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Mathematics of a Golden Ratio Ellipse:

the Minoan 5-priestess gold signet ring from

the Griffin Warrior tomb at Pylos. Update 1

Dimitrios S. Dendrinos

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

In residence, at Ormond Beach, Florida, US

Contact: cbf-jf@earthlink.net

December7, 2017. Updated December 11, 2017

The Golden Ratio. Source of diagram ref. [3.1]

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Table of Contents

Abstract

Introduction

Mathematics of a Golder Ratio Ellipse

The Minoan 5-priestess Ring

Conclusions

References

Acknowledgments

Notice on Legal Copyrights

Figure 1. The Minoan 5-priestess gold signet ring from the c 1450 BC

Mycenaean Griffin Warrior tomb at Pylos, Messenia, Greece. Source: ref. [3.2],

used with permission from the University of Cincinnati, Dept. of Classics.

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Abstract

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.

Introduction

The instigation into the Mathematics of an ellipse that has its major to minor axes at a Golden

Ratio was the discovery that this was very likely the case governing the Minoan 5-priestess signet

gold ring (found in 2015 by the archeologists of the Classics Department from the University of

Cincinnati) inside the tomb of the Mycenaean so-called “Griffin Warrior” at Pylos, in the South-

Western part of the Peloponnese, in Greece, A detailed description of this finding is reported in

ref. [1.1], a follow-up to ref. [1.2] on the topic. Here, the mathematical details are explored of an

ellipse obeying the condition that its major and minor axes are in a Golden Ratio relationship.

This is an interesting mathematical problem, associated with an interesting type of ellipses.

Hence, regardless of whether or not the Minoan artist who made this extraordinary signet ring

by the middle of the 2nd millennium BC, was aware or not, the problem of a Golden Ratio Ellipse

requires analysis on its own right, and this is what is attempted in the first section of the paper.

There, the basic attributes of what is termed as a Golden Ratio Ellipse are given, namely the

algebraic statement of quantities such as: the linear eccentricity, eccentricity ratio, semi-latus

rectum associated with an ellipse’s focus and its complement (the distance of the corresponding

point on the ellipse’s perimeter to the other focus of the ellipse), and the Golden Ratio Ellipse’s

total area. Against this theoretical Mathematics backdrop, the gold signet ring from Pylos is

examined, as to how well it matches these lengths, ratios and area. In other words, to what extent

was the maker aware of the Mathematics of the true ellipse (s)he was trying (quite diligently as

it turns out) to approximate by the ultimately ellipsoid the maker was able to produced. The

quantitative evidence seems to suggest that the creator of this exquisite ring was attracted to

the Golden Ratio and to a true ellipse on Aesthetics grounds, as the maker didn’t seem to had

mastered the mathematical details, except possibly at an elementary level. This is a finding of

some import, as it gauges the level of mathematical knowledge possessed by the Minoans at the

time of the ring’s making. In conjunction with other artifacts discussed in ref. [1.1] and [1.2] by

this author, these findings reported here might be of interest to the historian of Mathematics.

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Mathematics of a Golder Ratio Ellipse

The Golden Ratio Ellipse

In this section of the paper the mathematical basics of a Golden Ratio ellipse are set. The diagram

of an ellipse is shown in Figure 2.1, where the reader can locate all of an ellipse’s basic elements.

A Golden Ratio ellipse (GRE) in particular is an ellipse that, in addition to all other conditions of

an ellipse, also satisfies the condition that its major (2a) and minor (2b) axes (a > b) are such that

(“^1/2” is an expression meaning “square root” in the exposition that follows):

a/b = {1 + 5^1/2}/2 =

where is the Golden Ratio given by the arithmetic expression {(1 + 5^1/2)/2}; is an irrational

number, approximately equal to: 1.6180339887… Two lengths (a, b, where a > b) are said to obey

the Golden Ratio when the following condition holds:

(a + b)/a = a/b. (1)

so that in the case where these quantities (a, b) represent the semi-major and semi-minor axes

of the GRE correspondingly, one axis’ length can be stated as a function of the other:

a = b = b{1 + 5^1/2}/2. (2.1)

Or, equivalently:

b = a/ = 2a/(1 + 5^1/2). (2.2)

It is also recalled that: ^2 – 1 = , see ref. [2.1] and [2.1]. This property will be extensively used

in the formulae and in the proofs of theorems and lemmas that follow.

A fundamental requirement for the formation of an ellipse (by mechanical means, as it is

impossible to do so using only the classical means of a compass and a ruler) is that, see ref. [2.1]

and [2.2], the perimeter of an ellipse consists of all points on a 2-d plane such that the

fundamental condition of ellipses is met, namely that the sum of their distances from the

ellipse’s two foci, points F1 and F2 of Figures 2.1 and Figure 2.2, is a constant and equal to the

ellipse’s major axis (2a). It is also known that any ellipse’s linear eccentricity, that is the distance

of either of the two foci from the ellipse’s center (the “omphalos” of the ellipse, or the origin of

a Cartesian 2-d space) c is given by the expression (easily derived using the Pythagorean Theorem

for the ellipse’s covertex – the point where the minor axis of the ellipse, or where the vertical y-

axis of an orthogonal Cartesian set of coordinates, x and y, intersects the ellipse’s perimeter):

c = {(a + b)(a – b)}^1/2. (3)

For a GRE type ellipse, and on the basis of condition (3), the first set of Theorems can be proven;

namely that, the linear eccentricity c’ is obtained as a function of the minor semi-axis, b, and the

Golden Ratio since (a=b), Theorem 1a:

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c’(b) = b(^2 -1)^1/2 = b^1/2 = b{(1 + 5^2)/2}^1/2, (4.1)

and the corresponding expression casting it in terms of the semi-major axis, a, and the Golden

Ratio (since b=a/) is, Theorem 1b:

c’(a) = a{[1 – 1/(^2)]}^1/2 = a/(^1/2) = a/(1 + 5^1/2}^1/2. (4.2)

Figure 2.1. The fundamental components of any ellipse. Source of diagram: ref. [2.1]

Any ellipse’s eccentricity ratio, e, is given by the expression, see ref. [2.1] and [2.2]:

e = c/a. (5.1)

Hence, and as Theorem 2, one can state that the eccentricity ratio e’ for all GREs is independent

of the GRS’ major and minor lengths, and a constant linked to the Golden Ratio :

e’ = 1/(^1/2). (5.2)

Another basic component of the ellipse (any ellipse) is the length referred to as the semi-latus

rectum, shown in Figure 2.2 and Figure 3.1. It simply is the vertical distance of the focus (in the

cases of Figure 2.1 is focus F1) to the ellipse’s perimeter. The expression of the length is, for the

general case of an ellipse, see [2.2]:

p = (b^2)/a. (6.1)

Hence, in the case of the GRE, the length is:

p’ = 2b/(1 + 5^1/2) = b/. (6.2)

Thus, as a Theorem 3, one can state that in a GRE, the ratio of the ellipse’s semi-minor axis, b, to

the ellipse’s semi-latus rectum, p’, is equal to the Golden Ratio . It follows that, Theorem 4:

ap’ = b^2. (7)

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Figure 2.2. The basic elements of any ellipse, and the semi-latus rectum p. Source of diagram

ref. [2.2] and credit to ref. [3.3].

If by P one designates the distance of the point on the perimeter to the other focus, F1 (at right

in Figure 2.2 and in Figure 3.1) that is a length that can be considered to designate the semi-latus

rectum’s compliment, then from the fundamental condition applicable to any ellipse one has:

P + p = 2a (8)

it follows that for all GREs, Theorem 5, one obtains directly that:

P’ + p’ = 2a = 2b = b(1 + 5^1/2). (9.1)

Substituting from previous relationships, one derives Theorem 6, that for a GRE:

P’ = 2b(3 + 5^1/2)/(1 + 5^1/2), (9.2)

Given that:

a/b = (3 + 5^1/2)/(1 + 5^1/2) = (1 + 5^1/2)/2 = , (10)

since a basic relationship valid at the value of the Golden Ratio holds:

(3 + 5^1/2)/(1 + 5^1/2) = (1 + 5^1/2)/2. (11)

Finally, In general for the case of any ellipse, the area of the ellipse is given by the formula, see

ref. [2.1] and [2.2], where is the irrational number approximately equal to 3.1415926…:

A = ab. (12.1)

In the case of a GRE, the area A’ is such that two irrational numbers are involved when expressed

as a function of either the semi-major or semi-minor axis, deriving the set of two Theorems. In

the case of expressing the area of a GRE as a function of semi-major axis a, one has Theorem 7a:

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A’ = (a^2)/ (12.2)

Or, equivalently, expressing the area as a function of the semi-minor axis b, one has Theorem 7b:

A’ = (b^2). (13)

It is noted that in the above exposition, the Geometry and Algebra of the ellipse were discussed,

and not the ellipse’s vector representation, whereby the ellipse is in general defined as: (x/a)^2

+ (y/b)^2 = 1 in Cartesian 2-d space with its omphalos (center) as the origin. This presentation is

left to the interested reader(s).

The Golden Ratio Cones

A uniquely defined elliptical base right cone with a base being a GRE on a plane E, and a vertex

(or apex) at any point V such that the vertical to the plane Y-axis goes through both the vertex V

and the center (omphalos, or origin in the Cartesian 3-d space) of the GRE, point O, and at a

height H from the base of the cone (or H distance from the GRE), produces at its intersections

with any plane E’, parallel to E, and at any height h, GREs, Theorem 8. The proof is straight

forward and it employs the property of proportions from Geometry.

Lemma 1: The intersections have semi-major/minor axes at a Golden Ratio. In addition, the semi-

major/minor axes of the two GREs at the base (distance H from the apex) and at h are such that:

a(H)/a(h) = H/h = b(H)/b(h). (14)

Lemma 2: This is the only cone that contains all GREs in all of its intersections with planes running

at 90 to its vertical axis going through its apex V and to the base GRE’s origin (center).

In addition to that uniquely defined elliptical base right cone, to be defined as a er-GRC, there is

an oblique circle-based (on some plane E”) non-right-angle cone, with apex V* at distance D*

from E”, such that - stated as Theorem 9 – given a unique plane E* (at distance d* from V*)

intersects the cone and produces the specific GRE. This cone will be designated as cone C* with

radius of the base circle r* and a height H*. The uniqueness condition, that is the unique

specifications giving rise to that cone, are straight forward and left to the interested reader.

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The Minoan 5-priestess Ring

In ref. [1.1] and [1.2] the background to the ring that will be analyzed here are found. Formal

permission to use the image of the ring is found in Note 1. The background will not be repeated

to any extent here. Only absolutely essential elements of that background will be presented and

repeated (or amended and expanded) in this paper. The reader is strongly advised to access at

least ref. [1.1] to obtain a complete picture of the Minoan signet ring under analysis here.

Figure 3.1. The basic components of the true ellipse, the Golden Ratio Ellipse, superimposed on

the 5-priestess signet ring from Pylos. The expected location, R, and length, p’, of the true ellipse’s

semi-latus rectum (F2R) from focus F2, and its complement, P’, of length (F1R) are shown. Source:

the author from a photo in ref. [3.2]. Permission for the use of the ring’s image is in Note 1.

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In Figure 3.1 the basic components of the Golden Ratio Ellipse (argued in this paper as the true

ellipse the maker of the ring intended to approach) on the top surface of the bezel of the circa

middle 2nd millennium BC Minoan 5-priestess ring from Pylos are shown. The lengths a, b, c’, p’

and P’ along with the eccentricity ratio e and the area A of the ellipse are supplied in the paper

next, and are shown in the diagram as expected lines or points. Then, comparisons between the

expected from the analysis and actual quantities are made. The degree to which the maker

proved to be successful in replicating a true GRE is determined, as is the degree to which the

maker was familiar with the fundamentals of an ellipse, the Golden Ratio, and ultimately the GRE.

Given the analysis of the previous section, and the data reported in ref. [1.1] the true ellipse’s

(that is the ellipse with major and minor axes as specified by the actual measures from the ring’s

bezel) quantities are as follows:

Length of a (the semi-major axis of the ring’s GRE) is 4.47 centimeters; b (semi-minor axis of the

ring’s GRE) is 2.77 cm. Their ratio is approximately 1.6137, less than three tenths of one percent

of the Golden Ratio (which is approximately 1.618). Such proximity can’t be random; hence, the

author in ref. [1.1] and [1.2] argued that this is in fact a deliberate attempt by the ring’s maker

to approximate at the microscale of the ring, the Golden Ratio. Deliberate was also the effort by

the ring’s maker to approximate the true shape of an ellipse with its major and minor semi-axes

being of the magnitude just listed. All this is documented in the author’s papers cited. It is now

suggested (as a further expansion of the prior analysis by the author) that the creator of the ring

was aware that this was a special type of an ellipse he was trying to approximate, what is

referred to here as the Golden Ratio Ellipse that enjoys the conditions elaborated in the previous

section of the paper.

With the values of a (2.235 cm), and b (1.385 cm), and using the results conditions (4.1) or (4.2)

of the previous section, one obtains the length of the approximate linear eccentricity c’ = 1.76

cm and hence the expected distance between the two foci F1 and F2 of the GRE is 3.52 cm; from

condition (5.2) the expected eccentricity ratio is e’ = .786; the semi-latus rectum, the expected

distance, that is, of a point R on the ellipse’s perimeter from focus F2 of the GRE and shown by

the length (RB2) in Figure 3.1, from condition (6.2) is p’ = .86 cm; the expected distance from

focus F1, (RF1) and from condition (9.2), is P’ = 3.75 cm. Finally, the total area of the bezel’s

surface (the ellipsoid that closely approximates a Golden Ratio type ellipse) is from the

theoretical condition (13) approximately: A’ = 9.75 cm^2.

All of these expected quantities from the GRE as estimated above, under the assumption this

was in fact a true GRE and supposedly the ring’s maker was trying to approximate, were off by

less than two percent on the average from actual measurements. For example, the largest

deviation was found on P’, which was read as the actual (on the photo of Figure 3.1) value of 3.67

cm versus the theoretically estimated (from the GRE conditions stated in the previous section)

value of 3.75 cm. This represents a difference of .08 cm or 2.1%. It is concluded that this is in fact

the ellipse the maker was trying to approximate with this artifact, albeit largely based on

Aesthetics, although (s)he possessed some elementary knowledge of ellipses.

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Figure 3.2. The Golden Ratio at a particular direction (among a number of ways in which it can

be brought to bear on the ring’s iconography) is shown. The Fibonacci sequence is applied

running towards the upper left-hand side section of the ellipse. Source: the author from a photo

in ref. [3.2]. Permission for the use of the ring’s image is in Note 1.

The conclusion is arrived at because, as the next Figure 3.2 will make clear, the actual Golden

Ratio is not the ratio that the artist had mathematically derived, either computed or roughly (but

analytically) estimated. Otherwise, the artifact’s creator, would had attempted to design the ring

bezel’s iconography around that ratio in more explicit ways. This conclusion is also based on the

locations of the two foci, F1 and F2, relative to the entire iconography. These two points are

found on rather routine points on the iconograph, rather points of some extraordinary

importance in (or meaning for) the entire composition.

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A careful look into the ring’s iconography, shown in Figure 3.2, and the superimposed Golden

Ratio proportions and conditions seem to suggest that the ring’s maker didn’t follow on either

the critical points corresponding to the two foci or the Golden Ratio’s key lines and points as

shown in a specific representation of the Golden Ratio spirals (associated with the Fibonacci

sequence 1, 1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21…) in any discernable manner. Hence,

a fortiori one must conclude that although the artist had an intuitive understanding of the critical

value of both the Golden Ratio and the Golden Ratio Ellipse, on grounds of Aesthetics, the maker

didn’t yet possess full mathematical knowledge to handle either.

It is apparent, on grounds that the author presented in ref. [1.1] and [1.2] that the maker of the

ring wanted to meet the fundamental condition of ellipses, that the sum of the two distances of

any point on their perimeter to the two foci of the ellipse be equal to a constant (and equal to

the ellipse’s major axis), the maker didn’t quite exactly locate the two foci on the ellipsoid’s major

axis. Or, at least, if the ring’s creator had located them, (s)he didn’t recognize their true and full

mathematical import. The maker had just a very strong but intuitive understanding of the issues

surrounding an ellipse, the Golden Ratio, and ultimately their intricate connections through a

GRE, but neither the means or knowledge base to accurately replicate it, only roughly

approximate them.

Conclusions and Suggestions for Further Research

Similar to the Golden Ratio Ellipses (GREs) class of ellipses, other classes of ellipses can also be

analyzed along similar lines the GRE was analyzed in this paper. Such are those ellipses that have

their major to minor axes ratios equal to other irrational numbers of interest, such as

(approximately 3.141592…), that is an ellipse in which the major semi-axis has the length of a

circle’s perimeter, the circle having a radius equal to b; the Silver Ratio Ellipse based on the Silver

Ratio = 1+2^1/2 = 2.4141135…; or the e-Ellipse, an ellipse based of the base of the natural

logarithms, e, approximately equal to 2.718281…

In the paper’s first section, where the formal Mathematics in Algebraic form, were presented, a

number of suggestion have been made on how the analysis can be both completed and extended.

In so far as the work on the Minoan ring is concerned, there are numerous (and obvious)

suggestions, including a computer simulation based analysis of the ellipsoid as presented in the

photo found in Figure 1. Such analysis would not only pinpoint the possible deviations of the

ellipsoid from the true ellipse that it corresponds (with a and b semi-axes), but also it would

provide an exact measurement of the ellipsoid’s area that then can be directly compared with

the expected area from the theoretical model of condition (13).

Although the maker of the Pylos 5-priestes signet ring was very successful in approximating the

very special Golden Ratio Ellipse discussed here, (s)he fell short of fully recording on it the impact

of a fuller understanding of both true ellipses and the Golden Ratio and their combination. To

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that end, nonetheless, the artifact is of extreme interest to the historian of Mathematics (of

course beyond the Art historian). Humanity must be also grateful to the Griffin Warrior who

appreciated its worth and kept it for posterity and humanity’s benefit.

Note 1.

Formal permission to use the image of the 5-priestess gold signet ring from the Pylos’ tomb of

the Mycenaean Griffin Warrior, excavated by the archeologists Professors Jack Davis and Sharon

Stocker of the Classics Department of the University of Cincinnati, was granted to the author on

December 11, 2017. The author is indeed very grateful to the Classics Department of the

University of Cincinnati that holds the copyright to this specific ring’s image (ring #2, excavated

from the tomb), and to Professors J. Davis and S. Stocker. The image of the ring is included in the

paper by Professors Davis and Stocker cited in ref. [2.3], on p. 641, Fig. 10: ring #2 “female figures

flanking a shrine”.

References

Author’s work

[1.1] Dimitrios S. Dendrinos, December 5, 2017, “The Ellipse and Minoan Miniature Art; analysis

of the 5-priestess signet ring from the Mycenaean Griffin Warrior’s tomb at Pylos. Update 1”,

academia.edu The paper is found here:

https://www.academia.edu/35345812/The_Ellipse_and_Minoan_Miature_Art._Analysis_of_th

e_5-priestess_signet_ring_from_the_Mycenaean_Griffin_Wariors_tomb_at_Pylos._Update_1

[1.2] Dimitrios S. Dendrinos, December 2, 2017, “The Ellipse and Minoan Miniature Art; analysis

of the 5-priestess signet ring from the Mycenaean Griffin Warrior’s tomb at Pylos”, academia.edu

The paper is found here:

https://www.academia.edu/35317508/The_Ellipse_and_Minoan_Miniature_Art_Analysis_of_t

he_5-

priestess_signet_ring_from_the_Mycenaean_Griffin_Warriors_tomb_at_Pylos._A_new_version

_of_this_paper_is_out_12_5_17_

Work by others

[2.1] http://mathworld.wolfram.com/Ellipse.html

[2.2] https://en.wikipedia.org/wiki/Ellipse

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[2.3] Jack L. Davis, Sharon R. Stocker, 2016, “The Lord of the Gold Rings: The Griffin Warrior of

Pylos”, Hesperia: The Journal of the American School of Classical Studies at Athens, Vol. 85, No.

4, pp: 627-655.

Credits/Sources for diagrams/photos

[3.1] https://www.pinterest.com/pin/551339179353455048/

[3.2] http://www.news.com.au/technology/science/archaeology/pylos-combat-agate-is-a-

3500yearold-masterpiece-of-ancient-art-technology/news-

story/74b886dc0b87dedcf0f4581e0d421137

[3.3] By Ag2gaeh - Own work, CC BY-SA 4.0,

https://commons.wikimedia.org/w/index.php?curid=57428275

Acknowledgments

The author wishes to thank Professors Jack L. Davis and Sharon R. Stocker of the Classics

Department of the University of Cincinnati, as well as Ms. Carol R. Hershenson, Curator, Classics

Study Collection and Excavation Records for granting him permission to use the image of ring #2

from the Griffin Warrior’s tomb at Pylos.

Further, the author wishes also 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 on Facebook. Their posts and comments have inspired him in his

research over the past three years, since he opened his Facebook account.

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.

Notice on Legal Copyrights

© The author, Dimitrios S. Dendrinos retains full legal copyrights to the contents of this paper.

Diagrams provided in this paper were made by the author on a processed image of the ring’s

bezel top surface. Formal permission for the use of this ring’s image was granted to this author

from the Department of Classics, University of Cincinnati as noted in Note 1 of the paper.

Reproduction in any form, of parts or the whole of this paper’s narrative and Figures, is prohibited

without the explicit and written permission and consent from the author, Dimitrios S. Dendrinos.