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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


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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 an 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.
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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
Dimitrios S. Dendrinos
Professor Emeritus, University of Kansas, Lawrence, Kansas, US
In residence, at Ormond Beach, Florida, US
December 2, 2017
The Minoan gold signet 5-priestess ring from the Mycenaean “Griffin Warrior”
tomb (c 1450 BC) at Pylos, at an approximately four-fold magnification.
Table of Contents
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
Legal Note on Copyrights
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 an 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 is put forward. It 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.
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.
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.
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].
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 has in fact an almost perfect elliptical shape, 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.
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 of approximating 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 a human eye, 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
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.
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 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 a number of days
around (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.
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].
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.
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 they approximated (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.
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 (unscaled) publicly available photos of the five Priestesses Minoan gold
signet ring from the circa 1450 BC Mycenaean “griffin warrior” tomb at Pylos, the Peloponnese,
Greece. The ring’s actual oval top surface is estimated by this author to be about 1.5 inches at
length. At the top, a side view of both the ring’s hoop (band) and the mounted part 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 is at bottom. Source of photo: ref. [3.2], and
provenance of the University of Cincinnati (Department of Classics), see ref. [2.1] and [2.5].
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 is a complex
structure, with its top surface consisting 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.
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. An attempt to answer this core question 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 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.
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, is 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.
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.
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.
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.
Close-up of a female face with oval eyes from a photo (in the public domain); the entire
photo is at the end of the paper, showing the presence of 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.
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.
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 the ancient designers of artifacts attempted to do is 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.
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
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 parabolawas 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].
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. 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 is 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, 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 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 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.
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
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].
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 shapes
that are neighboring to 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
random, especially occurring 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.
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 Norther
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 Phase III.
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, and 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 of a site
on the Euphrates River – both strong indicators and indicative of an older settlement. 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 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.
Gobekli Tepe. The site is a nodal one for Neolithic Architecture, with its megalithic monolithic
components within the stone enclosures of Layer III, and specifically 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] Gobekli Tepe
ushered the 6th millennium Neolithic Era Monumental Architecture. Studying its 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).
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].
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
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. It was found that the manner in which shadows
cast during daytime by the megalithic monoliths of the two monuments were incorporated into
their design proper. The Astronomy of both monuments was examined in detail in ref. [1.4].
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.
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.
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 linked 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
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].
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].
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. Source of the architectural 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.
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].
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 spatio-
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
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.
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 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.
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
somewhere in the 1.5 -1.7 inches range 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 why the author is not using an exact count on the ring’s
length, see Notes 1 and 5. A description of this gold signet ring is 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, by 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. These
differences are noted and highlighted. They are neither unexpected, as analysts in Archeology do
usually differ in what the evidence presents (and how it could be interpreted); nor are they
undesirable, as such differences 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 top 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 to which looms in the background. The five
priestesses in the two sets (2+3) are dressed differently, with their cone in form and the 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.
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.
In fact, her body posturing and head position seem to portray an arrogant and defiant look away
from the rectangular central structure. In addition, one notes that the two priestesses at her
flanks, all three in the left side of the ceremonial scene, must be her assistants (something
equivalent to deacons or apprentices, 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
is iconographic representation depicts an island. This is a second critical point of disagreement
between this author and the archeological team from Cincinnati.
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 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 a 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 depicted 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 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. And now we arrive 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, 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 azimuth. Counting of the number
of raw (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. 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.
Within this context, the meaning of the Door can now be ascertained: it may symbolize the
Entrance into the planning 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 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 (by the
carrier of it and others) from the hand’s back side, i.e., at the back side of the palm’s arch. Placing
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.
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
studies (that is studies where documentation of symbolism is 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.
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; whereas, W is the
minor axis, and the length (2b) in the discussion of Figure 1.1. At the maximum magnification of
the ring that the author worked was such that the major axis (shown in Figure 3.3.1) was about
seven inches (17.8 centimeters), and the minor axis was about four and a quarter inches (10.8
centimeters) for a conversion ratio of 2.54 centimeters per inch. The ratio of these two lengths
r*, where {r* = a/b = L/W} is found to be r*=1.647. It could be noted that this is a ratio somewhat
close to the Golden Ratio (1.6180……) One might speculate that this approximation could serve
as an indication that the Minoans were getting close to that Ratio (on aesthetic grounds), a Ratio
that was elevated to prominence by the Art and Architecture of Classical Greece about a
millennium later. These specifications (the 2a, and 2b axes) uniquely define a true ellipse. Hence
the question is, to what extent the ellipsoid we encounter in the 2-d surface of the Pylos 5-
priestess ring corresponds (or comes close) to this true ellipse.
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. 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, as 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
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 (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, although the
present 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. In any case, the author examined a number of selected points on the
ellipse’s perimeter, 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, 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
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 invalidates 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.
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. As the previous subsection proved, the
maker knew how to draw an ellipsoid that could closely approximate an ellipse. 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,
and since the actual size of the ring is not publicly known, and only inferences can be made on it
from the presentation in ref. [2.1], although it is presumed to be about 1.5 inches in length, 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
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. 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, and most likely 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
To systematically search the question (and test related hypotheses) one need set up the following
research program: 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 in this particular ring’s ellipse is under the 7/9 ratio, or: e = .777… In 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 or linear
eccentricity c of the ellipse. 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-priestes 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. Lengths L (approximately 1.5 inches in actual length) and W
(about .9”) 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].
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 .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].
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. That intersection of 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].
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].
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.
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.
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 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
on 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 address.
As 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) 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
The quest for our understanding of the ancients’ knowledge and their own level of understanding
must be fair to them: misrepresenting them is ultimately 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 by today’s standards, Astronomy, or Mathematics they possessed, as
some archaeo-astronomers would contend. But nor was it the outcome of random acts of carving
motivated by pure emotion and perceived Aesthetics 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. 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, and 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. That ring has 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 possibly commissioned it, certainly guarded and kept it for humanity’s
Finally, and on a different front, it is noted that the analysis carried out in this paper, although
not critically depending on the actual or on an extremely accurate estimate of the ring’s size (the
author roughly estimated the length of the Minoan Griffin Warrior 5-priestess ring to be about
1.5”) it does bring about 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 was
addressed a bit more extensively in Note 5 below, and it draws attention to 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. Above all, 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.
Six Notes
Note 1. The exact size of the ring is not known to this author; only approximate sizes are found
in the published reports cited. The author has (unsuccessfully) attempted to obtain the ring’s
exact sizes from the archeological team. Towards this effort, the author wishes to acknowledge
the contribution of his Facebook friends, Luci Philips and Laurie Pierce. 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. 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].
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 hat 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 the 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.
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 and ethical) means.
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. And it is these high profile archeological digs
that demand possibly a rethinking on 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 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 Archeology 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
(and specific) political entities or individuals. 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
contingent or contained by 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. 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 if not
immediately at the time it has been uncovered. Modern Cannons 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 proved to be futile.
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 where monuments were constructed by architects that drew ellipsoids employing
two circles (as all those shown in section 2 of this paper do), transitions 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”.
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.
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, almost time-constant durable symbols of fertility, virility and strength. Source of map:
ref. [3.17].
Appendix I. Ellipses, Circles and Cones
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.
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.
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
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
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].
Beauty of elliptical and parabolic shapes on a woman’s face, shoulders and arms
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” The
paper is found here:
[1.2] Dimitrios S. Dendrinos, January 21, 2017. “A Primer on Gobekli Tepe”, The
paper is found here:
[1.3] Dimitrios S. Dendrinos, September 9, 2016, “From Newgrange to Stonehenge: monuments
to a Bull Cult and origins of innovation”, The paper is found here:
[1.4] Dimitrios S. Dendrinos, March 14, 2017, “On Stonehenge and its moving shadows”, The paper is found here:
[1.5] Dimitrios S. Dendrinos, November 15, 2016, “In the shadows of Carnac’s Le Menec stones:
a Neolithic proto supercomputer”, The paper is found here:
[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:
[1.8] Dimitrios S. Dendrinos, November 11, 2017, Facebook post titled “The metal implement
responsible for the miniature Minoan Art: the bronze needle” at:
[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”, The paper is here:
[1.11] Dimitrios S. Dendrinos, July 17, 2015, “On the ‘HFAISTION at Kasta Hill’ hypothesis”, The paper is found here:
[1.12] Dimitrios S. Dendrinos, November 15, 2017, The Earth’s elliptical orbit around the Sun
and the Kasta Tumulus at Amphipolis”, The paper is found here:
[1.13] Dimitrios S. Dendrinos, September 19, 2016, “Dating Gobekli Tepe”, in The
paper is found here:
[1.14] Dimitrios S. Dendrinos, April 22, 2016, “Alexander’s Network of Cities and their Dynamics”, The paper is found here:
[1.15] The analysis is found in the Facebook group “Shadows and Monumental Architecture” with
a June 30, 2017 research post here:
[1.16] Dimitrios S. Dendrinos, May 14, 2017, “Nefertiti’s bust and Dynamical Geometry”, The paper is found here:
[1.17] Dimitrios S. Dendrinos, January 19, 2016, “The Mathematics and Astronomy on
Tutankhamun’s Mask”, The paper is found here:
[1.18] Dimitrios S. Dendrinos, November 25, 2016, “Gebekli Tepe: a 6th millennium BC
monument”, The paper is found here:
Work by others
[2.2] Alexander Thom, Archibald Stevenson Thom, 1978, Megalithic Remains in Britain and
Brittany, Clarendon Press, Oxford.
[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.
[2.10] 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.
References of Figures/Photos
[3.1] By Ag2gaeh - Own work, CC BY-SA 4.0,
[3.6] AP Art History Study Guide (2013 – 2014 Weber) in
[3.9] By Bernard Gagnon - Own work, CC BY-SA 3.0,
[3.17] By Original diagram by Samuel Ball Platner, scan by Felix Just, S.J., Ph.D., alterations by
Mark James Miller -, Public Domain,
Miscellaneous References
References on the Geometry and Astronomy of Ellipses
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 and Laurie Pierce in his efforts to obtain information on the ring from Pylos.
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 by the author, Dimitrios S.
Full-text available
A new class of ellipses is discussed in this paper, with the bezel ellipse of the Minoan 5-priestess signet ring from the 1450 BC Pylos’ tomb of the Mycenaean Griffin Warrior acting as the springboard to this new classification. The ellipse carries a strong mathematical interest. The elementary theorems governing ellipses that obey the condition that the ratio of their major to minor axes is at a Golden Ratio are stated, and to that end nine sets of mathematical theorems are proven. Following the mathematical exposition of the Golden Ratio Ellipse, the ring from Pylos is examined as to whether its maker was aware of any of these Mathematics. It is concluded that although, and on purely Aesthetics grounds, the maker gravitated and tried to approximate the making of a true Golden Ratio Ellipse, (s)he was only partially aware of the underlying Mathematics. The findings are based on a detailed analysis of the ring’s iconography. This is an updated version of a previous article by the author under the same title, of December 7, 2017. In this version the formal permission by the department of Classics, University of Cincinnati, to use the ring's image is included
Full-text available
A new class of ellipses is discussed in this paper, with the bezel ellipse of the Minoan 5-priestess signet ring from the 1450 BC Pylos’ tomb of the Mycenaean Griffin Warrior acting as the springboard to this new classification. The ellipse carries a strong mathematical interest. The elementary theorems governing ellipses that obey the condition that the ratio of their major to minor axes is at a Golden Ratio are stated, and to that end nine sets of mathematical theorems are proven. Following the mathematical exposition of the Golden Ratio Ellipse, the ring from Pylos is examined as to whether its maker was aware of any of these Mathematics. It is concluded that although, and on purely Aesthetics grounds, the maker gravitated and tried to approximate the making of a true Golden Ratio Ellipse, (s)he was only partially aware of the underlying Mathematics. The findings are based on a detailed analysis of the ring’s iconography
Full-text available
The paper extensively expands on and revises a prior paper by this author on the subject of the Kasta Tumulus form and design. It advances the propositions that, first, the ground elliptical shape of the Tumulus in combination with the tomb's modulus records the Earth's elliptical orbit around the Sun; second, that the 3-d shape of the Tumulus stands for the Earth's ellipsoidal shape; and third, that either mathematician Menaechmus or Callipus were behind the Mathematics and Astronomy of the monument. The paper is a revised version of the author's February 17, 2016 paper:
Full-text available
The paper is a continuation of a trilogy of papers by the author on the subject of dating Gobekli Tepe. In this paper, two additional sites and some of their monuments are analyzed: Tell Qaramel and Tell Es-Sultan. They provide, along with some more evidence on the c-14 readings and structure B from Gobekli Tepe and with a more detailed examination of the Temple at Nevali Cori additional documentation and evidence as to why Gobekli Tepe is a 6th millennium BC site. In doing so, the author presents a Theory of Evolution of Early Neolithic Architecture. A new view of the burial phases of Gobekli Tepe is presented, that offers the possibility to re-examine the entire construction and burial historiography of the site.
Full-text available
The paper presents an early work by the author, regarding the Tomb at Kasta Hill, Amphipolis, MAKEDONIA, Greece. It was based on three hypotheses: that the tomb was completed at the last quarter of the 4th Century BC, and that the architect was Deinokratis are both assumptions attributed to Katerina Peristeri, the archeologist in charge of the excavation; the third assumption was based on Professor Theodoros Mavrojannis hypothesis that HFAISTION was the (single) person buried there. Further, the paper was based on information prior to a November 29th, 2014 conference given by K. Peristeri, where new evidence was presented (having to do with Roman coins and pottery fragments found in chambers 1 and 2 - although no 3-d specific location for these items was disclosed). As a result of these new revelations, and subsequent announcements about the skeletal bones from a number of individuals found in the funerary chamber, major revisions are needed. thus, this paper serves purely as a means in archiving and recording on-going research by this author.
Full-text available
This is the first update of the paper by the same title and author of July 10th, 2016. Here, some more information is incorporated about the mosaic, and especially his M+ and M- type meanders. This new information is instrumental in further analyzing the micro and macro mosaic design and construction imperfections and their underlying artistic and social conditions which gave rise to them.
Full-text available
This is an updated version of an earlier paper titled "Stonehenge, Durrington Walls, Newgrange: Monuments to the Egyptian Bull and Cow Cults and Origins of Innovation" by the same author. However, in this paper new material has been included. Thus, this paper marginally amends (in view of the summer 2016 retraction of the summer 2015 announcement regarding Durrington Walls) and considerably extends the previous paper. On September 19th, 2016 I revised the view presented in this paper regarding the date of Gobekli Tepe's oldest layer's construction with this paper: A new version of this paper is forthcoming to account for this update.
Full-text available
The paper analyzes the monument of Le Menec at Carnac, in Brittany, France. It advances a number of propositions, key among them being that the strings of stones at Le Menec are not linear parallel alignments but converging arcs. These stones and strings performed a variety of cultural functions. Arcs acted as a Theme Park for celestial objects and their orbits. It is suggested that Le Grand Menec, Kermario, Kerlescan, and Le Petit Menec represented the four seasons. The paper also suggests that the stones' shadows were used as part of sundials. Each stone was used as a computing device, and collectively they constituted a proto Neolithic supercomputer.
Full-text available
The paper analyzes the evidence regarding the dating of the Gobekli Tepe complex. First, it examines the C14 dating information supplied by the archeologist in charge of the Gobekli Tepe excavation, Klaus Schmidt, and a number of others. This is claimed as evidence that Gobekli Tepe is of the at least PPNB period. The evidence they analyzed was obtained from both the fill, as well as from the plaster at the surface of certain Gobekli Tepe structures. The paper also examines the lithic based evidence regarding the fill at the site. Clear evidence that counters these claims is presented is presented in this paper. Although the Gobekli Tepe site can be shown to be of much later construction date than PPPNB, the paper sets as a modest aim to show that the structures at GT so far analyzed are of a later than PPNB date. Evidence covering both C14 dating, as well as architectural, urban design, urban planning, demography and art evidence is offered to back this argument. Extensive use is made of architectural elements from PPNA Natufian settlements, as well as PPNA/B settlements Hallan Cemi and Jerf el-Ahmar.
Full-text available
The paper documents the date for the initial construction phases of Layer III of structures D (middle 6th millennium BC) and structure C (end of 6th millennium BC - beginning of 5th millennium BC) at Gobekli Tepe. It is a sequel to the author's September 19, 2016 paper "Dating Gobekli Tepe". It uses comparative Architecture and Design analysis from Catalhoyuk and Nevali Cori as well as Jerf El Ahmar for the dating process. It also employs Alexander Thom's schema of classifying stone enclosures, by appropriately expanding it and applying it to Gobekli Tepe. The paper also traces linkages between Gobekli Tepe, Carnac, Malta, Stonhenge and Menorca.
Full-text available
This paper is an effort to provide in simple and not technical terms the reasons why the monument currently referred to as "Gobekli Tepe" is a set of structures that were initially constructed in the latter part of the 6th millennium BC and were buried at the start of the Bronze Age, around the middle 3rd millennium BC. The paper briefly shows why the establishment view of a 10th millennium BC construction with an 8th millennium BC burial of the monument is simply erroneous.
Full-text available
The paper analyzes the Stonehenge Phase 3 II Architecture, modular structure, and their connection to the sun-induced cast-off shadows from its sarsens, and the motion of the shadows over the course of a day and throughout the year. It establishes that a direct link exists between the size of certain shadows and the design of the monument. It further documents that besides the summer solstice sunrise alignment, the vernal and autumnal equinox alignment was a major one in the monument's design. Moreover, the paper demonstrates that the Trilithons sarsens ensemble's quasi-elliptical form is a type belonging to an extension of the Alexander Thom's classification of stone enclosures.