. All of them involve two semicircles

in deriving the quasi-circular pseudo-ellipsoidal forms. In combination (scale and complexity in

floor plan), the conclusion must be drawn that the likely chronological sequence could be: A, B,

D, and C. This implies that the entire complex at Gobekli Tepe might had not been a one-shot

32

construction, but instead a phasing in and out of structures over time. This scenario is further

explored in the last section of paper in ref. [1.1] by the author. The placing of the four enclosures

as shown in Figure 2.2.2.d seems also to suggest that the area of structure C, a structure with a

double shell – more so delineated than in any other enclosure – was a special location around

which the other three enclosures were placed over time. However, a more complete formulation

of such evolutionary scheme must await the unearthing of the neighboring structures to the left

of enclosures D, A and B. It is underlined that the fact that only two semicircles form the pseudo-

ellipsoid at Gobekli Tepe’s C and D structures confirms that the schema employed represents an

earlier design to approximating ellipses than the 3-circle Genesis design of Figure 1.5.1.

Figure 2.2.2.d. Gobekli Tepe, excavated structures D (top), C (right), B (left), A (bottom). North

is up. Photo is found in the public domain (search terms: “Gobekli Tepe bird’s eye view”).

In ref. [1.4] this author traced the occurrence of a similar to Gobekli Tepe’s stone enclosure 2-d

pseudo-ellipsoidal shape in the case of the Phase 3 II Trilithons ensemble at Stonehenge (c 2400

BC). In addition to the shape of the enclosures, the role cast-off shadows played in the design of

both monuments was examined in that study. The common manner in which cast off shadows

during daytime by the megalithic monoliths (mainly the pillars) of the two monuments were

incorporated into their design proper and their Astronomy was examined in detail in ref. [1.4].

33

Hagar Qim Temple of Malta. Next, attention turns to a complex of structures comprising the

early 4th millennium BC Hagar Qim Temple at the South-Western corner of the island of Malta, in

the Maltese Archipelago, see Figure 2.2.3. The Temple, for which a description is found in ref.

[4.8], has been reviewed by the author in ref. [1.3] as to its design influences in the subsequently

constructed monuments at Newgrange (c 3200 BC) and Stonehenge Phase 3 II. Hagar Qim has

been an influential and nodal monument in Neolithic monumental Architecture. Its stone

enclosures’ apse shaped design is a shape that is also encountered in the case of the circa 5th to

4th millennium BC “egg” stone structure at Carnac’ Le Grand Menec Western end, see ref. [1.5].

Figure 2.2.3. The Hagar Qim Temple (circa first quarter of the 4th millennium BC) at the island

of Malta, in the Maltese Archipelago. An architectural drawing of the Temple complex is found

in ref. [2.12]. North is straight up. Source of photo: ref. [3.6].

The Temple of Hagar Qim is located at a central place in the Western Eurasian landscape of

monuments, and is on the road to Western Europe from the Levant and the Fertile Crescent, as

agriculture spread in a Northern and Western direction in that great so-called “demic” migration

and diffusion movement, see ref. [2.10], which along the later Kurgan (often also referred to as

the Yamnaya, see ref. [2.11]) migration from the Pontic Steppe apparently shaped the

Architecture and cultures of Europe in the Neolithic.

34

Stonehenge Phase 3 II. The fourth setting that will be very briefly mentioned in reference to the

oval shapes of the Neolithic monumental Architecture of Western Eurasia is the Phase 3 II,

Trilithons ensemble at Stonehenge. The inner stone enclosure, the set of Trilithons, also involves

a schema consisting of two semicircles. See a drawing and analysis by the author of the

monument’s Phase 3 II (c 2600 – 2400 BC) in Figure 2.2.4, from reference [1.4].

Figure 2.2.4. Stonehenge Phase 3 II site plan, showing the sarsens circle and the Trilithons apse.

Source of diagram: the author, in ref. [1.4], (p. 17, figure 5), with an analysis of the design.

35

2.3 Large Scale structures: the roman amphitheaters and forums

Two examples of a roman amphitheater and a forum. The ellipse appeared at its largest scale

rendition in Antiquity’s monumental Architecture in the form of roman amphitheaters. Being

places where spectators could find escapism from their daily lives, these spatially dominating

structures were arenas for entertainment and public display of brutality by the phantasmagoric

spectacles’ participants and dominance by the ruling elite. It was an era when sports events

acquired a different scale, nature and socio-political (cultural) purpose. Mass psychology was

exhibited at an unprecedented scale and intent. Strategies for optimal management and control

of crowds was tested and applied. The various impacts of the many “sports’ events staged in the

arenas, became grounds upon which the populace was profitably exploited by the ruling elites.

For the first time in history, sports arenas became instruments and mechanisms to effectively

exercise political control. In this cultural context, their ellipsoidal stands and elliptical form of

their arenas became an efficient design and space to house the events. Shows intended to shock,

create awe and be used as a vehicle for the audience to release its anger were staged, and the

elliptical shape (and the almost 45 angle in the cross section of its stands) proved to be

instrumental, efficient and successful in accentuating the intended purpose of the event. These

forums were used primarily as spots to assert dominance by the local ruler (or emperor in the

case of Rome). Roman amphitheaters became also spaces were architectonic creations

flourished. Amphitheaters where means to exhibit large-in-scale public fiscal policies,

construction engineering and architectural design innovations. Publicly, socio-political messages

were conveyed to the effect that a social system was on the move and in full control. Roman

amphitheaters were spaces where a class structure and social stratification were not only in full

public display, but carried the intent by the ruler for the plebeians to abide by and accept it.

Besides the best known Roman amphitheater, the Flavian Amphitheater in Rome (known as

simply the “Colosseum”, a 1st century AD construction where about 50,000 spectators could be

seated, see ref. [4.6] and to be briefly reviewed in a bit), there are numerous other elliptical

(actually ellipsoidal overall with their arenas being elliptical) roman amphitheaters. They became

abundant over the Roman Empire’s domain, at its maximum spatial extent, at the time of

emperor Marcus Aurelius, circa late 2nd century AD. One of them is the roman amphitheater at

the current city of El Djem, see ref. [4.7], in present day Tunisia, Figure 2.3. it could seat about

35,000 spectators, and it was built in the first third of the 3rd century AD. Elliptical in shape, in

their basic floor plan, the roman amphitheater played a significant role: it linked a primordial

fertility laden symbol, the quasi-ellipse (the female womb and/or the egg and the human eye’s

oval form) to basic biology based competition for survival and dominance. Hence, the shape (its

quasi-elliptical form) appeared organically connected to the multiplicity of socio-cultural

functions the amphitheaters performed. And it was displayed in the grandest possible scale. In

presenting these key places where pseudo-ellipse structures from the Neolithic were shown, and

perfectly elliptical in form structures, of a grander scale, in the Roman Era, a spatio-temporal

sequence is intended to be picked up: the smaller scale precedes the larger scale, and the less

than perfect in design ellipsoidal form is succeeded by a more perfect in construction and design

36

elliptical form (see Note 6 for more), an example of Evolution in Architecture and Design. Roman

amphitheaters were not however the only large in scale structures where elliptical or ellipsoidal

design was used, again under the intent to optimally and spatially manage social massive-in-scale

events. Roman forums were first designed in usually 2-d ellipsoidal forms, as for instance the

impressive late 1st century AD Roman Forum at Jerash, see Figure 2.3.2 in present day Jordan.

Figure 2.3.1. The Roman amphitheater at El Djem, Tunisia. Source of photo ref. [3.4].

Figure 2.3.2. Jerash Roman forum. For a brief history of the city see ref. [4.0]. The forum and

its design as a large in scale sundial has been discussed in ref. [1.15]. Source of photo: ref. [3.9].

37

The city of Jerash, also known as the Antioch on the Chrysorrhoas (or Barada) River, or Golden

Flow River, has an interesting history. Its foundation is attributed to Alexander III, as among the

first cities he founded in the Levant upon his return to the region from his trip to Egypt, see ref.

[1.14]. The city attained prominence in the early part of the 2nd century AD, under roman rule.

The Flavian Amphitheater. However, the most spectacular, interesting from a History as well as

Architecture, and largest in scale roman amphitheater is the Flavian Amphitheater, the Roman

Colosseum, shown in Figures 2.3.3.a (floor plan) 2.3.3.b (axonometric section) and 2.2.3.c (cross

section). The architectonic design of the structure will be the subject of an author’s future paper.

Figure 2.2.3.a. The oval in general shape Flavian Amphitheater, Rome, floor plan: basically, it

consists of six concentric ellipsoids-platforms/corridors placed at different heights from the

ground level, five of them above ground surrounding a central elliptical in shape arena at ground

level. There are numerous subsidiary and auxiliary ellipsoidal in form (and progressively higher

placed sub-platforms) embedded in the structure. Source of architectural drawing: ref. [3.14].

38

Although there are numerous ellipsoids embedded in the overall floor plan of the Flavian

Amphitheater’s design, six are the basic ellipsoids comprising the 3-d structure of the monument,

shown in the axonometric section of Figure 2.2.3.b. These ellipsoids define the spatial extent of

five corridors, set up at different heights, that identify six platforms above the level of the central

arena. These platforms are placed so that they form an approximate 45 angle, shown in the

drawing in Figure 2.2.3.c.

Figure 2.2.3.b. An axonometric section of Rome’s Flavian Amphitheater, showing the six basic

ellipsoids that comprise the 3-d structure of the monument. The five decreasing in width levels

of spectators stands at progressively higher levels are shown. Source of the drawing: ref. [3.15].

The Flavian Amphitheater has been extensively studied and its design analyzed in numerous

contexts, see for example ref. [4.13]. This extensive and spread into many fields literature will

not be reviewed here. What will be though presented is an aspect (not discussed in the literature)

of the Urban Design context within which the Flavian Amphitheater was placed by the architect

(and in conformity with then politics). The context is schematically shown in Figure 2.2.3.d.

39

Figure 2.2.3.c. Rome’s Flavian Amphitheater in cross section. The six basic platforms of the

amphitheater’s architectural design are shown, spaced in 3-d so as to form the arc of a chord set

at about a 45 angle. A cross-section of the entire structure along the minor axis offers an

approximation to a parabola. Six corridors-platforms for circulation were anchoring the

ascending, stepwise, seating (and standing) sub-platforms. These spectators’ seating sections

were placed so that the class structure of Roman society was not only obvious, but also

prominently and intendedly displayed. The emperor’s quarters were at the lowest level and

closest to the arena, at the covertex of the central arena’s elliptical minor axis. All other classes

of roman society were to be seated behind the imperial court’s seating section, and at its side.

Whereas, the poorest of the plebeians were to be seated or stand at the very top platform. The

income distribution was also on display, the less in number spectators of higher income

occupying the lower in capacity and lower laying ellipsoids. The progressively decreasing in

number and in height arcs from ground up, and from out inwardly of the amphitheater’s skeletal

form is also shown in this architectural drawing/cross section. Source of drawing: ref. [4.14].

40

Figure 2.2.3.d. Rome’s Flavian Amphitheater in the Urban Design context of central imperial

Rome’s Forum. The fertility related symbolism is apparent, as the entire ensemble of structures

along Rome’s main North-East to South-West axis peak at the egg-shaped ellipsoidal overall

structure of the Colosseum. Source of Urban Design site map: ref. [3.16].

Situated in the low valley between the Caelian, Esquiline and Palatine Hills is where Rome’s

central area is found; within that area, and between the Palatine and Capitoline Hills is the Forum,

the central axis of the city, where the main municipal (in fact imperial) public (administrative and

religious) buildings were placed, a location designated for both imperial uses as well as municipal

public assemblies. The Forum underwent a large in scale transformation (relatively fast spatial-

temporal dynamics) from the Republican Phase to the Imperial Forum Phase, which is shown in

the site plan of Figure 2.2.3.d. The ubiquitous fertility implying design, and intended phallic

morphology of the Forum through the many phases of construction involved, is a research topic

still to be fully investigated. The numerous planners and architects involved in the design of the

structures, access roads and public spaces of the Imperial Forum Phase, seem to have followed a

schematic plan, the subject of an author’s forthcoming paper. In the period post the fall of the

Republic (past the Actium September 31 BC battle, and the dawn of the Octavian rule) till the

rule of emperor Vespasian (c 72 AD) many structures and events affected the overall form of

41

Rome’s Imperial Forum, as con- and destruction occurred in multiple phases. The amphitheater

(itself built in stages and under three roman emperors, Vespasian, Titus and Domitian) was built

on an artificial lake made under emperor Nero in the Domus Area site and by a colossal bronze

statue of his. The (unknown) architect(s) of the Flavian Amphitheater itself, placed it on, and most

importantly oriented it so that it is precisely lined up with, the major axis of orientation of the

Forum itself, the major components of which are the structures of Basilica Julia and Atrium

Vestae. A detailed diagram of the Imperial Forum and its constituent parts is in Map M.2, before

the paper’s Appendices.

Rome’s Flavian Amphitheater, view from the West. Public domain photo.

42

3. The Five Priestesses Minoan Ring from the “Griffin Warrior” tomb at Pylos.

3.1. Introduction

One of the fifteen or so seals and rings found in the circa 1450 BC Mycenaean so-called “Griffin

Warrior” tomb site at Pylos, in the Peloponnese part of Southern Greece, at an area close to the

Western shores of the Peloponnese’s West-most peninsula of Messenia and close to the Palace

of the Homeric legendary King Nestor, about eight miles North of the modern day city of Navarino

(closely situated to the ancient city of Pylos), and approximately 3.5 miles from the current

Peloponnesian shores on the Ionian Sea, see map M1 at the end of the paper, following the

Notes’ section, and during the excavations of 2015 and 2016 undertaken by archeologists from

the Classics Department of the University of Cincinnati, in cooperation with the American School

of Classical Studies in Athens and the Greek Ministry of Culture and Sports, is the gold signet ring

of Figure 1.3. In the analysis that follows scale and magnification play pivotal roles, as description

of the ring’s iconography and its embedded Mathematics are critically linked to both.

In this section of the paper, first a detailed and comprehensive description of the ring’s curved

top side iconography will be offered. It is intended to cover almost all and from many possible

angles the iconography’s many elements. The description will contain the author’s interpretation

as to what is depicted there, given a number of ring’s magnifications. The oval, almost a true

elliptical in form representation will be linked to certain mathematical as well as astronomical

features apparently embedded into the ring’ iconography. The astronomical feature in

combination with the ring’s thematic structure, supply strong evidence as to the iconography’s

title: it as a ceremonial scene taking place at a specific time of the year, lasting a particular

number of days as indicated by the ring’s imagery. Hence, it will be argued that this ring was the

product of an underlying relatively advanced in both Mathematics and Astronomy culture, with

considerable overall analytical sophistication. Moreover, certain aspects of the Minoan Bronze

Age miniature Art making process will be touched upon that seem also to imply a relatively, for

that era, high degree of dexterity in Metallurgical Engineering as well as in Art and Architecture.

For instance, the minimum size of a discernable component in the ring’s iconography will be

identified, and evidence will be presented as to the likely instrument used by the ring maker to

achieve such a minimum width in carving into the ring’s curved 3-d bezel’s surface.

This subsection will be followed by a more detailed Geometry based analysis of the ring’s surface

design. It will be attempted to establish (in fact to prove) that this surface is quite close to a

perfect or true ellipse. This subsection will be followed by an analysis of a set or key geometrical

features of the sing’s top sided surface. These features support the argument that the artist of

this miniature piece of exquisite Minoan Art was to an extent cognizant of certain basic

mathematical properties of the true elliptical shapes. He may not have had an ellipsograph at his

disposal at that time, but his dexterity coupled with the scale factor for the artifact’s surface in

question rendered the maker’s creation as close as it could possibly get to a true ellipse.

43

3.2. Description of the ring’s iconography and its Astronomy

At the outset, it must be noted that what this author was able to detect and describe was

obtained by a maximum and approximately fivefold magnification of the ring’s surface, which is

very close to 1.76 inches at length in actuality. This is a reminder that the detailed description of

an object is directly dependent (i.e., a function of) the scale (or magnification) at which the object

is observed. The magnification brought the size of the ring at about six inches at length, and 3.75

inches wide. On December 3, 2017 the author received an exact count on the ring’s length, see

Notes 1 and 5 for more on this part of the research. A description of this gold signet ring is also

available within the context of the presentation on the 2015 dig at Pylos by the archeologists in

charge, Shari R. Stocker and Jack L. Davis, the Department of Classics at the University of

Cincinnati, in reference [2.1]. The oral presentation had the catchy title: “The Lord of the Rings”.

For a reference on the archeological matrix of the ring analyzed here, see ref. [2.5]. The ring

obviously was meant to be worn by the warrior in his middle finger, anchored at the finger’s base.

Its band size indicates that the warrior was of a modest stature. The archeologists and forensic

analysts estimate his height to have been around 1.70 meters, and to have had strong oval and

dark brown facial features, which indicate a possible origin from the Levant, see ref. [2.1] and

[2.5] for more on this subject, and ref. [2.13] on the possible (and varied) DNA based origins of

the Mycenaeans and the Minoans. In the comprehensive description that follows, an attempt is

made to cover almost all elements shown on the ring’s surface as detected at the level of

magnification used. In a number of details this author’s description differs a bit from the

description supplied by the archeologists, not unexpectedly, as analysts in Archeology do usually

differ in what the evidence presents and how it could be interpreted. Nor are these differences

undesirable, as they give rise to debate and to a better understanding of the subject matter.

Furthermore, these differences give further credence to the argument by this author that the

field of Archeology is characterized by a quantum superposition state of affairs, whereby multiple

views on evidence and interpretations seem to co-exist at any point in time. It is a natural

phenomenon. This point has been brought up and elaborated in a number of papers cited already

([1.1 – [1.5) and it traces back to the 1991 article by the author in ref. [1.6]. The 5-priestess signet

ring from the circa 1450 BC Pylos tomb of the Mycenaean so-called “Griffin Warrior” apparently

depicts a ceremonial scene, involving five priestesses, three at left and two at right. Some

hierarchy is embedded and a ranking of priesthood is implied by the way these five figures are

displayed on the ring’s bezel curved surface, and the manner in which they are placed on the

ring’s surface ceremonial scene, forming a 1+2+3 ensemble, where the single element (the nature

of which will be addressed in a bit) is the element which looms in the background. The five

priestesses in the two sets (2+3) are dressed differently, with their cone cone-shaped lower part

of their gown decorated in two differing ways. The two sets’ hand postures are also different.

The three at left have their hand(s), the two short figures’ right hand as their left hand is not

shown in profile, while the middle priestess’ both hands, resting on their waist. Whereas, the two

priestesses at right have their left hand extended at their back, and their right hand bent and

pointing to their head. The leading priestess at right is also the tallest of the pair placed at right.

44

Furthermore, another potentially critical differentiation among the five figures is that the pair of

priestesses at right wear cone-shaped hats that touch the ellipse’s perimeter (as we shall see, at

two key points, see also Appendix II on those points), whereas the three priestesses at left wear

no headgear. All five priestesses wear typical Minoan (for the period, see Note 2 on this aspect

of Minoan chronology) attires. The head priestess at center left, wears a long skirt with eleven

ruffles, each ruffle layer consisting of folded pleats. These pleats are among the smallest in scale

elements carved on the ring. Their width is slightly less than half of a millimeter. This is possibly

the smallest size object carved on the ring. In reference [1.8] this author identified the implement

responsible for such miniature Art in the Minoan period under review here (the so-called “Neo-

Palatial period”, see Note 2 on Minoan chronologies). This implement is the Minoan bronze

needle, see Note 3 and Figure N.1, on this aspect of Minoan miniature Art making.

The (possibly assistant and younger in age, indicated by both her height and lack of pronounced

breasts) priestess to her right has a long dress containing nine ruffles, whereas the one to her left

(also younger in age than the middle priestess at the left of the ensemble) has on her dress seven

ruffles, possibly indicative of a still finer subdivision in the ranks of the priesthood hierarchy. The

two priestesses at right wear different type dresses, more glittering but simpler in design. There

are two sets of stripes of horizontal decorative pleats, on both of these priestesses; in the case

of the head priestess at left (but at the right side of the ring), her dress’ pleats carry two sets of

stripes, one at the top with two lines and one at the bottom with three lines. The priestess to her

right (the right hand most priestess of the ring’s surface) has also two pleats with two sets of

stripes on them: the one on top has three lines, whereas the one at the bottom has two (the

opposite of the pattern worn by the head priestess of the pair at right). Maybe, this was another

indication of rank, depending on such insignia. In fact, no two priestesses are alike, and there is

some aspect of differentiation among all five – indicative of a top-down linear totem pole type

of hierarchical structure in Minoan priesthood, rather than a pyramid type structure.

Notwithstanding these differentiations, the middle priestess at left is the tallest not only of the

three on that side, but also the tallest of all five involved in this ceremonial scene immortalized

on the gold ring, barring the height of the cone-type hats worn by the pair of priestesses at right.

That height differential might possibly be indicative of the implied ranking among all five, and

who is the real top figure. This ranking might also be further accentuated by the fact that the

middle priestess at left is the only one not facing towards the Door, the central structure in the

entire ceremony and the structure placed at the very center of the ring’s elliptical surface and

indicative of a background. Moreover, it seems to this author that the pair of priestesses at right

stare at the central priestess at left, whose two assistant priestesses (with the low stature) stare

at the two priestesses at right. This is a major difference in the iconography’s description between

this author and the archeologists’ description of the scene in [2.1], where they contend that ALL

FIVE face the central structure (more on this structure shortly). Hence, the five priestesses

posturing and body motion seem to grant a special role to the central female figure at left. This

author leans towards the view that this figure is the central figure of the iconography.

45

In fact, her body posturing and head position seem to portray an arrogant and defiant look away

from the rectangular central structure. With her posture she conveys an air of superiority in

reference to all other female figures. In addition, one notes that the two priestesses at her flanks

must be her assistants, who may, as already noted, be of a younger age given their lower height

and lack of breasts. The two at right could represent a slightly lower rank than that carried by the

dominant priestess at center left, in the hierarchy of Minoan female priesthood, simply because

they are portrayed as being shorter (exclusive of their hats). All five priestesses’ body angles as

well as their cone shaped dresses’ angles are of import in the Geometry of the ring, and these

angles will be explored in detail in the subsection which follows.

As mentioned, four of the five priestesses are apparently facing, according to the archeologists,

what seems to be a shrine (again, according to the archeologists, see ref. [2.1]), which may be a

tripod (with a very narrow left side) in the form of a doorway (according to this author). The

Door’s entrance is covered by a net, obviously preventing Entrance into the space behind it. The

Door is flanked by two leaning palm trees, connoting origins of this ceremony in Northern Africa

(possibly Egypt) and the lower Levant (possibly Mesopotamia and the Eastern coastal line of the

Mediterranean). It is noted however, that Crete and the Southern region of the Helladic space

are grounds for palm trees to grow as well. More detail analysis by botanists must be carried out

to determine the origin of the particular palm tree depicted in the scene. At a first glance, the

two palms seem to be of the date (Phoenix dactylifera) type (see ref. [4.1]). The two palm trees

could also be specimens of Phoenix Canariensis (the so-called “pineapple palm tree” – a tree that

grows to about 40 to 60 feet in height, see ref. [4.3]).

Height is a critical factor in determining the nature of the central structure at top center of the

signet ring’s iconography, and according to this author forming the background to the entire

iconography. The archeologists, see ref. [2.1], describe this structure as a shrine. The question

however is, what function would a shrine of this scale perform? The relative height of the five

figures, thus the scale of the structure, seems to suggest that this is a door to a space at the

iconography’s background, beyond the reaches of the five figures. Both palm trees and Door (or

shrine, if the reader still wishes to consider it to be a shrine dedicated to sacrifices) are positioned

on a two-layer pedestal and at a higher level than the ground on which the five priestesses stand.

That ground slopes on both sides of the central Door-pedestal ensemble, so that the five

priestesses “ascend” to it. The two-layer pedestal the Door stands on are filled with animals of

various types, as sheep, goats and ibex, possibly griffins, although further analysis by a biologist

is needed to determine what exactly is the species of each of these animals, if real, and whether

they were native to Crete at the time. As for the level of miniature detail and resolution needed

to carve (by the maker), observe and record (by an analyst) all of these elements shown on the

ring’s top surface are still topics to be addressed as possible extensions to this paper. What is

noted is that the archeologists “see” rocks out of these figures, in ref. [2.1] and wonder whether

this iconographic representation depicts an island. Hence, this is a second critical point of

disagreement between this author and the archeological team from Cincinnati.

46

Of interest is also what is placed by the artist in this explicitly three-dimensional iconography at

both above the five priestesses and at the ground levels. And in the description of these items

implanted by the artist and maker of this marvel of miniature Art onto the ring’s bezel top surface

much of the real meaning (let alone the symbolism) of this iconography is pegged. At the ground

level and on both sides of the ring’s ceremonial scene two python type constrictors with their

skin clearly scaled are shown crawling, their heads pointing towards the Door. Pythons are not

native to Crete; thus, these snakes must represent South-East Asian or African species, see

reference [4.2], further lending support to the view that this ceremonial scene must have had

Asian or African roots. In effect, these two snakes seem to be playing a major part in this

ceremony. At a small in scale versions, and held by priestesses in their fists, snakes constitute a

major element in Minoan mythology and Art. Both snakes are shown in this iconography moving

in the general direction and ascending by crawling towards the summit, where the Door is placed.

And now we arrive at potentially the major difference between the archeologists and this author

in the description as to what is depicted at the lower part of this signet ring, and in fact at the

iconography’s foreground. The archeologists in [2.1] contend that what the miniature artist did

was to design a sea with waves (and they allude to an island scene). This is not what this author

“sees” in the pattern we observe in this ceremonial scene. To start with, waves can’t be uniform

squares in a 3-d depiction. A 2-d representation of waves as an eternity symbol in Greek mosaics,

a topic which this author has extensively analyzed in a number of papers, see for instance ref.

[1.9], appear at a totally different context (framing mosaics, for example, where they play a

marginal role – not a central one as they do here). See Note 4 for more on the wave as an eternity

symbol, and why it is rather difficult to accept the view that these squares are part of a net

forming pattern and that they depict a “sea” with uniform waves cresting all in unison and

uniformly. In any case, the intent of the maker in this iconography is certainly not to depict a 3-d

wave structure as an eternity symbol. Instead of a sea, the author sees solid ground and Mother

Earth depicted in this iconography, possibly identified by the head (middle) priestess at left. The

way the Earth is depicted in the ceremonial scene’s foreground, involving the 5-priestess in this

specific Minoan miniature gold signet ring in the middle-ground/stage, is in fact of extreme

interest. The Earth’s soil provides a unifying theme for the entire iconography, and a deeper

message, as it puts a name on the ceremony performed and it identifies a possible season

associated with it. The Earth is shown as the darkest section of the ring’s iconography, possibly

indicative of the fertile soil’s color during the planting season. The darker hue could be in the

carats in gold sheets used. The pattern on its surface closely resembles the netted pattern found

blocking the Entrance at the Door’s passage. The net’s interlocking chain has ripples which run

parallel to those of the Door’s net. These ripples form squares, as those of the Door do. They also

resemble the snakes’ skin leather cover pattern for some species of snakes. This grid could be a

hint at a possible calendar function of the ring’s iconography. At the extreme left and right parts

of the iconography’s Earth-covered-in-soil section, there is a single square; whereas at maximum,

there are ten squares formed by the net in the direction from the North-West to the South-East

(and pointing at the sunrise spot of the local azimuth at Winter Solstice). There are slightly less,

47

eight, at maximum in the net, running from the North-East to the South-West (and at a direction

pointing towards the local sunset azimuth during Winter Solstice). In the middle of each square

formed by the net’s interlocking “wires” hangs a mushroom looking plant. Notice that the

Northern part of these Earth linked net pattern is blocked with the podium supporting the Door

(or Shrine), thus clearly indicating that the observer of the ring’s iconography does not deal with

the extensions of these two directions, whereby the extension of the sunset at Winter Solstice

towards the North would identify the point on the azimuth of the sunrise at Summer Solstice

(and the corresponding one at the other end of the sunrise at Winter Solstice would correspond

to the azimuth point of sunset at Summer Solstice). Hence, the maker of the ring wanted to

clearly indicate and record the Winter Solstice azimuths on the iconography.

The top part of the squares in the Earth’s net seems to be linked with a “seed”, possibly intended

to be planted in the soil. This might offer some additional indication of the time of the year this

ceremony was taking place: late autumn, early winter. Hence, in the direction of the Earth’s net

interlocking squares, one may detect some Astronomy implanted there: the winter solstice. The

“seeds” are attached to the section of the wiring that runs from the North-East to the South-

West, and these corresponds to the lesser number of squares in the Earth’s soil covered surface.

This “seed” does not appear in the archeologists’ description of these squares forming a net as

“waves”. Whereas, in this author’s description the squares-producing grid pattern closely

resembles regular planting patterns on a field (and/or a snake’s skin decorative pattern). In

addition, however, to just designating the Winter Solstice as the temporal marker of this

ceremony depicted on the ring’s surface, the manner in which the set is placed on the ground

may also imply a duration for this ceremony, in terms of number of days involved. As already

noted, there are two directions along which the Earth’s squares containing net is deployed,

indicative of the sunrise and sunset directions on the location’s Winter Solstice sunrise and sunset

rays. Counting the number of rows (or columns) carved in these two directions indicate that there

are eighteen rows in both directions. The first row in each direction (at both ends) contains just

a single square. Hence one might deduce from this netting the duration of the festivities, or

ceremonies – eighteen days, anchored on the Winter Solstice day of the year. The layout of the

net seems to imply that the key event took place at the day of the Winter Solstice, and that was

preceded by nine as well as succeeded by eight days of celebration (although it could be 8+1+9

rather than 9+1+8).

This author finds no other reasonable explanation as to why the size of each square in the net is

what it is on the ring’s bezel top surface. In terms of Architecture, this grid pattern of the Earth’s

soil as depicted on the ring might also be a hint of a modulus on the ring. Moreover, a detailed

look at the Earth’s net, and a plausible interpretation of their slight differences in the two

directions the squares’ sides are set is to detect a “day” (on the North-West to South-East running

lines, identifying sunrises) and “night” (on the North-East to South-West running ones, identifying

sunsets). Hence, the manner one “reads” the iconography (whether from left to right, or from

right to left) would determine whether it is an 8+1+9 ceremonial even calendar days, or a 9+1+8

event, by “1” designating the calendar day of the Winter Solstice.

48

Within this context, the meaning of the Door can now be ascertained: it may symbolize the

Entrance into the planting season. Above all these earthly bound elements-participants in this

ceremony, the artist has placed certain hovering figures, that involve fruits, leaves and branches

of an important ceremonies’ related (possibly with religious connotations) plant. That plant

seems to have three stems, one vertically positioned and the other two leaning correspondingly

at left and right. A botanist might be able to analyze the nature of this plant, which to this author

seems to be a lotus plant. All three stems of the lotus(?) plant lay on the Door’s lintel.

In concluding this description of the ring’s iconography, it is an inescapable conclusion to derive

that this is a ceremony that is related to Earth’s fertility. It is noted that the Earth in this imagery

takes about a third of the entire area of the ring’s almost elliptical surface. All items (humans,

animals and plants) of this ring’s iconography are related to the Earth, which is portrayed as a

unifying theme, with its goddess the middle figure at left. The very meaning of using the ellipse

as the signet ring’s bezel shape – the form of the female womb and at the same time the shape

of an egg – might be the way to approach the ring’s fertility-linked message and the unifying

theme of the entire artifact’s structure. After all, even the very act of wearing the ring, the finger

penetrating the ring’s band, can be interpreted as a fertility-linked symbolic act (copulation).

Finally, one may now safely speculate as to how was the ring worn, regardless of whether its

carrier had placed it on his right or left hand: the ring’s Earth section (bottom part of the

iconography) would always point towards the hand’s index finger, and it was to be seen (as the

finger would point West) from the hand’s back side, i.e., at the back side of the palm’s arch, the

ring’s ellipse so that the major axis was in a due East-West orientation, the ring could be used as

a carry-on calendar for seasons by depicting the sunrise/sunset sunrays’ orientation.

A point on architectonic morphology must also be brought up in reference to this Minoan ring,

and a number of other Minoan rings and seals that have an ellipse-like form. The typical Palatial

Minoan Architecture is of the rectangular type, at an overwhelming degree. These rings and seals

belong to the oval, arc, apse, cyclical and quasi elliptical design strains that are so dominant in

the Architecture of the Levant, the Natufian rooted Architecture. Rectangular Sultanian typology

Architecture was an innovation also introduced during the late 7th, early 6th millennium BC time

period in the Levant, see ref. [1.1]. Could it be that the elliptical design was an import from the

Levant as well, and not a purely Minoan endogenously derived design? This question will be left

for the interested reader to ponder. Unless evidence of ellipse-like built structures are found on

the Island of Crete, and dated to an era prior and quite close to the start of the Neo-Palatial

period (circa 1700 BC), it could be that this design (and possible knowledge associated with it)

may had been imported from the Levant. Moreover, since this is not a paper on Minoan Religion,

symbolism, mythology, etc., and hence partly a study on the meaning of this iconography, the

possibly many descriptions and interpretations of these elements (as for example, what

priestesses – or possibly goddesses – are depicted in this iconography from the Minoan extensive

list of priestesses and goddesses, see ref. [2.6]) are left to the interested reader(s). Here, the

Geometry of the ring and its iconography is the main focus, which might assist in offering hints

to the expert on Minoan iconography and mythology related symbolic interpretations.

49

Consequently, all symbolic and non-Geometry based descriptive aspects of the gold signet ring

are to be thought of and considered as tentative. As is well known in Archeology, and especially

so for pre-classical subjects for study (that is subjects where documentation of symbolism is not

available in written records), propositions on symbolism are statements that can neither be

proved or disproved. Since formal statistical analysis isn’t possible in these cases, so that tests on

rejecting the null hypotheses are next to impossible to carry out, these statements on symbolism

are simply propositions potentially carrying insights on iconographies, at best.

3.3. The ring’s Geometry

Is the ring a real ellipse? It is as close to a true ellipse as it could possibly be under the

circumstances. In this subsection of the paper, and in the sequence of the five Figures provided

(Figures 3.3.1 – 5) the case will be made that the artist had an “intuitive understanding” of the

true elliptical shape, and that the maker approximated that shape as closely as it could possibly

do under the circumstances (s)he faced, namely that the full Mathematics of the ellipse were not

known back then to the maker and neither did the maker have access to an ellipsograph. The

argument of an “intuitive understanding” strengthens if the artist carved it without previously

drawing it. However, analysis makes it clear that a drawing of the ellipsoid preceded its molding.

First and foremost, in the context of this analysis, one must convincingly demonstrate that the

ring under investigation is in fact elliptical, and since it isn’t a true ellipse, to what extent is it as

close as possible to one. Initial tests would require to ascertain that the maximum length of the

surface’s shape under scrutiny does indeed occur at where the major axis of the ellipse ought to

be; and correspondingly the maximum width of the shape under investigation does occur where

the minor axis of the ellipse ought to be located. In fact, they do. The reader can easily verify this,

by a simple visual inspection of Figure 3.3.1. However, this preliminary test isn’t sufficient by itself

to make the firm determination that the shape is indeed an ellipse. More tests are needed. To

that end, by establishing the precise location of the two main axes of the ring’s surface, the major

(long) axis x, and the minor (short) axis y, the center (origin) of the ellipse, designated as point B

in Figure 3.3.1, is determined. There is a point to be made about a line slightly to the left of the

y-axis in Figure 3.3.1, and quantity shown, that will be expanded on later.

Lengths L is the major axis, and the length (2a) in the discussion of Figure 1.1; in the case of the

ring in question L=4.47 centimeters or 1.76 inches (conversion rate: 1 cm = .3937 inches).

Whereas, W is the minor axis, and the length (2b) in the discussion of Figure 1.1; in the case of

this ring, W=2.77 cm (1.09 inches). The ratio of these two lengths r*, where {r* = a/b = L/W} is

found to be r*=1.6146. It is thus noted that this is a ratio extremely close to the Golden Ratio

(1.6180……) The extremely close approximation is a strong indication that it is not random.

Hence, it is concluded that the Minoans were getting very close to that Golden Ratio (on aesthetic

grounds), a Ratio that was elevated to prominence by the Art and Architecture, as well as the

Mathematics of Classical Greece about a millennium later. These specifications (the 2a, and 2b

axes) are necessary and sufficient to define a true and unique ellipse.

50

Thus, the question is, to what extent the ellipsoid we encounter in the bezel of the Pylos 5-

priestess ring corresponds (or comes close) to this true ellipse with major and minor axes equal

to 1.76 and 1.09 inches. From the Geometry of a true ellipse, the length referred to as the “linear

eccentricity” (designated as c in the discussion of Figure 1.1), of the ellipse defines the distance

of each focus, points F1 and F2 in Figure 3.3.1, from the center B. In the case of the true ellipse it

is directly derived from the application of the Pythagorean Theorem: that length c is equal to the

square root of the difference of the a^2 minus the b^2. In this case c=1.754 cm (.69 inches). For

any scale of magnification of the ring’s surface one can locate the points where these two foci

must be. Once these two foci have been located, one can check whether the fundamental

property of ellipses applies, namely, that all points on its perimeter must have their sum of

distances from the two foci a constant equal to the length 2a (in this case of magnification equal

to seven inches or 17.8 centimeters) does apply or not. The author found that in general it does.

A large number (although not all) points on the ellipse’s perimeter (checked as to their pair of

distances from these two foci) seem to comply very well with the fundamental property of

eclipses, as stated in the paper’s section discussing Figure 1.1, as their relevant sums are

extremely close to seven inches in this scaled version of the ring. Any point on the perimeter of

the ellipse in Figure 3.3.1 can be tested as to whether it is part of the true ellipse as specified by

the pair of axis (2a,2b) or the pair of semi-axes (a,b). Obviously, in Figure 3.3.1 these lengths have

been scaled. At this point it must be noted that photographic distortions, from the source in ref.

[3.2], to the Figures below need be considered in making the determinations and propositions

advanced in this paper. Outmost care has been taken by this author to rule out such distortions,

although a formal statistical analysis (that could be carried out by randomly selecting points on

the perimeter and then run regression tests on them to check whether they meet their expected

length total) was not carried out (it is left to the interested reader).

There are obvious sections on the perimeter, like for instance the pair of symmetrically as to the

y-axis placed small arcs on the perimeter’s top side, the right-hand side of which is shown at the

vicinity of the point where line Z(r) intersects the perimeter. This is an anomaly that could

possibly be caused by the wear and tear over the three and on half millennia of the ring’s life in

the tomb, or prior to the burial events associated with the griffin warrior’s life. However, the

presence of its twin anomaly at the upper left-hand side may be ruling out this possibility, hence

strengthening the suggestion that some inscribed circle was the cause of this local approximation

to the true elliptical shape. The author examined a number of selected points on the ellipse’s

perimeter, among them points where the drawn green lines intersect the ellipse’s perimeter.

For each point the distances from the two foci, F1 and F2, distances which in sum, as already

stated in the earlier section, must equal 2a, met the fundamental property of the ellipses rather

satisfactorily and within ranges of at or less than 5%. More on this in a bit. It is recalled and

restated that, knowing a and b are the necessary and sufficient quantities to draw the ring’s

corresponding true elliptical shape. Parenthetically, and in a first attempt to link this subsection

of the paper to the previous one that contained a comprehensive description of the iconography,

51

it is noted that this pair of, critical to the Geometry of an ellipse, foci (F1 and F2) fall very closely

to the right-hand side border of the two priestesses’ (at the extreme right and extreme left sides

of the scene) dresses. The Geometry of the ring’s iconography, hence, seems to imply some

special designation for, or recognition of, these two priestesses, and especially for the dresses’

exact cone shaped spatial extent.

The test of whether the schema of the ring’s surface is an ellipse or not, as already noted, was

carried out for a number of points on the ellipse’s perimeter. As an example, the case of a critical

point is shown next. This point (at the right-hand side of Figure 3.3.1) is where the line H1 (the

line which contains the lintel of the Door) intersects the ellipsoid’s perimeter. This particular

point is also the point where line z5 (the line which can be thought of as identifying the general

direction of the body of the priestess located at the extreme right-hand side of the ceremonial

scene, and where the focal point F1 is located – actually about 1/36 the length L to the right of

z5) intersects the perimeter of the ellipsoid.

The sum of these two distances was found to be exactly 2a. Specifically, in this scaled version of

the ring’s ellipsoidal surface, where the length was about seven inches, the sum is {1.5” + (5.5” +

1/16”)}. Hence, in this approximately fivefold magnification the computed count is off by less

than 1% of the actual (expected) count, at the ring’s actual size. One to five percent was the range

of error encountered in almost all of the tested cases involving points on the ellipsoid’s perimeter

– thus fully proving the contention that this shape is in fact an extremely close approximation

to a true ellipse. The miniature scale consideration on top of the above-mentioned analysis,

considerably strengthens the argument, the paper’s main thesis as just stated.

A comment is needed at this point, and before the analysis proceeds any further. The comment

has to do with the line Z(r) of Figure 3.3.1 – where the local anomaly in the ring’s surface occurs

- and what it represents, as well as how it was drawn. The point brings up the quantum nature

of the line, meaning that different observers would draw the line differently, when asked to draw

a line representative of the priestess’s body. Drawing of that line is critically also pegged to the

magnification at which the observer observes the ring, since the resolution of the line (its

thickness) is a function of the magnification of the image used on which to draw it. This applies

not only to the priestess at the extreme right of the scene, but to all five figures (priestesses or

goddesses).

The author exercised his best judgement in drawing the line Z(r), fully cognizant that someone

else might had drawn it differently as magnification changes, to some extent, or the heuristic in

finding the body’s axis of symmetry may have different solutions. However, with this caveat in

mind, it is underscored that the slight differences in drawing this line (and all other lines to be

discussed in the following part of this subsection), in no way invalidate the main conclusion and

the proof that this ellipsoid is an extremely close approximation to a true ellipse. Nor does it

negate the basic propositions to be advanced next. It only serves as a reminder that these aspects

of an iconography (especially in the case of miniature Art) carry with them inescapably some

degree of uncertainty.

52

What exactly did the maker know about the Geometry of an ellipse? The maker very likely did

know some elements of the fundamentals of an ellipse, whether drawing or crafting it, and most

likely drawing it before molding and carving it. As the previous subsection proved, the maker

knew how to draw in 2-d and craft in 3-d an ellipsoid that could closely approximate an ellipse in

2-d. The artist placed the maxima along both axis at where there ought to be (at halfway points),

along axes that the true ellipse’s foci are found. Moreover, and to a great extent, many (although

not all) points on the ellipsoid’s perimeter fully satisfy the fundamental principle of ellipses. It is

obvious that the maker of the ring was trying to experimentally (through trial and error) estimate

the exact location of the two foci on the major axis, being intuitively aware of the fundamental

property of ellipses. This is the definition of the notion of “intuitive understanding” of ellipses

claimed by this author the maker of the Minoan ring had at the time.

Next in this subsection, an attempt is made to find out the full extent of that intuition. Namely,

how much intuitive understanding the creator of the ring’s iconography had of the other

fundamental components of an ellipse, namely its two director circles and the two directrices

(see Appendix II on their definition). We conclude that the maker was not aware of their

existence, as the design bears little relevance to them. Hence, it is concluded that the artist must

have known (intuited) some but not all of the ellipses’ fundamentals. In the analysis that follows,

a number of magnifications of the ring’s 2-d ellipsoidal surface are used, varying from about 1.5,

to approximately five times the actual length size of the ring, shown in 6” wide photos in scaling

due to photography in Figures 3.3.1 – 5, in Figure 1.3, and at the paper’s cover page preamble

photo. It is very reasonable to assume that the maker had an intuitive understanding and

perception of the linear eccentricity c, and the eccentricity (ratio c/a) e=.785. The maker did not

know how to algebraically compute the length c and the ratio e, possessing an intuitive

understanding of their existence, quite likely on aesthetic grounds. Most likely, (s)he used

approximations to both of these counts in the design of the iconography. In this subsection, the

author attempts to provide evidence from the ring’s iconography to support this contention.

However, it is also noted that this corresponds to a “heuristic” manner in approaching the

subject. A more systematic way to deal with this research question would be through an

exhaustive computer-based simulation search. At the start, the superposition of the true ellipse

to the actual ellipsoid would offer an exact degree of deviation between the two,

However, to systematically research the question (and test related hypotheses) one need set up

the following research program to search the degree to which the maker accounted for the

critical Golden Ratio, linear eccentricity, and ratio e: for all elements of the scene depicted on the

ring, for example the five female figures, the Door (or Shrine), the two palms, the two snakes,

the plant at the top, etc., to record their height and width. Then produce for the total number of

elements (say, ) pairwise comparisons – thus combinations on both counts, hence 2, height

and width – and obtain their ratios. Statistical tests then can be run on the distributions of these

ratios about a mean value, and test whether that mean value is close enough to relevant ratios,

such as e. The eccentricity e (.785) in this ring’s ellipse is very close to the 7/9 ratio of .777. In

53

addition, one can draw lines depicting the location of the elements (such as the set of lines Z

in Figure 3.3.1, some of which will be addressed momentarily) and derive their pairwise ratios of

distances from both the origin B as well as the foci F1 and F2, and compare their ratio to either

semi-axes a, b its linear eccentricity c (1.754 cm). In the heuristic search below, some specific

lines associated with the major elements of the ring’s iconography will be studied in turn, starting

with the central and dominant element in the entire iconography, the Door, see Figure 3.3.1.

Figure 3.3.1. The Pylos 5-priestess Minoan gold signet ring, at an approximately three-fold

magnification (the photo’s width is 6”). The two (major and minor) axes lengths are shown of the

ring’s oval (ellipsoidal) surface. Length L (approximately 1.76 inches in actual length) and width

W (about .1.09 inches) are such that they come extremely close to the Golden Ratio (deviating

by less than two tenths of one percent. L and W uniquely identify the equivalent true ellipse’s

shape. The ring comes extremely close to the true elliptical shape corresponding to these two

axes. Key lines are shown here, as well as the two foci (F1 and F2). Source: the author, from

photos given in ref. [3.2], and ref. [2.1], [2.5].

54

The Door’s vertical axis of symmetry (drawn from the extreme right vertical line of its right

orthostat, to the extreme left vertical line of the left orthostat) differs by a small quantity from

the ring’s vertical (and minor) axis y. The Door’s lintel (horizontal line H1) is at a distance at a

ratio of about .14 of the 2b minor axis from the top covertex (.86 from the bottom covertex). Line

H1 intersects the z2 line (identifying the leading priestess’ at left axis of symmetry) at a point

precisely on the ellipsoid’s perimeter, and a point tested by this author as to whether it fulfills

the fundamental property of ellipses – and it does, within a margin of error of less than one

percent. The ground slopes along lines k1 and k2 in Figure 3.3.1, and is traced by the bodies of

what seem to be apparently two serpents crawling towards the Door, the bottom part of which

falls on line H2, which is about halfway between H1 and H3.

Figure 3.3.2. The true ellipse’s two director circles (see Appendix II) are shown, superimposed

in a photo of the signet Minoan 5-priestess gold ring’s 2-d ellipsoidal surface (a proto ellipse). In

this 6” wide photo, the image corresponds to an approximate 1.5 times magnification of the ring’s

length. Source: the author from a photo in ref. {3.2] in conjunction with ref. [2.1] and ref. [2.5].

55

The equal angles of the two palm trees, see Figure 3.3.1, from the y-axis - formed by lines Z(r)

and Z(l) and the y-axis - are close to 23.5 (the right is close to 23 and the left close to 24, the

24 angle being exactly the 15th of 360 angle). The combined angle (close to 47) is about 1/7.7

of the 360 angle. The intersection of lines Z(r) and Z(l) is on the y-axis, at the level of line H3, and

close to one third of 2b on the y-axis from the bottom covertex (hence two thirds from the top

covertex). Finally, on Figure 3.3.1, it is detected that all three axes (z1, z2, z3) representing the

three priestesses’ (at left) angles of leaning form parallel lines at about 8 angles from the vertical

y-axis (1/45 of the 360 angle). On the other hand, and at right, the corresponding lines z4 and

z5 for the two priestesses are not parallel. Analysis of the iconography now turns to an

examination of the two director circles (see Appendix II) of an ellipse, and the pondering of the

question: to what extent was the maker of the ring’s iconography aware of their existence.

Figure 3.3.3. The Minoan signet gold ring photo, corresponding to a two-fold magnification of

the ring’s actual length and width in this 6” wide photo, has been superimposed on the

corresponding true ellipse’s two directrices, the vertical lines at the right (D1) and left (D2) sides

of the ring. Source: the author, from a photo in ref. [3.2] in conjunction with ref. [2.1] and [2.5].

56

Figure 3.3.4. The Minoan signet gold ring from Pylos, at an approximately two-fold

magnification in this 6” wide photo. Four circles at the two pairs of vertices and covertices are

shown. Source: the author from a photo in ref. [3.2] in conjunction with ref. [2.1] and [2.5].

57

The answer will of course hinge on whether elements of the iconography can be linked to their

presence. From Figure 3.3.2 one can safely conclude that the artist was not fully aware of their

existence, since apparently no elements of the ring’s iconography are linked in any discernible

way to them. The only possible exception is line L in Figure 3.3.2, (the likely axis of inclination for

the priestess at left closest to the Door). This line intersects the two director circles exactly at

their southern point of intersection. However, this single occurrence can be construed as a

random coincidence. Chance events of geometric interest are likely to occur in any iconography.

The determining factor is whether enough of them occur to allow one with some degree of

confidence to determine that a Geometry does appear to guide the iconography’s specifications.

Although a close relationship between the actual ring’s surface and the true ellipse may be

apparent by examination of Figure 3.3.2, analysis involving the ellipse’s two directrices (see

Appendix II on them) make the connection look rather tenuous, Figure 3.3.3. No part of the

iconography seems somehow strongly and directly connected to their presence. Consequently,

it is concluded that the maker was not aware of these two fundamental components of an ellipse.

However, the stage is different and connections become stronger when four particular circles

enter the picture, see Figure 3.3.4, when two sets of circles are drawn: one set with centers at

the two vertices of the ellipsoid, and a radius equal to the major axis; and the other set drawn

with centers at the two covertices of the ellipsoid and a radius equal to the minor axis.

It is apparent that the maker of the ring took these four circles far more under consideration in

the design of the ring’s iconography, than the mathematically important director circles and

directrices lines shown in Figures 3.3.2 and 3.3.3 correspondingly (and in Appendix II). These four

circles, of mainly aesthetic value, but not fundamental mathematical interest, do identify four

perimeters and four enclosures on and within which particular points and lines emanating from

the ring’s iconography do relate; see for instance lines L1 and L2, as well as M1 and M2. Hence,

they seem to play a somewhat pivotal role in the deployment of the iconography’s pattern.

Finally, the part of the ellipsoid that reveals the seasonal (calendar) function of the ring is

analyzed, through a detailed look at the square grid of the soil’s composition comprising the

lower section of the ring’s iconography. Lines N1 and N2 of Figure 3.3.5 identify the directions on

the Earth’s square grid pattern with the maximum number of squares in both directions (ten on

N1 and eight on N2). The directions themselves point at the sunrise of the Winter Solstice (N1)

and the sunset at Winter Solstice (N2) on the azimuth plane. Eighteen is interpreted by this paper

as the number of days the ceremonies lasted around the day of the Winter Solstice – this being

the only explanation this author could attribute to the size of the grid’s modulus.

Although the ring’s other important dimensions, associated with the band’s sizes and

morphology was not a part of this research effort, their sizes could be of import to researchers

of the ring’s total morphology. As obtained from the archeologists, these sizes are as follows: the

(also ellipsoidal) band’s width is 1.08 cm (.425 inches) i.e., about a quarter of the ring’s length.

The inner major axis of the ring’s ellipsoidal band (hoop) is 1.81 cm (.713 inches), hence the ring’s

58

length is about 2.5 times its band’s interior major axis. The ellipsoidal band’s inner minor axis is

1.48 cm (.5827 inches), so that the ratio of the major to minor axes is 1.22. The total height of

the ring (band plus bezel) is 2.32 cm (.913 inches); hence its ratios to the bezel’s length and width

are 1.92 and 1.19 respectively. The ring’s weight is 37.2 grams.

Figure 3.3.5. The Minoan gold 5-priestess ring’s calendar function, in a three-fold approximate

magnification in this 6” wide photo. The inscribing rectangle of the actual ring’s proto-elliptical

surface, and the key lines N1 and N2 (directed towards the sunrise and sunset points on the

azimuth plane) are at precisely the corresponding projections of the true ellipse’s two foci F1 and

F2. Source: the author from a photo in ref. [3.2] in conjunction with ref. [2.1] and [2.5].

This finding concludes the geometric analysis of the Minoan 5-priestess signet gold ring from

Pylos. Many topics for further research have been identified, which could shed additional light

on this magnificent miniature Art piece from the first half of the 2nd millennium BC. Although

more can certainly be learned by additional and more extensive analysis and search, the basic

components and findings about this ring’s Geometry and iconography have been set. Of course,

tentative, as all research endeavors are, the author is satisfied that the findings reported can

withstand scholarly scrutiny to a large extent.

59

Concluding Remarks

At about point 32’ into their presentation, see ref. [2.1], the archeologist of the excavation at

Pylos, J. L. Davis brings up an Emily Vermeule (see ref. [4.5] on her life and work) quote: “Most

prehistoric art is not really understandable. There is no convincing way to relate designs on gold

to burial rites or to religion or community symbols of belief. This is always true in a preliterate

world ….. Yet rational understanding is not necessary when confronted with so much that is

beautiful beyond reason” (in bold and italics emphasis by this author).

There is much one can agree with this quote. Yet, there are also some parts to it that one can

either mildly or strongly disagree. Yes, some of the artifacts of the so-called “prehistory” (i.e.,

points in time falling mostly in Neolithic of Upper Paleolithic periods) are indeed “beautiful

beyond reason”. What one may mildly disagree with is that “most prehistoric art is not really

understandable”. This author espouses the view that some of it is understandable and becoming

more so as time moves on and more systematic studies are undertaken on them. But what finds

this author in total and strong disagreement is the use of the term “preliterate”. Humanity, over

the course of the past fifteen millennia or so has moved way past “preliterate’. Abundant

archeological evidence, in both its Art and Architecture as well as Engineering, seems to indicate

that from the Epipaleolithic and early Neolithic down to the Metals Age (Copper, Bronze and Iron

Ages) the stock of knowledge gradually being acquired by the inhabitants of Eurasia (and possibly

Africa and the Americas – the author is in no position to offer any guidance on these settings, as

his study has primarily focused on Eurasia and North-East Africa) has been on the increase.

At times, given the extraordinary Art and Architecture we come across monumental sites in the

contexts of Western Eurasia and North-East Africa, some of them already mentioned in earlier

sections of this paper, evidence seems to strongly contradict the term “preliterate” attached by

E. Vermeule to the inhabitants of these regional settings and of that Era. Besides, from the angle

and standards of 5000 AD, we today may be considered “preliterate”. The term is obviously a

relative term. It can be applied to any civilization, at any point in space-time, in reference to any

subsequent cultural context. But there is also some grain of absolutism in that term. “Preliterate”

artists do not produce the miniature Art, Architecture, Engineering and Mathematics found in

the gold signet ring, the subject of this paper’s analysis. Yes, the mathematical knowledge base

of the artist who made this ring was not that of Ictinus or Apollonius. Yes, the process involved

in the acquisition and production of knowledge and understanding has not been smooth and

painless over the millennia. But it has been incessantly evolving and increasing in stock size.

One might legitimately pose the question whether is this stock of knowledge from the Neolithic

or the Bronze Age, as is the case of the ring under study here, comprehensible to us, analyzing

these cultures today. Whether is it comprehensible to us today as the result of a rational process

at both ends; and whether the rationality of that time obeys the same rules (or cannons) of

current day rationality, even though the manifestation of these cannons by individual and

collective action may have been different, are obviously questions not for this paper to fully

60

address. But ss researchers, we must answer that, very possibly yes, it is comprehensible; and at

least as individuals dedicated to reason, we have neither the inclination nor desire to think

otherwise. This of course doesn’t render us, or them back then, always correct in our (or their)

assessments. But as evidence seems to suggest that they tried and experimented, so do we today.

But there are qualifiers to this author’s contention, that need to be brought up. On the one hand,

as E. Vermeule (a scholar on ancient Greek culture nonetheless) has done, one can significantly

underestimate the stock of knowledge and the “reason” behind the monumental Art and

Architecture of the Neolithic, Bronze and Iron Ages. On the other hand, as many archaeo-

astronomers have done, one can significantly overestimate that stock of knowledge, at times at

absurdum. This author espouses the strong belief that one must pursue a balanced approach, a

route between the two extremes.

The quest for our understanding of the ancients’ knowledge and their own level of understanding

must be fair to them: misrepresenting them is ultimately doing a disservice to them and a

misspecification of their real capabilities and struggles in the pursuit of knowledge. By examining

this magnificent piece of middle 2nd millennium BC Minoan miniature Art, one can possibly gauge

well the mathematical as well as engineering sophistication of its maker(s). It was neither

advanced Nanotechnology, Astronomy, or Mathematics by today’s standards they possessed, as

some archaeo-astronomers would contend. But nor was it the outcome of random acts of

drawing and carving, motivated by pure emotion and perceived Aesthetics and politics of the

day, by an illiterate or preliterate Cretan peasant as Vermeule would have it.

Instead, the ring’s design and iconography were the outcome of a methodical process that

evolved over time periods that spanned centuries and with input by individuals from many

locations and cultures. In its final configuration the ring’s ellipsoidal form, approximating the

shape of a true ellipse, was attained as the result of a dynamic design process that commenced

by using initially three (the Genesis of the ellipse event, which is unclear when or where it took

place), to employing in this case of the Minoan ring from Pylos four circles. Down the road, the

designer employed five circles – the case of Borchardt’s ellipsoid at Luxor. In addition, this proto

elliptical in shape ring was the product of some efforts by the maker to incorporate onto its

iconography some information related to the local Astronomy, associated with certain

ceremonial and religious observances at the time. All that copious effort was coupled with a

struggle to better handle what back then must had seemed a formidable task and challenge: the

mastering of the Mathematics and the drawing of true ellipses.

At the end, this was an artifact of a nodal character in both Archeology and Mathematics. By

using a number of circles, most likely four, to draw his/her ellipsoid and approximate the shape

of a true ellipse, the maker of the 5-priestess gold ring from the Griffin Warrior’s tomb at Pylos

at the boundary between the Bronze and the Iron Ages, followed in the steps of the architects of

the stone enclosures of the Neolithic, made an advance and thus glazed the path for Thutmose’

Bust of Nefertiti, for the maker of Tutankhamun’s Mask, for the artist of the ellipsoid at the

Temple of Luxor, and for the architects of the Roman amphitheaters and forums. That ring has

61

been imbued with a fascinating story of 2nd millennium BC Mathematics and Astronomy. It must

had been considered a significant artifact by the standards of that era, so much so that it

attracted the attention of possibly one of the major political and cultural figures of the

Mycenaean world, the “Griffin Warrior”. His tomb preserved for posterity this exquisite

miniature artifact. To the Minoan ring’s maker(s), humanity owes a great deal of recognition and

praise, as it does to the Griffin Warrior who might had commissioned it. Certainly, he guarded it,

and kept it for posterity and ultimately, for humanity’s benefit.

Miniature Art, and hence this ring’s form and iconography, bring up an interesting question

having to do with scale and drawing. In architectonic construction, architectural drawings at a

smaller scale precede the actual construction of a structure at a larger scale than the drawings.

In miniature Art and artifacts, the direction is reversed, as the sequence follows in the opposite

direction: from a larger in scale drawing the maker proceeds to a smaller scale creation. How

exactly this occurred back then needs further pondering and analysis.

Finally, and on a different front, it is noted that the analysis (initially, and reflected in the first

version of the paper of December 2, 2017 uploaded on academia.edu and researchgate.net the

same day) carried out in this research, without knowing, and having estimated from publicly

available photos the approximate measurements of the ring (in the range of 1.50”-1.70”). On

December 3, 2017 the author received the almost exact measurements used in this version of

the paper (Update 1) from Luci Philips, as she had received from the University of Cincinnati,

Classics Department and prof. Jack Davis. The author is extremely grateful to both for this

information. The measurements received are reported in full at the concluding part of subsection

3.3 of this paper. In their utilization here, among other findings, the information received on

December 3, 2017 confirmed the author’ original estimated ratio of length to width of the ring’s

bezel to be close to the Golden Ratio.

The issue of how the author obtained the exact measurements of the ring’s bezel and band does

bring about, however, some critical issues associated with archeological excavations and making

public the hardcore data associated with the product of the archeologists’ digs. The associated

serious need for an in-depth overhauling of current archeological practices is addressed a bit

more extensively in Note 5 below. The Note draws attention to potentially some significant

shortcomings involved in the current practices of archeological exploration and discovery. An

analogy to planetary explorations makes the point clear. In space explorations, the differing roles

and involvement of spacecraft engineers and planetary scientists are rather clear. Clear are also

the formal procedures to be followed by the agency responsible for the exploration program.

Explicitly spelled out are issues of data collected provenance and expected practices by all

involved in the program regarding their use. Above all, formal procedures are laid out regarding

the availability of the data obtained through the space explorations to the scientific community

and the public at large. It is high time that similar procedures be set forward by the archeological

community regarding the matter of its exploring the humanity’s past.

62

Six Notes

Note 1. The exact size of the ring was not known to this author until December 3, 2017. Only

approximate sizes were used from the published reports cited to produce the initial version of

this paper of December 2, 2017. The author obtained the ring’s exact sizes (from Luci Philips who

obtained them from the archeological team) on December 3, 2017, and on the basis of these

counts this version of the paper was produced. See also Note 5.

Note 2. This author will not attempt to put firm chronologies on the Minoan artifacts discussed,

except to remark that the artifact in question belongs to the 1700 – 1450 BC period. When

chronologies can’t be cited with any advanced degree of confidence and usually margins of error

in dating artifacts and structures range in the 10 – 15% range of the base (thought as some “likely”

chronology), to provide for Minoan Civilization finer time periods than the margin of error seems

futile if not misleading. Some have insisted on using more refined subcategories and time periods

to these categories: Pre-Palatial (2700 – 1900 BC), Proto-Palatial (1900 – 1700 BC), Neo-Palatial

(1700 – 1400 BC), and Post-Palatial (1400 – 1150 BC), Sub-Minoan (1150 – 1100 BC), and Doric

(post 1100 BC), see for instance a description of these classification in [2.3]. As another example,

see the case involving Minoan chronology in ref. [2.4]. The author finds these time frames

indefensible. Hence, the signet ring under investigation here ought to be simply referred to as

“Neo-Palatial”. It is little to be gained by trying to decide whether it is MMIIIA, MMIIIB, LMIA, or

finally LMIB, the classification found in [2.4] – these are in fact meaningless for the case in hand.

This topic is further expanded in a forthcoming paper by this author on “The evolution of Minoan

Miniature Art”. Fuzziness in dating is a complex topic addressed in a research post by the author

in ref. [1.7].

Note 3. In a Facebook research post, see ref. [1.8], the author identified the implement

responsible for the miniature Minoan carving, the Minoan Bronze needle. A photo of it is shown

in Figure N.1 below. It identifies a 2nd millennium BC six centimeters long Minoan Bronze Needle

– an exhibit at the New York Metropolitan Museum of Art (item # 26.31.476). As the author noted

in ref. [1.8], the needle shown is about half of a centimeter at top, while its working edge reached

down to about a third of a millimeter. This width is slightly less than the width of the priestesses’

pleats at their dresses’ ruffles, possibly the smallest in scale item in the Minoan miniature artifact

under consideration.

Figure N.1. The Minoan Bronze Needle. Source: ref. [3.3].

63

Note 4. The author has created and administers a Facebook scholarly group where the wave and

the meander, as eternity symbols, are analyzed. In specific, their origins (which, in the case of the

meander, have been found to be in the Balkan 6th millennium BC South of the Danube and on the

Strymon River Valley in current day Bulgaria, as well as appearing concurrently at the central

Thessaly region of current Greece, all in the form of four-legged frogs) are studied in that group.

Furthermore, how these two eternity symbols, possibly in combination, have evolved in their

iconographic representations from the Pre-Classical Greece (circa 7th century BC) to the

Hellenistic and Roman periods found in ref. [1.10] is also a subject of analysis of that group. The

group explores the form and the structure of closely associated with the meander eternity

symbols, and key among them is the double-flowing deign of waves, usually appearing as framing

a variety of mosaic iconographies. In the iconography under analysis here, it is hard to accept the

view that the net pattern shown in the signet Minoan ring iconography represents identically

flowing and equal in size “waves”, especially under a 3-d perspective. The double and

counterflowing waves as an eternity symbol was a feature not known yet to the Minoans, and it

is not encountered in frescoes of Minoan Art as far back as the middle of the 2nd millennium BC.

Note 5. This Note deals with Ethics and proper conduct in Archeology by archeologists regarding

the evidence they had the high privilege of being granted the authority to unearth, and on how

properly to dispose of the information they have gained and to which they enjoy primary and

exclusive (but time limited) access. It also addresses some issues involved on the proper disposal

of hardcore information (data) describing their findings, as well as who is the most qualified to

analyze these findings. This is clear from an analogy to Planetary Science and Space Exploration.

When an Space Agency (NASA or ESA) carry out a particular space exploration, through the use

of a spacecraft, the roles of the spacecraft construction engineers and the planetary scientists

responsible for the processing and analysis of the data collected are rather clear. They are

different in each case. Moreover, how the data collected by the spacecraft are to be used and by

whom are explicitly stated in the program’s procurement contract. Legal and ethical, as well as

engineering and scientifically established and acceptable guidelines are stated as to the differing

roles various entities are called to play in the process. Adherence to those guidelines is in

accordance with formal procedures governing any scientific Code of Conduct. It is high time for

similar guidelines to be set up and govern archeological explorations as well, accounting of course

for the obvious differences that distinguish the two cases. One of these differences is that in the

case of archeologists in the process of excavation they are (unavoidably and maybe also

unfortunately) guided on how to proceed at any point in time-space by their own theories about

the evidence, thus affecting the evidence obtained. This condition may be ameliorated by having

simultaneous, contemporaneous, feedback by outside agencies in the process of excavation, a

situation that may entail additional provisions in the Code of Conduct recommended to be

drafted by the international archeological community and adopted by practicing archeologists.

64

Worldwide access to the Internet by a wide range of users, is now making access to archeological

findings to a far greater audience, far easier and considerably quicker than the pre-Internet era.

This increased access has brought up certain issues associated with archeological evidence and

associated information, and how is this information diffused or, better stated, how it should be

diffused and appropriately dispensed in space-time and through proper, legal, ethical, formal

channels and other means of communication. Issues of access to archeological information are

becoming of paramount import and concern, concern particularly acute in cases involving high

profile excavations, where eagerness to quickly access information pertinent to the findings (both

by the public at large as well as by scholars) is relatively elevated. Rush to access information is

as expected creating bottlenecks. It is of course perfectly understandable that such bottlenecks

would appear in such extremely important archeological cases, attracting worldwide attention.

These high profile archeological digs demand possibly a rethinking by the archeological

community about the archeologists’ responsibilities on how to optimally handle the information

to which they have been granted limited exclusive rights. And, of course, how to optimally

manage information bottleneck conditions in the information traffic process.

Examples where the issues of optimal diffusion and dissemination of information to the public

has been brought under question, in the past three years or so, are the high-profile digs at Kasta

Tumulus at Amphipolis, the Orkneys’ Ness of Brodgar, and the Griffin Warrior tomb at Pylos. In

all these three archeological excavations that this author has expressed strong interest in

obtaining raw evidence-data (involving key measurements and architectural drawings) he has

met considerable reluctance and or delays on the part of the archeologists involved to divulge

specific and concrete information on the evidence and particularly some basic measurements

from important evidentiary material. Making available upon request such evidentiary material

would had considerably facilitated analysis not only by this analyst but also by many others not

directly associated with the digs. Such an almost “proprietary” attitude by archeologists on

evidence is a topic that need be addressed by the archeological profession. Some reasonable

rules of conduct in making material available to the public, and through public domain forums on

the Web, must be adopted and be strictly adhere to by archeologists.

Evidence produced through archeological excavations does not “belong” in any sense to anyone

in specific, except to the public that has not only incurred (at least in part) the social costs of the

dig but holds the “legal rights to ownership” of such artifacts and structures, their ultimate

provenance is the Human Heritage. Ownership of archeological finds does not to belong to either

(specific) political entities, agencies or individuals, even to those with fiduciary responsibility over

digs. Of course, credit for their discovery duly belongs to the persons who uncovered the

evidence and the agency (and rather, agencies) that made it possible to uncover and unearth the

evidence. To that specific end, certain archeologists were granted limited duration exclusive right

to dig and reveal the discovered evidence. But analysis of that evidence doesn’t have proprietary

limits, and it should not be either constrained or contained by or contingent upon whatever

temporal limits of monopoly power are granted to the excavators. Exclusive rights to analyzing

archeological evidence should not be granted to the archeologists for any length of time.

65

Archeologists must be compelled to reveal to the public the evidence they uncover (with key

details about it, especially photographs, architectonic drawings and related key measurements –

be them structures or artifacts) under perfect transparency. The public (including the academic

community at large) should not have to wait to access vital information on the evidence, till the

archeologists feel “comfortable” in offering (publicly or in professional conferences, scholarly

publications, and related forums, or through press conferences) their “explanations” and

“descriptions” of what they uncovered. In fact, archeologists are as qualified to “explain” what

they found, as the space probe engineers are in explaining and interpreting the date their

spacecraft obtained by exploring the planet it was launched to explore. The evidence, and the

hard-core data (architectural drawings, accurate and precise as possible measurements on

structures and artifacts, and photographic material) on that evidence must be made available to

the public on the Internet in a very timely manner, possibly very shortly after, if not immediately

at, the time it has been uncovered. Modern Cannons and Code of Conduct must be drafted by

the professional branch of the international archeological community. Archeologists throughout

the World must strictly comply to such conduct. This author has first raised the issue of a socially

desirable and morally proper Code of Conduct by archeologists in his paper on Kasta Tumulus,

see ref. [1.11] pp; 64-68, Appendix 1, “A Note on archeologists’ ethical responsibilities”. In the

context of this Note, and in reference to the Minoan ring which is the subject of this paper, it

must be mentioned that the author attempted directly, and also indirectly through the assistance

of two Facebook friends of his (Luci Philips and Laurie Pierce), to obtain from the archeologists

that excavated the Pylos tomb a measurement as to the ring’s top surface actual length. The

effort came to fruition on December 3, 2017, and the author is now grateful to all involved.

Note 6. The “Genesis” schema of Figure 1.5.1 may depict a phase transition between the era of

a pre-three circle effort to approximating a true elliptical shape, by the pseudo or quasi

ellipsoidal forms they designed and built their monuments, and the era post this genesis event,

an event to which we do not yet know the specific point in space-time that it took place. Certainly,

it must be post Stonehenge Phase 3II (c 2400 BC) and pre-Minoan 1700 BC. The location could be

the Levant, Dynastic Egypt or the Helladic space. Archeological evidence, at present, can’t narrow

down more the search in space-time for this “genesis” event. What is clear, however, is the

recognition that the “genesis” of the effort to produce ellipse-like ellipsoidal schemata marks a

point in time, a boundary, whereby prior to it monuments were constructed by architects that

drew pseudo-ellipsoids employing one or two semicircles (all those shown in section 2 of this

paper employ two). Beyond that boundary, a transition occurred to monuments/artifacts

designed to approximating true ellipses with ellipsoids employing three, four or five circles. This

phase transition must had been associated with the acquisition of what this paper has referred

to as an “intuitive understanding” of a true ellipse, although the Mathematics, Algebra and

Geometry of it were yet not mastered by the mathematicians, architects and artists of the day.

In this context, one is justified to refer to the schema according to which three circles were used

to obtain a first approximation to a true elliptical design as a “proto ellipse”.

66

Two Maps

Map of the archeological site at Pylos

Map M.1. Map of the “Palace of Nestor” archeological site in the Messenia region of the

Peloponnese, Greece. The yellow line depicts an approximately 7.5 miles distance of the site at

North from the modern-day city of Navarino (ancient Pylos, although the main part of the ancient

city was not where the modern city is located at the Southern part of the Bay). Of interest is the

quasi elliptical shape of the Bay at present, although the exact way the Bay looked three and one-

half millennia ago (as well as the blackwaters Lagoon to the North of the Bay) is largely unknown.

For an informative description of the broader (and nodal in the context of the Ionian Sea)

Navarino/Pylos area and its rich history and natural ecology, see ref. [4.4] (in Greek). Since the

early Neolithic, many settlements have occupied this area over the course of the past ten

millennia, in a soup of superposition of cultures over time. Associated with this multiple and

diverse occupancy, significant historical events took place there, from the Mycenaean (Homeric)

period and the thirteen century BC, to the Peloponnesian War of the last third of the 5th century

BC, to the modern times (notably during the 19th century, including the 1827 naval battle of

Navarino, associated with the Greek War of Independence, where British, Russian and French

naval forces fought the combined Ottoman and Egyptian navies). Source of the map: the author,

from the available in the public domain Google Earth map search program.

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Map of Rome’s Imperial Forum

Map M.2. Map of Rome’s Republican (red) and Imperial (black) Forum. An informative

description of the Forum’s Architecture and History is in ref. [4.15[. An introductory citation is

ref. [4.16]. The ellipsoidal-elliptical shape of the Flavian Amphitheater, coupled with the phallic

symbol shape of the Roman Forum under imperium, are testament to the lasting effects over the

long haul, as almost time-constant durable symbols of fertility, virility and strength. Source of

map: ref. [3.17].

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Appendix I. Ellipses, Circles and Cones

The Mathematics of ellipses is as fascinating as it can be. A. Wile’s 1994 proof of Fermat’s last

theorem employs ellipses, see ref. [2.14]. In this Appendix, a number of points will be made on

the subject of topological transformations or geometric equivalences under perspective. These

topics are not explored in any detail here, as this isn’t the appropriate forum to do so. However,

they do present some innovative perspectives on both, ellipses, circles and cones. Under the

point of view of “perspective” the vertex of a cone can be thought of as a point of viewing a circle

from some distance from its plane. This is a condition which might had something to do with the

advent of ellipses in the Late Neolithic and Bronze Age. An ellipse, in mathematical terms, can be

thought of as a topological transformation of a circle drawn on a (2-d) plane E, seen from a

general 3-d perspective, namely at some distance from its center O and from an angle to that

plane E formed by the point of perspective to point O, and a vertical to E line at the circle’s center

O. One can fit any ellipse to any circle (including the “unit circle” where {x^2 + y^2 = 1}, or any

other circle with any radius {including that where r = 1, and where the unit is any arbitrary length

taken as a scaling measure}, as long as the distance to that circle and the angle to the circle’s

plane is chosen appropriately (the topological transformation implied and a simple mathematical

equivalence). This is a subject falling under the mathematical Theory of Cones, see ref. [5.7].

The topic is also related to the mathematical property that has an ellipse being the affine image

of a unit circle, see ref. [5.3] and [5.7]. The two foci of an ellipse can be seen as the split up of a

circle’s center, the base of a cone with vertex (perspective) at some point P, as the point of

perspective P” moves closer to surface E, sliding along the perimeter of a circle (of any radius r’)

on a plane E’ perpendicular to that of E, and with a center at O. That plane contains the

transformed inscribing square of the original ellipse that is now an inscribing rectangle under the

special perspective which has parallel lines meeting at infinity and where the forming ellipse’s

minor axis is the line where the equal rectangles join and the point of perspective (the cone’s

vertex) lies. Looking at a circle from a 45 angle to its plane E, i.e., forming a cone with vertex at

some point P” in which E intersects it forming the circle at E, the resulting ellipse as traced on a

plane E’ perpendicular to that of the perspective’s line of sight to O, and at any distance from

point O, thus producing an ellipse with a major axis that is double that of the minor axis on E’ and

where the inscribing rectangle contains four equal squares.

Circles are 2-d shapes on a plane E that contain symmetry in reference to any axis, as long as this

axis goes through the center of the circle, O. On the other hand, ellipses lose that property, and

their mirror symmetry is restricted in only two axes, their major and minor axes., see Figure 1.1.

Of course, they retain the symmetry property of all points on their perimeter in reference to their

center (origin O). Hence, circles can be thought of as special cases of ellipses. in summary, ellipses

can be transformed into circles and vice versa. A special case (where in this specific perspective

the parallel lines meet at infinity) of the above-mentioned equivalence (or topological

transformation) is shown in Figure 1.2.

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Appendix II. Ellipses’ directrices and director circles

Directrices. From reference [5.2] the drawing of the directrix associated with the right-hand side

focus point of an ellipse is shown in Figure A.II.1. Line L is the directrix at a distance x=a^2/c from

the ellipse’s center, where a is the semimajor axis of the ellipse and c is the ellipse’s linear

eccentricity (the distance of the focal point from the ellipse’s center (origin). It is noted that e is

the ellipse’s eccentricity ratio (c/a). For any point on the ellipse’s perimeter the ratio of its

distance to the directrix over its distance to the ellipse’s focus is equal to e. Hence, in the diagram

below, x=a/e.

Figure A.II.1. The directrix associated with the right-hand side focus point F of an ellipse,

according to ref. [5.2]. An equivalent directrix exists for the left-hand-side focus of the ellipse at

the left-hand side part of the above diagram, as well.

In the paper’s Figure 3.3.3, the two directrices (D1 and D2) are drawn which belong to the true

ellipse that corresponds to the ring’s ellipsoidal form with major axes 2a and 2b. The case of a

point on the ellipsoid’s perimeter, that is located at its upper right-hand side, between the points

where the couple of priestesses’ (at right) hats touch the ellipsoid’s perimeter, was checked as

to whether it fulfills this condition. It was found that it is off by about 5% deviation from the

expected count. A more complete examination of this specific property of ellipses, in so far as

the ring’s ellipsoid is concerned, is an interesting research question and a good suggestion for

further research.

70

Director circles. Next, the elliptical properties of the two director circles associated with an ellipse

are shown in Figure A.II.2 from reference [5.3]. The two circles have centers at the two foci of the

ellipse, and radii equal to 2a (the ellipse’s major axis). The basic property of the ellipse in

reference to the director circles (in the case of Figure A.II.2 it is the director circle belonging to

the left-hand side focus F2) is such that: (Pc2) = (PF1) – namely that the distance of any point on

the ellipse’s perimeter from an ellipse’s focus {in this case (PF1)} is equal to the line segment

drawn from the other focus onto its director line, in this case (Pc2).

Figure A.II.2. The director circle associated with the left-hand side focus of an ellipse. Source

in ref. [5.3], with credit to [5.14].

In the paper, see Figure 3.3.2, an effort was undertaken to check whether this property was met

by a number of points on the ellipsoid’s perimeter, specifically the point at the upper right-hand

side section of the oval shape. This is the point between the critical points where the priestesses’

(at right) hats touch the ring’s oval perimeter. It was found that the deviation in this case was

about 5% from the expected length. Again, a more exhaustive search is needed, and this would

be an excellent computer based research project to undertake and a good suggestion for further

research.

71

Appendix III. Ellipsograph

The ellipsograph is a device that can mechanically draw ellipses. A number of such devices exist

and are based on the Mathematics of ellipses. A special one, and possibly the oldest, is the

trammel – attributed to Proclus Lycaeus (a 5th century AD Platonic philosopher and

mathematician, see ref. [5.10]), but also to Archimedes. For references on a trammel, see [5.8]

and [5.9]. On ellipsographs in general, see [5.11]. There is absolutely no indication to date that

the way to draw an ellipse, or that the formal Mathematics of an ellipse, were known prior to

mathematician Menaechmus (of the 4th century BC), see ref. [5.12]. In ref. [1.12] the author

examined the evolution of the Mathematics associated with ellipses, and the role that

Menaechmus and Apollonius (a 3rd century BC mathematician, see ref. [5.13]) played in that

evolution.

A trammel is shown below, in Figure A.III.1. The reader can easily check that the fundamental

condition for ellipses (that the sum of any point on the ellipse’s perimeter from the two foci is

equal to the length of the major axis) holds.

Figure A.III.1. The diagrammatic representation of an ellipse drawing trammel. The blue part

at center moves along the y-axis, whereas the blue part at right moves along the x-axis. The end

point of the mechanism at right traces the perimeter of an ellipse. Source of diagram: ref. [5.9].

72

Beauty of oval and parabolic shapes on Tara Bank’s face, shoulders and arms

73

References

Author’s work

[1.1] Dimitrios S. Dendrinos, August 3, 2017, “Gobekli Tepe, Tell Qaramel, Tell El-Sultan: why is

Gobekli Tepe a 6th millennium BC site, and evolution in Neolithic Architecture” academia.edu The

paper is found here:

https://www.academia.edu/34125725/Gobekli_Tepe_Tell_Qaramel_Tell_Es-

Sultan_Why_is_Gobekli_Tepe_a_6_th_millennium_BC_site_and_Evolution_of_Early_Neolithic_

Architecture

[1.2] Dimitrios S. Dendrinos, January 21, 2017. “A Primer on Gobekli Tepe”, academia.edu The

paper is found here: https://www.academia.edu/31039270/A_Primer_on_Gobekli_Tepe

[1.3] Dimitrios S. Dendrinos, September 9, 2016, “From Newgrange to Stonehenge: monuments

to a Bull Cult and origins of innovation”, academia.edu The paper is found here:

https://www.academia.edu/28393947/From_Newgrange_to_Stonehenge_Monuments_to_a_B

ull_Cult_and_Origins_of_Innovation

[1.4] Dimitrios S. Dendrinos, March 14, 2017, “On Stonehenge and its moving shadows”,

academia.edu The paper is found here:

https://www.academia.edu/31884455/On_Stonehenge_and_its_Moving_Shadows

[1.5] Dimitrios S. Dendrinos, November 15, 2016, “In the shadows of Carnac’s Le Menec stones:

a Neolithic proto supercomputer”, academia.edu The paper is found here:

https://www.academia.edu/30164088/In_the_Shadows_of_Carnacs_Le_Menec_Stones_A_Ne

olithic_proto_supercomputer

[1.6] Dimitrios S. Dendrinos, 1991, “Methods in Quantum Mechanics and the Socio-Spatial

World”, Journal of Socio-Spatial Dynamics, Vol. 2, No. 2, pp: 81-109.

[1.7] Dimitrios S. Dendrinos, November 11, 2017, Facebook post titled “Miniature Minoan Art

Part 3: a theory of evolution in Minoan miniature Art, and some empirical evidence” at:

https://www.facebook.com/profile.php?id=100006919804554

[1.8] Dimitrios S. Dendrinos, November 11, 2017, Facebook post titled “The metal implement

responsible for the miniature Minoan Art: the bronze needle” at:

https://www.facebook.com/permalink.php?story_fbid=1804357303138246&id=100006919804

554&pnref=story

[1.9] Dimitrios S. Dendrinos, July 14, 2016, “Ostia Antica: the geometry of a mosaic involving a

meander with a rhombus and tiling of the plain – update #1”, academia.edu The paper is here:

74

https://www.academia.edu/26988886/Ostia_Antica_The_geometry_of_a_mosaic_involving_a_

meander_with_a_rhombus_and_tiling_of_the_plane._Update_1

[1.10] https://www.facebook.com/groups/1430150017052267/?ref=bookmarks

[1.11] Dimitrios S. Dendrinos, July 17, 2015, “On the ‘HFAISTION at Kasta Hill’ hypothesis”,

academia.edu The paper is found here:

https://www.academia.edu/14138924/On_the_HFAISTION_at_Kasta_Hill_hypothesis

[1.12] Dimitrios S. Dendrinos, November 15, 2017, “The Earth’s elliptical orbit around the Sun

and the Kasta Tumulus at Amphipolis”, academia.edu The paper is found here:

https://www.academia.edu/35166321/The_Earths_Elliptical_Orbit_around_the_Sun_and_the_

Kasta_Tumulus_at_Amphipolis

[1.13] Dimitrios S. Dendrinos, September 19, 2016, “Dating Gobekli Tepe”, in academia.edu The

paper is found here: https://www.academia.edu/28603175/Dating_Gobekli_Tepe

[1.14] Dimitrios S. Dendrinos, April 22, 2016, “Alexander’s Network of Cities and their Dynamics”,

academia.edu The paper is found here:

https://www.academia.edu/24667299/Alexanders_Network_of_Cities_and_their_Dynamics

[1.15] The analysis is found in the Facebook group “Shadows and Monumental Architecture” with

a June 30, 2017 research post here: https://www.facebook.com/groups/595173407341581/

[1.16] Dimitrios S. Dendrinos, May 14, 2017, “Nefertiti’s bust and Dynamical Geometry”,

academia.edu The paper is found here:

https://www.academia.edu/33039241/Nefertitis_Bust_and_Dynamical_Geometry

[1.17] Dimitrios S. Dendrinos, January 19, 2016, “The Mathematics and Astronomy on

Tutankhamun’s Mask”, academia.edu The paper is found here:

https://www.academia.edu/20392247/The_Mathematics_and_Astronomy_in_Tutankhamuns_

Mask._1st_update

[1.18] Dimitrios S. Dendrinos, November 25, 2016, “Gebekli Tepe: a 6th millennium BC

monument”, academia.edu The paper is found here:

https://www.academia.edu/30163462/Gobekli_Tepe_a_6_th_millennium_BC_monument

75

Work by others

[2.1] https://vimeo.com/187628186

[2.2] Alexander Thom, Archibald Stevenson Thom, 1978, Megalithic Remains in Britain and

Brittany, Clarendon Press, Oxford.

[2.3] http://ancient-greece.org/history/minoan.html

[2.4] https://en.wikipedia.org/wiki/Minoan_civilization

[2.5] http://griffinwarrior.org/griffinwarrior-burial.html

[2.6] http://www.rwaag.org/minoan

[2.7] Ludwig Borchardt, 1896, “Zeitschrift fur agyptische sprache und altertumskunde

(Berlin/Leipzig), Journal of the Egyptian Language and Archeology, Vol. 34, pp: 75-76.

[2.8] http://www.egyptorigins.org/luxorellipse.htm

[2.9] https://archeorient.hypotheses.org/3900

[2.10] A. J. Ammeerman, L. L. Cavalli-Sforza, 1971, “Measuring the rate of Spread of early Farming

in Europe”, Man, Vol. 6, No. 4, pp: 674-88. The Royal Anthropological Institute of Great Britain

and Ireland.

[2.11] http://www.nature.com/news/european-languages-linked-to-migration-from-the-east-

1.16919

[2.12]

http://www.odysseyadventures.ca/articles/malta_temples/maltaTemples03f_hagarQim.htm

[2.13] http://www.nature.com/articles/nature23310

[2.14] http://mathworld.wolfram.com/FermatsLastTheorem.html

References of Figures/Photos

[3.1] By Ag2gaeh - Own work, CC BY-SA 4.0,

https://commons.wikimedia.org/w/index.php?curid=57497218

[3.2] http://www.news.com.au/technology/science/archaeology/pylos-combat-agate-is-a-

3500yearold-masterpiece-of-ancient-art-technology/news-

story/74b886dc0b87dedcf0f4581e0d421137

[3.3] https://www.metmuseum.org/art/collection/search/252397

[3.4] Tunisia-travel-planner.com

76

[3.5] https://archeorient.hypotheses.org/3900

[3.6] AP Art History Study Guide (2013 – 2014 Weber) in studyblue.com

[3.7] http://www.bozar.be/en/activities/106411-gobekli-tepe-neolithic-site-rise-of-civilization

[3.8] https://nexusnewsfeed.com/article/ancient-mysteries/gobekli-tepe-s-vulture-stone/

[3.9] By Bernard Gagnon - Own work, CC BY-SA 3.0,

https://commons.wikimedia.org/w/index.php?curid=10714350

[3.10] http://texnografia.blogspot.com/2017/01/blog-post.html

[3.11] https://blog.education.nationalgeographic.org/2015/10/29/grave-of-griffin-warrior-

uncovered-in-greece/?sf14737939=1

[3.12] http://www.atozkidsstuff.com/egypt.html

[3.13] https://en.wikipedia.org/wiki/Tutankhamun%27s_mask

[3.14] http://etc.usf.edu/clipart/61000/61017/61017_colosseum_gp.htm

[3.15] https://www.pinterest.com/pin/295126581801529722

[3.16] http://www.bible-history.com/archaeology/rome/1-colosseum-bb.html\

[3.17] By Original diagram by Samuel Ball Platner, scan by Felix Just, S.J., Ph.D., alterations by

Mark James Miller - http://catholic-resources.org/AncientRome/, Public Domain,

https://commons.wikimedia.org/w/index.php?curid=4451077

[3.18] http://primaryfacts.com/2018/ancient-egyptian-gods-and-goddesses-horus-facts/

Miscellaneous References

[4.1] https://www.britannica.com/plant/date-palm

[4.2] https://en.wikipedia.org/wiki/List_of_largest_snakes

[4.3] http://homeguides.sfgate.com/fast-pineapple-palms-grow-67219.html

[4.4] http://pylos.info/gr/

[4.5] https://www.revolvy.com/main/index.php?s=Emily%20Vermeule

[4.6] http://www.history.com/topics/ancient-history/colosseum

[4.7] http://whc.unesco.org/en/list/38

[4.8] http://heritagemalta.org/museums-sites/hagar-qim-temples/

77

[4.9] http://almashriq.hiof.no/jordan/900/930/jerash/jerash.html

[4.10] https://plone.unige.ch/art-adr/cases-affaires/aidonia-treasure-2013-greece-and-ward-

gallery

[4.11] http://texnografia.blogspot.com/2017/01/blog-post.html

[4.12] https://dictionaryofarthistorians.org/borchardtl.htm

[4.13]

http://archive1.village.virginia.edu/spw4s/RomanForum/GoogleEarth/AK_GE/AK_HTML/AS-

005.html

[4.14] http://www.the-colosseum.net/architecture/numeri_en.htm

[4.15]

http://penelope.uchicago.edu/Thayer/E/Gazetteer/Places/Europe/Italy/Lazio/Roma/Rome/For

um_Romanum/_Texts/Huelsen*/home.html

[4.16] https://www.britannica.com/topic/Roman-Forum

[4.17] https://www.britannica.com/topic/Eye-of-Horus

References on the Geometry and Astronomy of Ellipses

[5.1] http://demonstrations.wolfram.com/topic.html?topic=Ellipses

[5.2] http://mathworld.wolfram.com/Ellipse.html

[5.3] https://en.wikipedia.org/wiki/Ellipse

[5.3.1] http://www.thefullwiki.org/Ellipse

[5.4] https://www.intmath.com/plane-analytic-geometry/5-ellipse.php

[5.4] https://www.britannica.com/topic/ellipse

[5.5] https://plato.stanford.edu/entries/epistemology-geometry/

[5.6] https://www.mathopenref.com/ellipse.html

[5.7] https://www.encyclopediaofmath.org/index.php/Cone

[5.8] http://library.wolfram.com/infocenter/TechNotes/6413/

[5.9] https://en.wikipedia.org/wiki/Trammel_of_Archimedes

[5.10] https://en.wikipedia.org/wiki/Proclus

[5.11] http://americanhistory.si.edu/collections/object-groups/ellipsographs

78

[5.12] http://www.robertnowlan.com/pdfs/Menaechmus.pdf

[5.13] https://www.britannica.com/biography/Apollonius-of-Perga

[5.14] https://commons.wikimedia.org/wiki/File:Ellipse-def-dc.svg

Acknowledgments

The author wishes to acknowledge the contributions made to his work by all his Facebook friends,

and especially by the members of his current twelve groups the author has created and is

administering. Their posts and comments have inspired him in his research over the past three

years, since he opened his Facebook account. In specific, in writing this paper, the author wishes

to thank Luci Philips (who obtained the exact ring’s measurements from prof. Jack Davis of the

University of Cincinnati) and Laurie Pierce.

But most important and dear to this author has been the 22 years of encouragement and support

he has received from his wife Catherine and their daughters Daphne-Iris and Alexia-Artemis. For

their continuing support, assistance, encouragement and understanding for all those long hours

he allotted doing research, when he could have shared his time with them, this author will always

be deeply appreciative and grateful.

Legal Note on Copyrights

© The author, Dimitrios S. Dendrinos retains full legal copyrights to the contents of this paper.

Diagrams and photos provided in this paper carry their own copyrights found in the sources cited

in the paper. Reproduction in any form, of parts or the whole of this paper’s narrative, is

prohibited without the explicit and written permission and consent from the author, Dimitrios S.

Dendrinos.