Content uploaded by Dimitrios S. Dendrinos

Author content

All content in this area was uploaded by Dimitrios S. Dendrinos on Nov 17, 2017

Content may be subject to copyright.

1

The Earth’s Elliptical Orbit around the Sun and

the Kasta Tumulus at Amphipolis

Dimitrios S. Dendrinos

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

In residence, at Ormond Beach, Florida US

Contact: cbf-jf@earthlink.net

The paper was originally written on February 14, 2016. It was revised on February 17, 2016. It

was expanded on November 15, 2017.

The Earth’s (exaggerated and schematic) elliptical orbit around the Sun.

2

Table of Contents

Preface

Abstract

Introduction

Kasta Tumulus and its almost Circular but indeed Elliptical base

The Earth’s Elliptical Orbit around the Sun

The Kasta Tumulus as an Oblate Ellipsoid

Philolaus the Pythagorean

Menaechmus, Apollonius and the Ellipse

On the Impossibility of a 26.40-meter high Tumulus at Kasta

Concluding Statements

References

Acknowledgments

Note of Legal Copyrights

3

Preface

This paper is the product of adding some material to a prior paper by the author by a similar title,

see reference [1.1]. Additional information is added with the new section on “Menaechmus,

Apollonius and the discovery of the Mathematics of the ellipse. It is of significant import to this

study because it links the Kasta Tumulus’ ellipse to the work by Menaechmus. The Greek

mathematician Menaechmus is credited as not only being the first to formally study the

Geometry of ellipses, but he is also linked to Eudoxus (a key person in the narrative here).

However, and possibly most importantly Menaechmus, historical evidence has recorded, was

associated with both Aristotle and Alexander III. This information, which is drawn largely from

references [1.2] and [1.3], is documented further, and it is added by inserting the new section

after that on Philolaus in the original version of the paper.

In addition, a number of editorial comments and corrections are made to the paper’s version of

February 17, 2016. They reflect developments in the author’s approach to the Architecture of the

monument, as well as a more accurate accounting of the Kasta Tumulus and its tomb’s

historiography in light of new evidence presented in 2015 and 2016.

Menaechmus contribution to solving the Delian problem, see last section of the paper on this

topic. Source of diagram, reference [1.4].

4

Abstract

With this paper a significant discovery and a possible but equally significant preliminary finding

are reported. Close examination of Kasta Tumulus’ exterior morphology reveals that the

Mathematics and Astronomy at the Great Amphipolis Tomb, as imprinted on the Tumulus

exterior wall and its possible overall 3-d shape, are far more advanced and complex than

previously thought. Specifically, a significant discovery is documented, namely that on Kasta

Tumulus’ ground perimeter and exterior wall Earth’s elliptical motion around the Sun is depicted.

The potentially significant finding that Kasta Tumulus’ overall 2-d shape was an oblate ellipsoid,

representing Earth is also advanced. Both of these findings overturn what is currently thought of

about both subjects, namely that Earth’s elliptical orbit about the Sun, as well as Earth’s oblate

ellipsoid shape were intellectual achievements of the late 2nd millennium AD.

In the last two sections of the paper, documentation regarding the intellectual mathematical and

astronomical knowledge and tradition prevailing at the time the Kasta Tumulus was shaped by

its architects and engineers is supplied. Further, it is also documented that behind the elliptical

shape of the Tumulus at its ground level, the near circular but in fact elliptical motion of the Earth

around the Sun is recorded. This motion is depicted by the Tumulus’ intended perimeter length

(about 497 meters) and the suggested by this author modulus employed by its architect(s) in its

construction (1.36 meters). The paper traces the intellectual base of the Mathematics of ellipses

and the Earth’s Astronomy all the way back to the philosopher, astronomer and mathematician

Philolaus the Pythagorean and mathematician Callippus, as well as to the mathematician

Menaechmus. The later, historical evidence ties to both Aristotle and Alexander III, with

Alexander being the personality behind the funding and order to build this Tumulus and tomb in

it, to be used as the funerary temple and resting place for his deputy commander Hephaestion.

Introduction

In a set of previous papers by the author, see reference [1.5] for a summary of the theory and

the references to prior work by this author, the form and nature of Kasta Tumulus’ perimeter

wall as intended to be by the tomb and Tumulus designers, architects and engineers (although

probably never finished as designed, and having undergone major phases in its construction and

a variety of transitions till uncovered in August 2014, all described in detail through the prior

publications by the author) was extensively discussed and analyzed. It was suggested that, what

it initially appeared to be an almost circular outline of a perimeter wall of 497 meters in length,

identified a calendar. Based on the length of the monument’s modulus (a 3-d modulus with 1.36

meters in length, identified by this author and extensively elaborated in the prior publications

cited and summarized in ref. [1.5]), the exterior perimeter wall at Kasta Tumulus depicted an

approximation to the exact number of days in a calendar year (365.44 versus the actual 365.25).

5

Although most likely the architect of the monument knew the 365.25 count, and that was what

he may have intended to plant on Kasta’s Tumulus exterior wall on by its perimeter, either due

to construction imperfections and/or inaccurate measurements at present (that result in the

365.44 approximation) prevent us from quite likely obtaining this exact count (365.25).

Moreover, it was suggested and indicated in that series of papers cited in [1.5] that quite likely

the mathematician and astronomer Callippus from Cyzicus was the person behind the

mathematics of the wall. Further, it was suggested that the Tumulus at Kasta, Amphipolis, was

well represented as the planar intersection with a sphere, at an angle to the vertical axis equal

to the (precisely yet unknown but slight) ground level angle, that is the hill’s North-South slope.

That sphere, it was further argued, represented the shape of the Earth as a solid.

On January 29th, 2016 the head of the archeological team, Mrs. K. Peristeri, made a presentation

at the University of Cyprus, see ref. [1.6], where some new evidence was presented in the form

of a site plan of the Tumulus at Kasta, together with three cross sections of it. Along with this

new evidence, some interesting additional commentary was added, regarding the monument

and the Tumulus. For example, it was claimed that in its final form, upon the conversion of the

natural hill into a Tumulus by the architect of the monument (argued by the archeological team

in charge of the excavation and at the beginning of the dig as being Deinokratis, although in

subsequent communications this possibility has not been either pursued further or substantiated

by the team) the mound reached a (revised from 30 meters) height of 26.40 meters, on top of

which a 15.80 meters or so Lion of Amphipolis statue (that included some yet unspecified in

form, but estimated to have reached about 10.56 meters in height base) was installed (seen at

the 17-minute point into the lecture in [1.6]). These comments about the monument remained

just at the level of “commentary” however, as only vague references to some “monument’s

Geometry” was afforded and dispensed by Mrs. Peristeri.

No solid proof was presented documenting that either the mound attained that height, or that

the Lion was meant for and was actually ever installed at the Tumulus’ summit. In addition, a

specific base for the Lion has yet to been formally shown by the archeological team. It has been

suggested by the architect of that team that the current base of the Amphipolis Lion and its

current location do not represent the actual Lion’s base or location where the Lion was both

designed for and intended to occupy. However, these comments by the archeological team will

not be addressed here in detail, as the focus of this paper is different. But it is of course

acknowledged that the essence of these comments touches in substance central issues

presented here. To the extent that these central issues are concerned, the comments will be

peripherally discussed. The issue of the Amphipolis Lion has been discussed in ref. [1.5] and [1.7].

It is noted, that the firm conviction of this author has been that the Amphipolis Lion statue was

neither intended to be located at the top of the Tumulus, nor was it ever located there in a finish

format. The author, in the series of papers (since October 2014) he has published on this subject,

has argued that the Lion was both structurally and aesthetically too heavy for such a location. To

conclude this discourse, the author has since published a paper on a book written by the

6

archeologist in charge of renovating the Lion of Amphipolis monument in the 1930s, Oscar

Broneer, see reference [1.7] in which ample evidence is presented to indicate that the current

location of this monument was the Kasta Tumulus architect(s) intended location (where actual

evidence from its intended base were unearthed by Broneer). The paper also provides enough

evidence to suggest (found in O. Broneer’s writing) that the Amphipolis Lion most likely was

never assembled in antiquity. Be that as it may, the suppositions by the archeological team,

regarding the Tumulus’ total actual height at the time of the Tumulus construction (c 323 – 320

BC) and the location of the Lion of Amphipolis on top of the mound, would have a Tumulus of

about 30 meters in height (with a 15-meter high Lion on top). In prior presentations, the

archeological team has also suggested a 24-meter high Tumulus, see the papers cited in [1.5] for

relevant citations. This total Tumulus height estimate was updated to a 26.40-meter high mound

with the Peristeri presentation of January 29, 2016. With the 15.80-meter high Lion of

Amphipolis on top, this would have meant that the total height of the tumulus (42.20 meters)

would had reached about half the total height of the nearby Hill 133 (which has at present a

height of about 100 meters), Figure 1.

Figure 1. A Google Earth map showing the round mound of the Kasta Tumulus to the center-

right, and the square base of the pyramid-like (frustum structure) Hill 133 at center left. The

archeological team suggests that the total height of the Kasta Tumulus plus the Lion of

Amphipolis sculpture on top would reach about half the height of the Hill 133. Such a proposition

is unsustainable on aesthetic grounds alone. Source: the author, see ref. [2.1].

7

Kasta Tumulus and its almost Circular but indeed Elliptical Base

Our main focus here, however, is the Kasta Tumulus’ site plan. A key feature of this site plan

(presented on January 29, 2016, see Figure 2) is the shape of the mound, in the form of a floor

plan. For the first time, the public was presented with what was suggested by that team to be

the exact shape of the Tumulus’ base, and this is the main focus here. Along with this site map,

three cross sections of the mound were also presented (see the author’s January 30, 2016

Facebook post about these cross sections, reference [1.8]. The Tumulus’ cross-sections raise

some questions regarding their accuracy in depicting the limestone support structure, the

marble blocks cover, and most of all the slope of the hill’s surface above them. However, these

concerns will also not be addressed here, as key components of these sections were not clearly

and in detail shown during the presentation. Thus, in Figure 2, only the mound’s floor plan is

presented and discussed. This floor plan is the main (albeit not exclusive) subject of the analysis.

In fact, it is the shape of this site plan which stimulated this work and underlies the main and

most significant discovery of this study.

Figure 2. The archeological team’s site map of the Tumulus at Kasta. Although it looks

approximately round, it is in fact slightly elliptical, elongated along the approximately North-East

to South-West axis, and with an approximate azimuth of 15 to 195. North is straight up. Source:

reference [1.6].

It must be remarked that the archeological team’s proposed overall 3-d shape of the Tumulus,

through the three drawn cross sections found in ref. [1.6] and discussed in Facebook research

posts of ref. [1.8] touches on a potentially significant finding of this study. It suffices to say that

these cross sections’ overall shape (that lay the ground for the 26.40 and 42.20-meter high claims

8

by the archeological team) find the author in disagreement, and the reasons why are explained

in previous papers, all cited in [1.2], and [1.7]. Further arguments are supplied in the section of

the paper that follows this paper’s last section.

The author has argued in the past, see ref. [1.5] that the shape of the mound as seen above

ground, if visualized as extended below ground it would represent in 3-d a complete spherical

solid. In effect, it was argued that the visible Tumulus morphology was intended to stand as a

section of a sphere cut by the ground’s plane. That complete sphere was interpreted as

representing the Earth, possibly the visible above ground section of the sphere as symbolically

representing the living, and the section below ground representing the dead. This argument will

be qualified here. It will be argued now that what the architect wanted to create at Kasta

Tumulus was a far more sophisticated view, in its Mathematics: a particular ellipsoid that

approximates a sphere. In effect, a hint is supplied that the architects and astronomers of that

era may have speculated not only that the Earth is orbiting the Sun in an elliptical manner, but

also that the Earth’s shape was an ellipsoid as well. What we were to observe then above ground

was just the upper part of the intersection of that ellipsoid by the ground’s plane. This

proposition constitutes the study’s potential significant finding. Why the author refers to this

proposition as a “potential significant finding” and not as a “significant discovery” will be

explained later in the paper.

However, the evidence supplied by the archeological team itself, prior to January 29th, 2016 as

well as during the January 29th 2016 lecture by K. Peristeri, very likely points to a significant

discovery. This discovery is associated with the shape of the Tumulus’ site plan at its ground level.

Two pieces of evidence are needed for this discovery to be established, both supplied by the

archeological team. Namely that remnants of a single but wide enough hole in the ground at the

top of the Tumulus has been located by the archeological team, likely used to draw the outline

and guide the forming of the Tumulus’ final shape. That piece of evidence pointed initially to the

conclusion by this author that it was used to insert there a pole to draw a perfect circle at the

base of the hill’s original ground. The archeological team’s finding was the strongest evidence

used by this author to initially conclude that a circle was intended to be drawn by the architect.

An ellipse would have required two foci to draw it, based on the elementary Geometry

associated with the drawing of an ellipse (by utilizing the property that any point at the ellipse’s

perimeter has the sum of its distances from the two foci equal to a constant). There are

numerous other ways to drawing an ellipse of course, but the mechanics involved make these

methods infeasible in this case.

When the site plan was produced, see Figure 2, and it showed a slightly deformed circle (and

possibly an ellipse) at the mound’s base, the possibility presented itself that in fact an ellipse was

to be drawn by the mound’s perimeter wall (or, more precisely, of the Tumulus’ perimeter,

whether walled or not at sections – an important issue to analyze but not for this paper to

address, see references in [1.5] for more on this specific issue) after all. Clearly, that ellipse was

intentionally meant to approximate a circle. Immediately then, this realization begged the

9

question “why?” Why would an architect want to produce a tumulus in the form of an ellipse at

its ground base which would look at first glance like a circle? Why would the architect embed

into the Tumulus design such an ambivalence? In conjunction with the fact that three cross

sections of the Tumulus (as least as drawn and presented by the archeological team) seem to

suggest elliptical shapes as well that resemble (but do not look exactly as) arcs of a circle (cross

sections that is of a sphere), the set of questions posed acquires further potency.

See the Google Earth map of Figure 4 for the shape of the mound from above ground. This isn’t

a clear-cut shape, it’s an architectonically ambiguous shape. To draw such perplexing shape the

architect must have had some very strong reason(s), to counter the produced ambiguity.

Moreover, this single-hole, circle producing process, was also a point of concern. The mechanics

and Geometry involved pointed rather strongly towards an effort to draw a circle. Why and how

would an elliptical (but in actuality looking as if it is almost cyclical) shape be the product of this

ambiguous (and as it proved to be very ambitious) construction process?

In trying to address these design questions, one must contemplate the motivating factors, as well

as ponder the practical solutions sought after by the architect to derive and explain the

occurrence of that site plan shown on January 29th, 2016. It turns out, there are deep seated

reasons on that Hill as to why this was the form chosen by the architect of that site plan, and the

mathematicians and astronomers behind is. This Tumulus is not a run of the mill Tumulus, as it

potentially hides within it significant knowledge and understanding of Mathematics and

Astronomy by the Alexander III architects, engineers, mathematicians and astronomers. In

search for these reasons, and in arriving at the solutions offered by the site plan in front of us,

we will not only suggest the underlying rationale architect (maybe Deinokratis, but also maybe

architect Parmenion, on Alexander III staff during his Asian/African campaign - this author has

explored that possibility in the research post of April 22, 2016 in the Facebook group he

administers on subjects related to Kasta Tumulus and its tomb in ref. [1.9]] may have had in mind

to draw this apparent (albeit circle resembling) ellipse. But we will also slightly modify and extend

our prior suggestions to accommodate the now obviously slightly more complex, albeit far more

telling elliptical shape of the Tumulus floor plan. And in doing so, we will come to the realization

that the Mathematics and Astronomy of the Greeks of that Era were far more advanced than

what we currently think they were. The reader is reminded that this monument was intended

for Hephaestion, the deputy commander of Alexander III, the closest friend of the most powerful

person in Eurasia then.

As drawn, see Figure 2, the site plan of the Tumulus seems to convincingly indicate that, no

matter how close to a circle, in fact an ellipse was produced at the ground level of the Tumulus

by intent and not by any design flaw or construction failure. Careful analysis of this site plan

reveals that at approximately an azimuth of a 15-degree angle to the right (East) of the North-

South axis of the drawing in Figure 2, the long “major axis” of an ellipse is found. That angle of

course is related to the tomb’s orientation (today but not back then, 23 centuries ago, a topic

addressed in the various references in [1.5] but will not be further elaborated on here). Hence,

10

this study is an effort to first propose a reason why would the architect design such a floor plan,

with this slightly elongated form; and second, how this ellipse, which resembles very close a

circle, could have been drawn by the use of a single hole, but not necessarily a single pole. And

by doing so, we come to a significant discovery and arrive at a potentially significant finding.

Before we do that, however, two central questions will be posed: first, is this difference in the

mound’s floor plan (being elliptical rather than circular) perceptible at all? And second, by

whom? In answering these “primordial” type questions, that is questions which can always be

asked in regards to any monument in which the architect has designed and implanted subtle

elements of form, one could take two distinctly different perspectives. One perspective is to look

at those differences from the ground level; namely, what difference would it really make to

someone at the ground if the floor plan were to be a perfect circle rather than a (closely

resembling a circle) ellipse? The other perspective is to ask the same question if one were able

to obtain a bird’ eye view of the mound. Or, put it in more exact terms, what if the intended

observer by the architect was the view from above (supposedly, the domain of the gods).

By taking these two perspectives, the student of the Tumulus floor plan would immediately ask:

what was the architect’s objective in designing a mound with a base so close to a circle, yet an

ellipse in actuality? What was it that the architect was trying to accomplish by installing such a

subtle difference in the floor plan? As Geology was not the key factor here, after all the

conversion of a natural hill into a tumulus was an attempt to take care of geological anomalies,

the answer must be found in purely design factors, that is complex architectonic considerations.

In effect this subtle difference (stating it in different terms, the implied Mathematics and

Astronomy of it all) was done not for any functional but purely ceremonial, aesthetic and as a

repository of the stock of knowledge possessed back then purposes.

Looking at the mound from the ground level, the slight difference between a perfect circle and

an ellipse would not be even discernible. The monument is so big, with a perimeter almost half

a kilometer long, hence from any ground level perspective it would show (depending on the

distance of the observer from it) at most half of its diameter’s length, i.e., about 249 meters. The

difference between a perfect circle and a geometric form which resembles “almost a circle” at

its base would simply be unnoticeable. The ground perspective would simply be overwhelming,

on such a curved surface. In both form and function, the differences are simply too small to be

detectable. Of course, virtually unnoticed subtleties on the ground, further point to pure design

considerations, with aesthetics and Mathematics cum Astronomy lurking behind this

approximation. Before we turn to these analytical considerations, the bird’s eye view

perspective needs some more elaboration.

This second perspective, the bird’s eye view, is where we find the answer. It is noted that Greeks

did design their Temples with a view to and with a compass the Celestial sphere. They were of

course not the only ones, as that was the case since the Neolithic with all monuments built by

humans, in all regions of the World. Of course, the Greeks of the Classical and Hellenistic periods

11

did not use architectural monuments to observe the Celestial Sphere and make recordings. They

had advanced from that stage of humanity’s gathering knowledge and information about the

motions of celestial bodies. They used their monuments as depositories of knowledge already in

their possession. And this is what the Great Tumulus at Kasta close to Amphipolis is all about.

It’s along these lines that the answer must be sought, in the Astronomy embedded in the Greek

monuments of that time. In fact, this is where the answer lies, looking for the reasons behind the

extreme proximity of the monument’s elliptical base to a circle. Maybe, one might think that the

proximity of the two planar shapes (circle and a circle-looking-like ellipse) is a superficial

resemblance, or even a coincidence, construction imperfection, or a random event – at a first

glance. The author contends that this particular shape of an ellipse is not haphazardly derived or

by chance drawn, and perfunctorily constructed. Its resemblance to a circle is by no means

unintentional. Something quite profound is embedded in it.

One might interject that this issue (the difference between a circle and an ellipse that almost

resembles a circle) might be thought of as an obscure and insignificant matter of architectonic

minutia. Even if these minute differences are detectable at all, one might think that the actual

choice of an ellipse of this type was simply a matter of Architecture and Design (offering added

ground level perspective, enhanced by the Tumulus curvature). But it isn’t so, and to understand

why we need to reflect on the time this monument was built. We are at a point in time where a

profound transition is underway in the World of Astronomy and Mathematics. Kasta Tumulus is

constructed during a critical period in the History of Humanity, where indeed dramatic events

are underway.

The Earth’s Elliptical Orbit around the Sun

Slight differences in architectonic form are not always that insignificant. This difference

embedded in the Kasta Tumulus ground floor map might look small in size and lost in scale, but

it is in fact enormous in meaning and implications. It is something quite deeply reflective of

changes underway in the stock of knowledge in Mathematics and Astronomy of that Era in

Greece. For at that time in Greece a paradigm shift is about to occur. Two fundamentally

different astronomical systems are involved in this transition. Beyond simply an approximation

of two planar shapes (a circle and an ellipse that resembles a circle), and by the difference

between two solids (a sphere, and a sphere- looking ellipsoid) the shift in paradigms is recorded.

For underneath these slight differences in the two planar shapes and the two solids, two

significantly different astronomical systems hide. On the one hand, one finds the geocentric

system, with circles and spheres in it. On the other hand, one accounts for the heliocentric

system, where circles but also ellipses that are very close to circles are found.

12

The monument at Kasta was built at the cusp of astronomical work, when geocentric Callippus

of Cyzicus (370 – 300 BC) was about to be succeeded by heliocentric Aristarchus of Samos (310

– 230 BC). It turns out that on that exterior wall of Kasta Tumulus and its perimeter, we find a

resolution of that transition and possibly the first recording of the new paradigm in monumental

Architecture. This may constitute the major contribution, by far, the architect at Amphipolis’

Tumulus and Tomb implanted on its perimeter and, through it, endowed the monument.

In our detective’s search to explain the closeness in the shape of the Kasta Tumulus’ site plan

between a circle and an ellipse, we need to locate and identify the mathematician and

astronomer behind it. But in that search, we may also have to revise (if necessary) the

construction date of the initial Phase in the building of the tomb and forming of that monument’s

Tumulus. By answering the question as to who was behind this construction, we may also find a

more precise date as to when both, the Tumulus was made out of a pre-existing hill, and the

Tumulus perimeter was covered by marble blocks (to an extent) offering the exterior wall we

now observe, and thus when what this author has identified as Phase II of the monument did

actually take place, and the reader again is directed to the author’s papers in ref. [1.5] on that.

Pursuing an answer to the detective’s story, in the February 17, 2016 version of this paper, the

author made the following observation, and posed a research question:

“Alternatively, we may find that some, not yet identified, astronomer and mathematician existed

in the last quarter of the 4th century BC who was a hybrid between Callippus and Aristarchus (and

known to Aristotle), and that he worked with Deinokratis {or Parmenion now} at Kasta Hill. For

the 365.25 (or the 365.44 approximation) was in the grasp of Callippus. Although History informs

us that the circular orbit of the Earth around the Sun was an element of Aristarchus’s heliocentric

system, the elliptical nature of Earth’s orbit was not, we are told, in the ancients’ radar screen

yet. And neither was Earth’s ellipsoid shape. Yet the magnificent Tumulus at Kasta and its

magnificent wall strongly tempt us to revisit these two assumptions”.

In this version of the paper, indeed the sought-after mathematician alluded to in the previous

passage is found: it is mathematician Menaechmus. The life and work of this extraordinary

individual will be very briefly presented in the new added section of the paper, after the short

discussion on Philolaus the Pythagorean. However, before the paper turns into locating the

pedigree of the ellipse-related arguments, attention shifts to the central matter of this paper,

which is the elliptical orbit of the Earth around the Sun, and how was this orbital motion

implanted into the Kasta Tumulus ground perimeter.

At the outset it must be noted that all variables offered here about the movement of the Earth

around the Sun are approximate and ever changing over time scales involving centuries and

millennia. The basic components of this orbital motion are found (as an introduction to a very

elaborate and complex astronomically topic) for example in [1.10], from where the preamble

photo of the paper has been taken. It’s recalled that an ellipse is drawn by points with the sum

of their distances from the ellipse’s two foci remaining equal to a constant. Again, for an

13

introduction to this geometric subject (which also is elaborate and complex in the field of

Modern Mathematics) see reference [1.11]. However, before we discuss the Earth’s elliptical

motion around the Sun, the basic equation describing the Geometry of an ellipse is offered.

Expressed in a Cartesian coordinate system, x and y, the algebraic equation of an ellipse is given

by the algebraic expression:

(x^2/a^2 + y^2/b^2) = 1,

where a and b are its major and minor semi-axes correspondingly, when the ellipse is plotted

with its center (the point of the two axes’ intersection) as the origin.

For the purposes of this paper, which examines the Earth’s motion around the Sun, two key

elements of that elliptical orbit are: the approximate distance of the Earth from the Sun (in

around 2000 AD) during perihelion (or periapsis) of about 147 million kilometers (88.2 million

miles). The corresponding distance during aphelion (or apoapsis) is about 152 million kilometers

(91.2 million miles). The reader is referred to the schematic drawing of that elliptical movement,

shown as the preamble Figure of this paper.

Given that the Earth orbits the Sun in an elliptical fashion, where one of the foci is the center of

the Sun itself (the Sun has a diameter of about 523K miles; whereas that of the Earth, itself an

ellipsoid as we shall discuss later but resembling a sphere, is about 4,100 miles, and both radii

are not significant enough to alter the results of this study) one can easily compute the orbit’s

linear eccentricity.

The Earth’s major axis of motion (what we shall be referring to here for simplicity as the “long”

or the major axis of the ellipse) is thus approximately 299 (147+152) million kilometers (178.4

million miles). Consequently, the center of the elliptical orbit, that is the point of intersection of

the long axis with the “short” (or minor) axis of the ellipse, is at about a distance of 149.5 million

kilometers (about 89.7 million miles) from the Earth, when the Earth is at either aphelion or

perihelion. Thus, elliptical linear eccentricity (the distance of either elliptical foci from the orbit’s

elliptical center) is about 1.5 million miles (or about 2.5 million kilometers). Further, simple

“eccentricity” of the elliptical orbit is the ratio of the two foci’s distance over the length of the

long (major) axis; this is about 5 million kilometers over 299 million kilometers. It corresponds to

about 1.67%. A linear eccentricity of 2.5 million kilometers corresponds to about .837% of the

long (major) elliptical axis (half of the simple eccentricity measure).

At the scale of our analysis, regarding the Kasta Tumulus’ approximate “diameter” of about 158

meters (to be considered here as the ellipse’s “long” or major axis), this linear eccentricity would

correspond to about 1.32 meters (to be designated as f). Thus, if one were to precisely replicate

an elliptical motion equivalent to that of the Earth’s around the Sun at the perimeter (walled or

not) at Kasta Tumulus, one would need to draw an ellipse with a distance of its two foci on the

long (major) axis close to 2.64 meters. The total length of an ellipse is a quite mathematically

involved function, where integrals are included along with the length of the long (major) axis and

14

the ellipse’s eccentricity. The computation is left to the interested reader, as is the comparison

of that ellipse’s perimeter length measure to the 497 meters in length of the Kasta Tumulus’

perimeter (all of it walled or not, walled as announced by the archeological team, partially so

argued by this author).

The reader must note that any other (greater) linear eccentricity would increase the Kasta

Tumulus’ perimeter length. Thus, this count of 2.64 meters acts like a ceiling. Computing the

Tumulus’ “short” or minor axis, b, with a being 158.3m (taking the perimeter length’s given by

the archeological team, 497m, as an accurate and exact measure), and 2.64m as the ellipse’s

eccentricity, one by applying the Pythagorean Theorem obtains an estimate of the Tumulus’

“short” axis.

The formula is:

(b/2)^2 = (a/2)^2 – (f/2)^2.

This equation results in an estimate for b (79m) with semi-axis a being 79.2m. The difference is

less pronounced than that shown by the graph containing the Tumulus’ site plan, and far closer

to the Google map photo of Figure 4.

A final note on the count of 1.32 meters, the linear eccentricity’s length: it’s quite close to the

modulus of the monument (1.36 meters) and the basis of the 365.44 number.

To close this part of the analysis, one now can mention that the two and a half meters or so wide

hole, which the archeological team uncovered on top of the mound, can explain the drawing of

this almost circular ellipse by sticking two poles at that distance to form the ellipse at the ground

level. And with this piece of evidence we conclude the mechanics and analytics of the elliptical

design. The fact that we now have evidence, in support of the architect having drawn the ellipse

as defined, constitutes the significant discovery of this study.

Was the objective of the monument’s architect to replicate the Earth’s elliptical motion about

the Sun in the perimeter of the Kasta Tumulus? The question is now answered in the affirmative.

For this author, this is precisely what the architect of the monument at Kasta wanted to

accomplish. There’s simply no other credible reason to suggest Deinokratis (Parmenion, or any

other architect) would opt for such an otherwise ambiguous solution (an ellipse that is almost a

circle at its ground plan).

We now turn to the more central question: who could be behind this closely, resembling a circle,

ellipse? Apparently, astronomer and mathematician Callippus of Cyzicus either had figured this

out late in his life (and he left no records to so indicate); or a student of his and also

knowledgeable of the so-called “Aristarchus system” was present at Kasta at around 323 BC,

decades before Aristarchus work. It is known since the early part of the 20th Century, see for

example the work by Thomas Heath [1.12], that some, among the Greek mathematicians and

astronomers of the 5th and 4th Century BC, were aware of the fact that the Earth wasn’t the

15

center of the Universe – way before Aristarchus formalized it. Moreover, some may have even

speculated that the motion of the Earth around a central core (they referred to it as “fire”) wasn’t

even perfectly circular. In subsequent section of this paper, more information is offered along

these lines. In specific, the work of Philolaus the Pythagorean is discussed, as a potential

intellectual forebear of Earth’s elliptical motion.

Alternatively, one must consider that the Kasta Tumulus (and thus the parts of the exterior wall

that are covered by marble blocks - hence the whole of Phase II) was built much later than we

have thus far considered (323 BC). In such a case, the monument then would definitely not had

been constructed for Hephaestion. However, this alternative would necessitate the

abandonment and overthrow of basic archeological and historical evidence, a parting that this

author is unwilling to undertake at this point. What is for certain, given the evidence on the wall

in that part of the Tumulus perimeter that are standing covered by marble blocks, is the fact that

somehow the architect (be that Deinokratis, Parmenion or someone else) was exposed to these

astronomical developments. The architect felt likely quite strongly about them, and wanted to

immortalize the occasion by imprinting them on the tomb’s exterior wall and the perimeter of

the Tumulus. It was a unique opportunity in the History of Science, as it was a unique opportunity

to build a monument of that size to commemorate a very significant personality at the time that

had just passed away: Hephaestion, Alexander III deputy commander and close personal friend.

By doing so, and implanting on the exterior wall of the Great Tomb at Amphipolis what back then

was a major scientific innovation, the estimation of the exact number of days in a year, and the

elliptical orbit by the Earth around the Sun, (be that Deinokratis or anyone else) designed a

monument for the Ages. Its constituent elements were not the Art by which the monument

(Tumulus and tomb) were furnished with, but instead the Science and Mathematics imprinted

on it. Still, as impressive as all that might be, it was not all one reads at the perimeter, the exterior

wall coverage of Amphipolis’ Great Tomb, and the overall shape of the Kasta Tumulus. More

pioneer Science is still potentially to be found on this Hill, a mound most likely dedicated to the

then contemporaneous Mathematics and Astronomy, however potentially lost by human action,

benevolent and malevolent, that has been reshaping the Tumulus over the intervening millennia.

The Tumulus at Kasta as an oblate ellipsoid

We’ll now move a step even further in trying to explain the 3-d form itself of Kasta Tumulus. In a paper

found in ref. [1.5] the author argued that the architect’s objective in shaping the 3-d form of Kasta

Tumulus was to create a sphere. Further, it was argued that the observed shape of Kasta above ground

was to represent the intersection of a plane with that sphere at an angle equal to the ground’s North-

South slope there. Furthermore, it was argued in that paper that this sphere stood for the Earth,

symbolically representing the dominance of Alexander’s Empire over the known World. Here, it has also

16

ben alluded that the above ground level part of the Tumulus represented the world of the Living, whereas

the perceived below ground part of the Tumulus’ sphere stood for the world of the Dead.

Figure 3. The geometric shapes of ellipsoids in 3-d Cartesian coordinates. Top is a prolate

ellipsoid (with the vertical, along the z-axis, c-semi axis longer than the two, equal horizontal, x

and y, semi axes a); whereas the bottom ellipsoid is an oblate one (with c being shorter than a).

Earth is an oblate ellipsoid. Source of diagram: ref. [1.13].

But, as already mentioned, on January 29th, 2016 the archeological team presented some new

evidence, with three cross sections of the Kasta Tumulus. Although no details were presented,

17

these Tumulus cross sections alluded to an elliptical form. In view of the new evidence the

author’s prior work found and indicated in the paper cited in ref. [1.5] will now be slightly

modified as well in its Mathematics, but considerably expanded in interpretation of these new

findings.

It will be argued that the mathematician and astronomer behind the design of the Tumulus

elliptical shape at its cross sections, and thus in the overall shape of the mound above ground,

knew that the Earth itself is not a perfect sphere, but instead an ellipsoid that closely resembles

a sphere. For a brief description of an ellipsoid, the key equation will be offered here, expressed

in Cartesian coordinates (x, y, z), and as a function of its semi-axes (a, b, c), see Figure 3:

{x^2/a^2 + y^2/b^2 + z^2/c^2} = 1.

In Geodesic Science, Astronomy and in general the Geosciences, approaching the shape of Earth

as an ellipsoid has been traditionally considered and argued as having been a rather recent

endeavor. It was the middle of the 18th century and the work by Pierre Louis Maupertuis when

the first modern-day suggestion was made that the Earth is an ellipsoid, see ref. [1.16]. The first

ellipsoid models of Earth were suggested (and measurements were taken based on them) in the

late 19th century, see ref. [1.17]. Not much is known about Greek astronomers and

mathematicians regarding their views about the shape of the Earth, other that they knew fairly

early on that it was “spherical”. The bold suggestion is made here that apparently the specific

mathematician-astronomer working with the architect on the exterior wall of Kasta Tumulus as

well as in the shaping of the entire Tumulus contemplated that the shape of the Earth was an

ellipsoid in 3-d. And the suggestion is advanced that these talented individuals had derived some

estimates of it. The suggestion is thus made that the Tumulus at Kasta is a scaled down version

of the Earth’s ellipsoidal shape.

This author finds no other compelling reason to explain why the Tumulus’ architect would design

such a shape for the mound, given that this author rejects the idea that the Amphipolis Lion was

intended (and of course by implication, it never was installed) at the Tumulus’ summit. An

alternative explanation that this author would consider is that the shape of the Tumulus as a

section of the plane with an ellipsoid would enhance the view of the hill if it possessed the head

of a bull. Within this context, it is possible that the shape that the architect intended for the

monument was equivalent to that offered by the monument at Newgrange, as argued in

numerous papers by the author cited in ref. [1.5]. However, the more daring hypothesis is that

the ellipsoid implied then by the shape of the Tumulus was meant also as a proxy for Earth’s

shape, for purely symbolic reasons. Moreover, one could put forward the argument that this

ellipsoid was in effect at the same time the shape of a bull’s top section of its head and the Earth’s

ellipsoid. Such an assumption would link the historical Bull Cult symbolism and representation,

to that of the advances in Mathematics and Astronomy implied here.

The specific ellipsoid we now know Earth resembles is shown in Figure 3, bottom part, the oblate

ellipsoid, given by the formula (in Cartesian coordinates, x, y, z):

18

(x^2/a^2 + y^2/a^2 + z^2/c^2} = 1,

Where the two horizontal semi-axes a and b are equal (a=b), and: c < a. Although semi-axis c is

less than a, it isn’t less by much. The degree to which an ellipsoid in shape form deviates from a

sphere is called “flattening”, and it is an algebraically complicated function of the sizes a and c.

In now suggesting that the Great Kasta Tumulus is itself built to be a part of an oblate ellipsoid,

we are now appropriately revising and extending our previous findings reported in the various

papers cited in ref. [1.5]. To be able to say whether Kasta Tumulus oblate ellipsoid is a scaled

version of Earth’s ellipsoid, we need far more accurate measurements than what we currently

have in our possession. It is still hoped (more three years after the excavation at Kasta,

Amphipolis was completed) that in the near future such measurements will be forthcoming by

the archeological team, thus enabling more precise and exact replication and modeling of the

Tumulus’ morphology.

It is now suggested that whoever was the mathematician-astronomer working with the Tumulus’

architect (whoever that architect was) was well aware of the fact that the Earth, although close,

it was not exactly a sphere but an oblate ellipsoid. The sphere to which the ellipsoid was a close

approximation has been computed in one of the papers supplied in ref. [1.5]. The major semi-

axis, a, must be very close to the radius found in the paper by the author, estimated there to be

about 220 meters. An elementary introduction to the Mathematics of ellipsoids is found in [1.13],

from where Figure 3 has been taken. It is suggested that the c semi-axis at Kasta Tumulus could

hover around the 200-meter mark. Since we can’t penetrate any closer this solid, we announce

this as a potentially significant finding.

As already mentioned, the archeological team on January 29th, 2016 offered the public three

cross sections of the Kasta Tumulus. These cross sections were representing both actual current

Tumulus irregular soil compositions, and simulated configurations of an elliptical extrapolated

shape. These sections showed a Tumulus with shapes in effect resembling “ellipses” with the top

point of the ellipse located off center and towards the North side of the Tumulus. The

archeological team’s actual description of the Tumulus’ shape as a solid is still unknown, as no

member of that team has communicated the team’s views on it yet (to this author’s knowledge).

One might conclude that the shape they produced by the three cross sections of the Tumulus is

a solid which would accentuate and enhance the view of their “Lion-of-Amphipolis-on-top-of-

the-Tumulus” vision. It must be noted that such an argument would be to a large extent almost

vacuous, since the gain in enhancement by such a ground-driven perspective obtained from the

front (Southern part) of the mound would be countered by an unappealing sight of this very

structure from its hind (Norther) side. However, as mentioned earlier, this paper is not intended

as a review and evaluation of the archeological team’s cross sections of the Tumulus.

The paper now turns to the world of Mathematics and Astronomy in the Helladic space by the

time the Great Tumulus at Kasta was constructed, the last quarter of the 4th century BC. Here,

the effort is made to outline the intellectual currents in these fields, as they relate to the

19

Mathematics of ellipses and ellipsoids, as well as the Earth related Astronomy at the time and

the possible names that could had indirectly and directly affected and been involved in the

construction of the Kasta Tumulus and its tomb.

Philolaus the Pythagorean

One element, someone might call it “characteristic”, of Classical Greek Civilization was the

simultaneous presence of difference Schools of Thought, in all branches of human endeavor.

From Mathematics to Astronomy to Philosophy to the Arts, Greeks were not monolithic in their

approach to topics on Mathematics, Science, Philosophy, Politics, and more broadly on issues

involving human affairs. Although, like today in all fields, there may had been “dominant” or

“establishment views”, diversity of thought was ubiquitous in all subjects mentioned. A good

example of such variety in viewpoints we find in the various philosophical schools that sprang

out from various schools and locations throughout the Helladic space, during the period between

the 6th and 3rd Centuries BC. From Pythagoras, to Socrates to Plato and Aristotle, to the Stoics,

Epicurus and the Cynics, we come across a wide spectrum of intellectual traditions in Greek

Philosophy. To an extent, these traditions existed within a highly complex intellectual ecology,

where both evolution of thought occurred, as well as fierce was the competition among various

strains of intellectual perceptions and the persons behind them. It was a diverse, rich and fertile

Ecology of Ideas. Quite likely that kind of complex intellectual ecology was the World of Mathematics

and Astronomy by the final quarter of the 4th Century BC in the Helladic space, from the Western shores

of Asia Minor, Ionia, to the shores of Sicily and Magna Graecia in the Apennine Peninsula.

A relatively obscure figure in this highly competitive field of Greek Astronomy back in the second

half of the 5th to the early part of the 4th Century BC was philosopher-astronomer Philolaus the

Pythagorean (470 – 385 BC). Philolaus was born most likely at the Greek colony of Croton, in

Sicily, although his exact place of birth (as are the dates of his birth and death) is not exactly

known. It’s hypothesized that it was likely Croton, although Tarentum or Metapontum are also

possibilities. For his life and work, see ref. [1.14]. Philolaus, a Pythagorean in the mathematical

and philosophical sense, had ties to all three major philosophical figures of Classical Greece,

Plato, Socrates and Aristotle.

He lived during the period when the Geocentric System was dominant and in full swing in the

Greek World of Astronomy. But he expressed a dissenting view. We come across his contribution

to Astronomy and his views, in a passage of a letter Copernicus wrote to Pope Paul III (see Heath

ref. [1.12], p. 301), quoting from Cicero and Plutarch as well as work from Placita: “Philolaus the

Pythagorean asserted that the Earth moved round the fire in an oblique circle, in the same way

as the sun and the moon”.

20

That was a stroke of genius.

In a more detail passage, p. 97 in Heath [1.12], a full description of this system by Philolaus is

given. One detects an effort by Philolaus to present his theory of “Earth’s motion around the

fire” in a religious context, a context that Copernicus subtly exploits in writing to the Head of his

Church. The passage clearly shows how a revolutionary idea can be “packaged” to “sell” into the

“establishment” (religious Establishment at the time, in both Philolaus and Copernicus’ times).

Notice the term “oblique circle”. This is a key term, which alludes to an “elliptical” orbit and to

someone who at the beginning of the 4th Century BC perceived a motion by Earth that was not

perfectly circular. Of course, Philolaus considered that a “fire” (not the Sun as we know it today)

was the center of motion, and that the Sun was itself revolving around that “fire”. Nonetheless,

we now know that the seeds of Earth’s elliptical motion were present at the beginning of the 4th

Century BC, a good three quarter of a Century before Kasta Tumulus’ construction. And that

certainly was a long time for the idea to ferment.

One might also convincingly conclude that the “idea” about heliocentrism belongs neither to

Copernicus nor Galileo, although more exact measurements and observations on it do. For sure,

the matter also is indicative of how long revolutionary ideas take to be accepted by the

“establishment” and society at large, at times. A case can possibly be made that potentially the

Earth’s orbital motion as an idea doesn’t even belong to Aristarchus, (c 310 – 220 BC, see ref.

[1.20]) but to Philolaus the Pythagorean. It is of interest to note that even at the time of the great

Astronomer Ptolemy of Alexandria, see ref. [1.15], as late as the 2nd century AD, and in spite of

all his great observations and recordings of celestial bodies, he was not a heliocentric system

follower. Although Pythagoras (570 – 495 BC) in the 6th century BC proposed that the Earth is

spherical (on purely aesthetic arguments, see ref. [1.18]) and Eratosthenes (c 276 – 194/5 BC)

was the mathematician and astronomer who at Syene (current day Aswan, Egypt) computed the

Earth’s circumference, the view of a spherical Earth was neither universally accepted or widely

known even in the Greco-Roman world. In many cultures, from the Neolithic down to the Roman

Era, sundials would be made – although the makers didn’t fully understand or knew their

underlying Mathematics and Astronomy. This is subject this author has specifically address in the

Facebook research note under reference [1.19].

The point is that innovations take time to be spatially spread and adopted (a major topic of work

in the field of Human Geography). Moreover, the person who was the carrier of that innovation

may not always be the person who became known as the innovator; usually it takes time to sort

this aspect of an innovation out. Innovations seem to always be the product of combining existing

ideas, making the finding of the real innovator quite difficult to determine. Finally, it could be

that a particular innovation was independently derived by different individuals, at possibly

different points in space-time. With all these being relevant caveats to keep in mind, the issue of

who was likely the mathematician-astronomer behind the design of the Tumulus is approached.

21

In the paper of ref. [1.1] the author initially attributed the Mathematics of the Kasta Tumulus

(the elliptical ground base of the Tumulus) as those representing the views of Callippus of Cyzicus

(c 370 – 300 BC, a student in the Philolaus school and a contemporary of Eudoxus of Cnidus – a

4th century BC mathematician, see ref. [1.23] - as well as a student of Plato), as a less likely option,

or to one of his students, a more likely option. It turns out that Eudoxus is a key link in our search

for the mathematician behind the Kasta Tumulus. With the current version of the paper this

author contends that, the person who most likely meets these specifications must be considered

to be mathematician Menaechmus, whose contributions to the study of ellipses, as well as life,

the analysis turns next.

Menaechmus, Apollonius and the Ellipse

In the History of Mathematics, the major figure regarding the topic of ellipses is Apollonius (c

262 – 190 BC, see ref. [1.21]). His work on conics was fundamental in the work regarding the Geometry

of ellipses, along with parabolas and hyperbolas, with the term “ellipse” (in Greek: ΕΛΛΕΙΨΙΣ) being a

term many currently consider that he coined. How the term came about is mentioned in his treatise on

Conics, Book I, Proposition 13, see ref. [1.22].

A lesser known figure in the History of Mathematics is Menaechmus (in Greek: ΜΕΝΑΙΧΜΟΣ), see

reference [1.24] as well as [1.25] and [1.26]. The diagram of this paper’s p. 3, from ref. [1.4] identifies a

major contribution by Menaechmus in an attempt to solve the so-called Delian Problem in Geometry.

This problem calls for the geometric derivation (i.e., by employing only a ruler and a compass) of a cube

which is double in volume from a unit cube. Of course, this problem has no solution if the pure geometric

means are to be employed, see for a simple presentation ref. [1.27]. However, Menaechmus employed a

method in an attempt to solve the Delian problem by using intersections of parabolas, thus mechanically

obtaining a solution to it. The method involved the ratios a/x=x/y=y/b that can be seen as representing

points (x,y) in a 2-d Cartesian space, corresponding to the intersection of two parabolas.

However, the reason why Menaechmus is of import here, in this paper, is that he is credited as being the

first who formally studied ellipses (and of course before Apollonius did), see [1.28]. Menaechmus lived

in the period 380 BC to 320 BC, thus he died very close to the time Phase II of the Kasta Tumulus was

shaped, the perimeter wall was partially completed and the main portion of the tomb’s interior was

constructed, see references on these various phases, as suggested by this author in the set of papers cited

in [1.5]. His death, assuming he was involved in the construction of the monument at Amphipolis

coincides with a significant turning point in the monument’s historiography.

Menaechmus meets certain key criteria for being considered a person of interest in the Kasta Tumulus

historiography. He was a pupil of both Eudoxus (a key personality in the section above on Philolaus the

Pythagorean, which also included Callippus of Cyzicus), and of Aristotle. A number of the key

mathematicians in this narrative revolve around the Cyzicus region of North-West Asia Minor, a region

relatively close to Amphipolis. Recorded evidence seems to suggest that Aristotle introduced

Menaechmus to Alexander III, see ref. [1.26].

22

It seems inescapable, that Menaechmus must be considered the prime suspect in the design and

Mathematics as well as the Astronomy of the Kasta Tumulus. This section completes the Mathematics

and Astronomy Part of this paper. Attention now turns to the Tumulus height, since this height is a key to

the arguments regarding the 3-d shape of the monument.

On the Impossibility of a 26.40-meter high Tumulus at Kasta

Figure 4. Kasta Tumulus, Google Earth map as of 9/30/2014. The site’s top elevation at present

stands at a height of about ten meters above ground (that is, from the ground level of the covered

Entrance, shown at bottom, and lightly off center). Source: the author, see ref. [2.1].

The suggestion made by the archeological team that a 26.40-meter high Tumulus stood on top

of the Kasta tomb is objectionable on a number of counts. Top in this list of counts is the team’s

23

own work during the excavation (which at this point doesn’t refer to the significant re-

arrangement of the Tumulus soil on top of the tomb). As soon as they entered the tomb, they

set up an elaborate internal metal rod-based support structure to prevent the monument from

collapsing. It is to be noted that such support structure was found to be absolutely necessary in

2014, with soil loads far lesser in intensity (vertically and laterally) on the tomb’s interior space,

from the hill’s soil on top of it. These loads were far less than the loads exerted on the tomb’s

structure back in the last quarter of the 4th century BC following the architect’s conversion of the

hill into a tumulus.

In addition, and this point partly relates to the issue raised earlier about the accuracy of the

Tumulus’ cross sections the archeological team produced to go along with Kasta Tumulus site

plan, the hill’s soil conditions above ground seem to indicate a variety of human intervention

over time. During the ensued two and one third millennia, the shape of the Tumulus has been

altered significantly, by many forces and for numerous reasons, some benevolent some

malevolent in reference to the tomb and the persons interred there. A key phase in that constant

and at times quite drastic reshaping was the burial of the exterior wall. It was undertaken very

likely by Cassander in the middle of the penultimate decade of the 4th century BC, under Phase

III (described in the references cited in [1.5]). Use of the hill by various entities and for a host of

reasons have in all likelihood totally altered the Tumulus original and intended configuration.

How the Tumulus looks today is shown in Figure 4. It isn’t particularly productive trying to fit a

curve to the existing hill conditions. And it is next to impossible to suggest that, by reshuffling

soil currently on the hill, one could derive the original intended Tumulus configuration.

The above statement on the Tumulus’ current conditions notwithstanding, doesn’t preclude a

derivation from educated guesses as to the form of this Tumulus, the original architect intended

to create. Undoubtedly, the form of the Tumulus was specifically drawn to represent some

symbols of that Era, some of which were alluded to already in this paper. By the size of the tomb

we found, we can work backwards and estimate the possible height of the Tumulus. And on the

basis of this estimates, the contentions regarding the Kasta Tumulus morphology were made, in

the series of papers mentioned in ref. [1.5]. A 20-meter high mound has been suggested, and an

ellipsoid (almost spherical) overall shape was been put forward by this author as a likely scenario

for the Great Tomb at Amphipolis. Engineering Statics and structural stability of the tomb, could

not have supported any soil loads greater than that height, which is about double what the

current height of the Tumulus (or rather what has remained of the original Tumulus).

From a structural, civil and architectural engineering standpoint, it doesn’t take much calculation

to recognize that the vertical loads and lateral forces and stress exerted on the tomb itself, on

the interior walls and the tomb’s arched roof, as well as on the covered by marble blocks wall

part of the Tumulus perimeter by an alternative scenario containing a 26.40-meter high Tumulus

with a 15.80-meter high marble Lion of Amphipolis statue on top, is unattainable. It would have

resulted in the implosion of the tomb, the collapse of the exterior perimeter wall and the tipping

over of the statue in short order. Adding to all this soil statics and dynamics, the long-term effect

24

of soil erosion, and one can easily speculate that the very likely effect would be a structural

failure of both the Tumulus and its tomb in the short term, if the Tumulus were to have been

constructed as suggested by the archeological team.

It should be remarked that it isn’t totally random the fact that the Tumulus’ tomb is only 15-

meter deep (the space in front of the two Sphinxes isn’t counted here, as it was open space not

dug underground). The Tumulus’ loads above a 20-metrer high mound (the estimated by this

author maximum height of the Tumulus’ structure) could not structurally sustain any tomb dug

deeper inside the hill than that. The almost 30% addition in the height of the mound from the

author’s estimate would increase loads in the 15-meter deep tomb exponentially, rendering the

structural stability of the tomb unsustainable.

Without any elaborate terracing preparation (as we see in other mounds, tumuli, passage tombs,

and dolmens of antiquity all over the World, from the Neolithic onwards) a preparation largely

absent from Kasta Tumulus, one can’t justify heights anywhere near the heights envisioned by

the archeological team (greater than 26 meters, plus the load of a marble Amphipolis Lion on

top reaching an additional almost 16 meters in height). This author strongly recommends that

the archeological team rethinks their hypotheses on these two matters. Further, the author

recommends that the detailed architectural and engineering studies undertaken by the

archeological team be made available to the public domain, so that architects and engineers can

carry out such structural analysis as suggested here. It has now been more than three years after

the completion of this excavation, and next to nil has been made public by the archeological

team regarding pertinent geological, mechanical, architectural and design features of the

monument and precise measurements.

Concluding Statements

Much still remains to be learned about this extraordinary monument at Amphipolis. To that end,

it is absolutely essential that more details and measurements about the Tumulus and its tomb

be made available by the archeological team and the Greek Ministry of Culture and Sports to the

public, or be published in professional and scholarly Journals. That would shed light on numerous

aspects tangentially addressed here, like for example the position of the tomb’s entrance in

reference to the Tumulus ground floor ellipse’s major axis.

The Great Tumulus at Kasta, in Amphipolis, Macedonia, Greece, is a monument not so much to

Hephaestion, and all others who probably got hold of the tomb and made use of it till the time

it was internally buried, possibly by Philip V in the 2nd century BC, as described in the author’s

papers cited in ref. [1.5]. The magnificent Tumulus of Amphipolis is a monument to the

Mathematics and Astronomy of the last quarter of the 4th century BC, and possibly Callippus or

25

most likely Menaechmus. Of course, the artwork of the monument, the Sphinxes, the Kores, the

mosaic floor (in both its rendition of the abduction of Persephone by Hades under Hermes’

watchful eye, as well as its double-meander and waves eternity symbols sporting frame), all

these artifacts that have remained there and have not been looted by the various tomb robbers

and raiders of antiquity as well as the passage of time and its wear and tear, are unambiguously

exquisite pieces of Art. But by and large, the masterpiece of this monument is its modulus and

especially its exterior marble blocks covered wall and the elliptical in 2-d base (and possibly

ellipsoid in 3-d) design of the Tumulus.

When time comes, and this monument will be called to present its case for designation as one

of UNESCO’s World Heritage Sites (WHS), it will be its exterior wall, its overall possible shape as

a solid, and the mathematics and astronomy implanted in its 2-d site plan which will be the basis

for its ascension to prominence. It will be Callippus or Menaechmus and the genius of the

architect (Deinokratis or Parmenion very likely) that will propel this site to archeological heights,

possibly equivalent to that of the Parthenon.

As the Parthenon marked Athens’ Golden Age, this monument (more so than even the Sanctuary

of Samothrace with its Winged Nike) will mark Macedonia’s Golden Age. This Age of

unprecedented achievements in military conquest under Philip II and Alexander III and the

unprecedented spurt in economic growth, social prosperity and the Arts attained at the closing

of the 4th century BC (not only in Macedonia and the rest of Greece, but throughout the Region

of Eurasia and Africa touched by the albeit short lived empire of Alexander III) were accompanied

by significant advances in Mathematics, Astronomy and Aristotelian Science. These latter

achievements are mainly depicted in what we came across in August 2014, a monument to

Humanity anchored deep in the Neolithic and made for the Ages to come.

The Great Tomb at Amphipolis and the Great Tumulus that contained it formed possibly the

greatest monument of that Era. The Sanctuary of Samothrace was possibly a close second. We

ought to be looking at this monument, its ground 2-d shape as well as its 3-d design, as the

depository of all that knowledge and understanding that were achieved and attained at that rare

moment in Human History, at that special place near the ancient city of Amphipolis.

Deinokratis or Parmenion, and Callippus or Menaechmus and all the mathematical, astronomical

and scientific tradition from Classical Greece to the post Alexander III years, those years that

constitute what’s being referred to as the “Hellenistic Age”, are all imprinted on Kasta Tumulus.

The sizes of the modulus and its derivatives, were meant to capture the rare moment of

Alexander, Hephaestion, Macedonia and Greece, and place their deeds in their proper

perspective. And that perspective was not only a global, all Earth encompassing, but in fact a

Universal perspective – a perspective big enough to reach the Heavens and the Celestial Sphere.

This is ultimately the message of the Great Tumulus at Kasta.

At the end, by reading and de-coding the morphology of this late 4th century BC monument at

Amphipolis, we are almost forced to recognize that the Greeks of that Era were far more

26

sophisticated in Mathematics and Science and Astronomy than what has been acknowledged so

far. The Antikythera mechanism’s Engineering and Computing, the Archimedes’ Differential and

Integral Calculus and Geometry, and the Aristarchus Heliocentric System didn’t appear out of the

blue sky in a vacuum during the 3rd century BC. They drew from deep roots in Mathematics and

Astronomy from the late 4th century BC, the Era of Alexander III, the time that the Great Tomb

and Tumulus at Amphipolis was constructed. This paper is a call for re-examination and close

scrutiny of that period and of that place of Macedonia, Southern Thrace and North-Eastern Asia

Minor.

Note. The ellipse, as a shape imprinted on artifacts, was not first applied in the time period

covered here (the 6th to the 3rd century BC). It is encountered for the first time in the elliptical

deigns of seals and rings of the Minoans in the middle of the 2nd millennium BC. This topic is

examined in a forthcoming paper by this author, on “Minoan miniature Art”, where issues linking

ellipses to artifacts is analyzed more in depth.

References

[1.1] Dimitrios S. Dendrinos, February 17, 2016, “The Earth’s orbit around the Sun and the

Tumulus at Kasta”, academia.edu The paper is found here:

https://www.academia.edu/22103391/The_Earths_orbit_around_the_Sun_and_the_Tumulus_

at_Kasta._Update_1

[1.2] http://www-history.mcs.st-and.ac.uk/Curves/Ellipse.html

[1.3] http://www-history.mcs.st-and.ac.uk/Biographies/Menaechmus.html

[1.4] http://www.takayaiwamoto.com/Greek_Math/Delian/Menaechmus_Delian.html

[1.5] Dimitrios S. Dendrinos, January 18, 2016, “The Tumulus at Amphipolis: summary of a

theory”, academia.edu The paper is here:

https://www.academia.edu/20339980/The_Tumulus_at_Amphipolis_Summary_of_a_Theory

[1.6] The K. Peristeri 1/29/2016 video presentation at the University of Cyprus is available here,

in Greek: https://www.youtube.com/watch?v=4LHsuhhzhmI

[1.7] Dimitrios S. Dendrinos, August 15, 2016, “A review and critical appraisal of Oscar Broneer’s

1941 book ‘The Lion Monument at Amphipolis’”, academia.edu The paper is found here:

https://www.academia.edu/27809420/A_Review_and_Critical_Appraisal_of_Oscar_Broneers_

1941_book_The_Lion_Monument_at_Amphipolis_

[1.8] https://www.facebook.com/profile.php?id=100006919804554

27

[1.9] https://www.facebook.com/groups/520506078128011/

[1.10] https://en.wikipedia.org/wiki/Earth%27s_orbit

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

[1.12] Thomas Heath, 1915, Aristarchus of Samos: the Ancient Copernicus. A History of Greek

Astronomy to Aristarchus together with Aristarchus Treatise on the Sizes and Distances of the

Sun and the Moon. A new Greek text with translation and notes, Clarendon Press, Oxford.

[1.13] https://en.wikipedia.org/wiki/Ellipsoid

[1.14] https://plato.stanford.edu/entries/philolaus/

[1.15] https://www.britannica.com/biography/Ptolemy

[1.16] https://www.britannica.com/biography/Pierre-Louis-Moreau-de-Maupertuis

[1.17] http://www.georeference.org/doc/the_earth_as_an_ellipsoid.htm

[1.18] http://www.astronomy.ohio-state.edu/~thompson/1101/lecture_aristarchus.html

[1.19] See the November 11, 2017 article, shared from the author’s Facebook timeline, on the

subject of a Roman 1st century AD sundial, at the archeological site of Interamna Lirenas, here:

https://www.facebook.com/groups/765342993596415/

[1.20] https://www.britannica.com/biography/Aristarchus-of-Samos

[1.21] http://www-history.mcs.st-and.ac.uk/Biographies/Apollonius.html

[1.22]

https://web.archive.org/web/20070715063900/http://mathdl.maa.org/convergence/1/?pa=co

ntent&sa=viewDocument&nodeId=196&bodyId=203

[1.23] https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus

[1.24] http://www-history.mcs.st-and.ac.uk/history/Biographies/Menaechmus.html

[1.25] http://www.robertnowlan.com/pdfs/Menaechmus.pdf

[1.26] https://en.wikipedia.org/wiki/Menaechmus

[1.27] https://en.wikipedia.org/wiki/Doubling_the_cube

[1.28] http://www-history.mcs.st-and.ac.uk/Curves/Ellipse.html

28

Google Reference.

[2.1], the Google Earth maps of Figures 1 and 4, were produced by the author from the Google

program in the public domain, a search program made available to any user free of charge.

Acknowledgments

The author wishes to acknowledge the contributions made to his work by all his Facebook friends,

and especially by the members of his current twelve groups the author has created and is

administering. Their posts and comments have inspired him in his research over the past three

years, since he opened his Facebook account.

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.

Note of Legal 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.

Dendrinos.