ArticlePDF Available

Abstract and Figures

The paper extensively expands on and revises a prior paper by this author on the subject of the Kasta Tumulus form and design. It advances the propositions that, first, the ground elliptical shape of the Tumulus in combination with the tomb's modulus records the Earth's elliptical orbit around the Sun; second, that the 3-d shape of the Tumulus stands for the Earth's ellipsoidal shape; and third, that either mathematician Menaechmus or Callipus were behind the Mathematics and Astronomy of the monument. The paper is a revised version of the author's February 17, 2016 paper: https://www.academia.edu/22103391/The_Earths_orbit_around_the_Sun_and_the_Tumulus_at_Kasta._Update_1
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 Tumulusperimeter
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.
Article
Full-text available
This is an update of the paper under the same title by the author. It contains editorial corrections and the formal permission by the University of Cincinnati Dept. of Classics to use the image of the ring.
Article
Full-text available
The paper is an updated version of the December 2, 2017 paper under an identical title by this author. It incorporates in it exact measurements of the ring (received by the author on December 3, 2017). The paper strengthens and expands on the findings of the previous paper, as well as it amends and extends the prior analysis. Editorial corrections are also carried out.
Article
Full-text available
The Mathematics and embedded Astronomy are explored of the almost elliptical in shape Minoan 5-priestess gold signet ring of the c 1450 BC Mycenaean “Griffin Warrior” tomb at Pylos found during the 2015 archeological excavation there. It is documented that the shape of the ring is extremely close, albeit not exactly identical, to the true ellipse of an identical major and minor axes. The likely knowledge of ellipses possessed by the ring’s maker is identified. In the paper, a detailed description of the ring’s iconography is also offered, which to an extent differs from the current archeologists’ based description. The iconography’s Astronomy, is found to be associated with a ceremony dedicated to the fertility of Mother Earth, that quite likely was taking place around the Winter Solstice. An estimate of the ceremony’s duration, eighteen days, is also obtained, as having been engraved onto the ring’s iconography.
Article
Full-text available
The paper presents a 15-page summary of a comprehensive theory regarding the Great Tumulus at Kasta, Amphipolis, Greece. It is based on a series of six papers and their various versions, written in the period October 2014 till January 2016 on the subject of Kasta's monument initially intended for Hephaestion, as commissioned by Alexander III and carried out by Imperial architect Deinokratis. The theory ties the monument to a local version of the Bull Cult.
Article
Full-text available
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, are far more advanced and complex than previously thought. Specifically, we document the significant discovery that on Kasta Tumulus’ exterior wall Earth’s elliptical motion around the Sun is depicted. We also document the potential significant finding that Kasta Tumulus’ overall shape as a solid, is that of an oblate ellipsoid, representing Earth. 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 2 nd millennium AD. In a Note at the end of the paper (Note 2), more on the possible intellectual tradition behind the elliptical motion of the Earth is provided. It traces back to astronomer and mathematician Philolaus the Pythagorean. Here we take an analytical approach, albeit algebraically simplified, to describe and understand the morphology of the Great Tomb at Amphipolis. It’s a follow up study to previous work undertaken by the author. That prior work is both mildly revised and considerably extended with this paper. Work reported here now presents the Tumulus’ form as potentially depicting significantly innovative ripples in the mathematics and astronomy of that Era (the last quarter of the 4 th Century BC) in Greece. The paper advances the thesis that on Kasta’s perimeter wall we “read” the Greeks’ understanding of Earth’s ellipsoid shape and the Earth’s elliptical motion around the Sun, beyond the recognition that this wall was shaped as well in the form of an annual calendar (a prior finding by the author). It is again suggested that mathematician and astronomer Callippus, or possibly one of his students with prior understanding of (what was later attributed to) Aristarchus heliocentric system was the intellectual force behind Kasta. The impetus behind this partial modification and considerable extension of the author’s previous work, presented here, was the site plan of the monument, as presented to the public by the head of the archeological team Mrs. Katerina Peristeri on January 29, 2016
Article
Full-text available
This is review of the 1941 classical book by Oscar Broneer about the Amphipolis Lion. It critically appraises the contribution made by the author of the book in recording the thought processes involved in both the Lion's reconstruction, as well as the efforts towards producing a proposed conjectured restoration. In evaluating the book, the author documents three propositions: the monument was not assembled in antiquity; the monument could not stand as conjectured; and that this is the reason why it was abandoned. The reviewer documents also that the Lion was reconstructed in situ, where it was intended to be raised by its original makers at the closing decades of the 4th Century BC. Further, the reviewer revises some of his prior suggestions about the intended location of the monument.
The Earth's orbit around the Sun and the Tumulus at Kasta", academia.edu The paper is found here: https://www
  • Dimitrios S Dendrinos
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
The Tumulus at Amphipolis: summary of a theory", academia.edu The paper is here: https://www.academia
  • Dimitrios S Dendrinos
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
Peristeri 1/29/2016 video presentation at the University of Cyprus is available here
  • K The
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
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
  • Google The
, 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.