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The modular structure of the tomb at Kasta Hill



In this paper an unexpected discovery is reported, regarding the modular structure of the edifice uncovered at Kasta Hill. Like all great buildings of antiquity, it is proved here by using relatively simple algebraic relationships, this temple/monument/tomb too was built with an underlying modular structure connecting it (and its occupant) to the Heavens, and the night sky. The use of an astronomical constant is detected, embedded in its design modulus. This modulus is shown on both the exterior and the interior walls' marble coverage. It demonstrates that the location of the temple/monument/tomb's entrance identifies a particular day of the year and even more accurately, a particular 6-hour segment of that day. Moreover, the entrance NorthEast to SouthWest axis may point to an astronomical alignment. In this regard, this edifice is a unique architectonic creation. In its complex modular design, among the numerous modular principles found in it, a "golden section" rule is shown to govern the Karyatides height to their base. However, the module of Kasta Hill's temple/monument/tomb which the skilled architect employed was far more elaborate and complicated than a simple rule based symmetry principle. Introduction.
The modular structure of the tomb at Kasta Hill
Dimitrios S. Dendrinos
Emeritus Professor, School of Architecture and Urban Design, The University of Kansas,
Lawrence, Kansas, USA. In residence, at the City of Ormond Beach, Florida, USA.
December 4, 2014
Email address for comments:
In this paper an unexpected discovery is reported, regarding the modular structure of the edifice
uncovered at Kasta Hill. Like all great buildings of antiquity, it is proved here by using relatively
simple algebraic relationships, this temple/monument/tomb too was built with an underlying
modular structure connecting it (and its occupant) to the Heavens, and the night sky. The use of
an astronomical constant is detected, embedded in its design modulus. This modulus is shown
on both the exterior and the interior walls’ marble coverage. It demonstrates that the location of
the temple/monument/tomb’s entrance identifies a particular day of the year and even more
accurately, a particular 6-hour segment of that day. Moreover, the entrance North-East to South-
West axis may point to an astronomical alignment. In this regard, this edifice is a unique
architectonic creation. In its complex modular design, among the numerous modular principles
found in it, a “golden section” rule is shown to govern the Karyatides height to their base.
However, the module of Kasta Hill’s temple/monument/tomb which the skilled architect
employed was far more elaborate and complicated than a simple rule based symmetry principle.
This paper reports a discovery involving an astronomical connection built into the modular
structure of a recently uncovered tomb-monument at Amphipolis, in the Northern Region of
Macedonia, Greece. The monument at Kasta Hill has a symmetry in it, far more complex than a
simple “golden rule” principle. Its built-in structural code is very complex, and this paper is a first
attempt to decode it. By even simple analysis of its modular structure, many topics emerge in
need of further study. They all point out the level of mathematical and engineering sophistication
characterizing the architect who built this tomb. He must have possessed advanced knowledge
of algebra, engineering skills, imaginative architectonic principles and astronomical knowledge.
Such skillful amalgamation of conceptual design and engineering technique is rarely
encountered in monumental Architecture.
The complex module used by the architect of the monument, both inside and outside this marvel
of architecture and engineering needs careful analysis, and some of its embedded relationships
are still to be discovered. This paper examines and identifies exactly only certain components of
the complex three-dimensional module, the KANABOS (or grid) used by the architect: the
module’ length and its width but not its height completely – due to lack of data. Thus the findings
will need further verification as more data become available. The undoubtedly significant
symbolic aspects of the entire module are not addressed at any length here, but are left to the
interested reader.
A few notes must be made at the onset. The monument has a complex modular (grid)
structure. The module’s length is the key in identifying the three-dimensional structure of both
the module and the building. It permits us to penetrate into the architect’s approach and
objectives in the design of this edifice. We do have enough data from the archeological team at
present to accurately define the length of the module, enough data to approximate its width,
but not enough to compute its exact application for all heights inside the monument, with
material that has been made public. It’s apparent now, as a result of this study, that the module’s
length is used both inside the tomb as well as outside, and it acts as the cornerstone in the
monument’s construction. Accurately estimating the grid’s length allows us to appreciate the
architect’s aesthetics, potential symbolism, as well as his overall construction objectives.
A persistent question, since the tomb’s discovery, has been why the architect built such a huge
tomb-mound with a 497 meter perimeter, only to house an edifice just 4.5 meters wide and
about 25 meters long? It becomes apparent by this study that the architect’s main objective was
not so much to achieve grandeur by building such a gigantic by any account monument. Instead
his aim was to design a structure that would replicate the annual cycle and within it pinpoint a
specific calendar date and time of the day, possibly related to a significant event in time for
the person this monument-tomb was intended. Most likely, the ritual element of an annually
repeated rite was the architect’s key objective. There is an impressive symmetry both inside and
outside the monument. However, symmetry was not the ultimate aim of the architect in building
this structure. It was something quite far more complex and elaborate than symmetry. It was an
attempt to establish a forum for an annual observance of an event, by gazing at specific point
into the Heavens at a particular date and time.
The Outside Modulus.
The approximation. We are told by the archeological team conducting the excavation at Kasta
Hill that the circular (in floor plan, elliptical in actuality) tomb’s perimeter is 497 meters long. The
estimated perimeter length should have been made available with at least two decimal digits
approximation – an accuracy demanded by the edifice’s modular structure. However, even
without this required level of accuracy, we can fruitfully work with this “497 meters length” rough
approximation. For details, see the archeological team’s photographic evidence in:
The ground’s slope. There is a North-South ground slope in the Hill’s surface area taken up by
the tomb’s perimeter wall. The precise slope, ψ, has not been made public. Casual look at some
photos of the perimeter’s marble wall covering (clad - ORTHOMARMAROSH) made available
lead to the conclusion that this ground slope isn’t significant relative to the perimeter’s length.
Obviously, on a natural Hill, one cannot expect a smooth and leveled surface. Equally unlikely is
the fact that all contour curves of a particular elevation (this Hill’s floor stands approximately 85
meters, on the average, above sea level at present) would form perfect circles or ellipses.
Nature doesn’t offer us such luxury. Engineers have to use creative means in order to obtain
appropriately leveled-bottom ditches to achieve smooth contour curves to their liking. On top of
this difficulty, the architect needs to provide escape routes with appropriate slopes to channel
water runoff, capable to handle waterfall conditions prevailing in the area. The manner in which
this complex undertaking is accomplished, indicates the level of sophistication characterizing
construction and its architect and engineer’s ability to handle and place a large in scale building
in its natural site’s setting.
The perimeter wall. Six parallel layers of masonry compose the exterior wall’s structure,
comprising six rings, resting on a ground ditch dug by the edifice’s engineer. The wall follows
the ground’s slope. In ground ring A is composed of limestone. Overlaying rings B, C, D, E and
F are marble clads, covering layers of limestone, and defining the visible section of the exterior
perimeter wall. Rings B, C, D are made out of the marble stone type most extensively used in
the monument’s internal and external structure and of central importance to this analysis: a
marble stone with dimensions, length (L) 1.36 meters, height (H) .19 m, and width (W) .72 m
lies either transversely against the limestone or face up. Rings B and D’s marble stones are
anchored inside the limestone slabs behind them, exposing their length and height to the wall’s
surface (that is a surface 1.36x.19), lying flat on their width (.72). Ring C exposes its length and
width to the wall (surface 1.36x.72), lying on its height (.19). Above ring D lies ring E which has
a length of 1.36 (and an undefined height) as well. Above it lies ring F, the wall’s cornice, also of
an undefined height; however, the oscillating pattern of the cornice top indicates a frequency of
half the 1.36 modular length (.68 m). Rings B, D, F are aligned, and so are rings C and E, at
midpoint of the stones above and below. The archeological team has said that the exterior wall
has a total height of approximately three meters; however, this approximation doesn’t allow
precise estimate of the rings’ E and F height. Ring B (the visible “base” of the marble clad) has
no relief. Rings C, D and E have a relief with an approximate 3 cm margin in all four of their
visible sides.
All five of the visible exterior wall’s rings, B, C, D, E, and F have identical lengths. It is concluded
that this length size must be the cornerstone of the building’s module. This length is
encountered inside the edifice as well (see section below). In fact, not only the stones’ length is
repeated inside but also the cornice pattern as well at the EPISTYLIO above the Karyatides (or
Klodones, or Kores, or Mainades – what these figures exactly represent isn’t a topic of interest
here) in chamber #1. Continuation of the length-wise as well as thematic patterns from the
outside into the tomb’s interior punctuates the continuous flow in motion the architect intended
to ascribe to the monument, demonstrated by the ubiquitous presence of the module at the
inside and outside spaces.
Before we take a look inside the edifice, a remark is needed: through visual inspection of
photos, one recognizes that the outside marble covered wall shows a slight incline towards the
Hilltop. It does so obviously to provide more support for avoidance of soil erosion (at the
artificially shaped mound), and for statics related reasons so that it can better withstand
pressure from the land mass behind and enclosing it. Let’s indicate this inward towards the
vertical line incline by the angle ω. We shall mention it again, in reference to another angle of
concern here, the ground’s slope ψ.
Further, a note must be made regarding the entrance to the tomb and its connection to the
outside wall. At the current state of the tomb, there’s a staircase leading to the entrance, where
the diaphragmatic wall with the two Sphinxes on top is located. The staircase starts at the very
top level of the perimeter wall. This indicates that at some point the perimeter wall was buried,
and the staircase made out of limestone was added. Since no architect would build such a
staircase (leaving unprotected the entrance from the weather conditions like rain, snow, flash
flooding, etc., and from unwanted visitors), it must be concluded that at some point well after the
perimeter wall and the interior marble clad were added to the tomb (the tomb itself possibly
made at some even prior time period) and for some reason which is not the subject of this
paper, a decision by some entity was made to bury the perimeter wall and build the staircase.
For this paper, the fact of interest is that the perimeter wall’s base and the entrance to the tomb
are at about the same level (possibly counting a slight incline towards the entrance for drainage
purposes). It also is of import here, that when the marble clad was completed, it was done
inside and outside at the same time period, and both are in a spatial continuum. More details on
this part of the tomb’s construction are forthcoming in a paper by the author.
The Inside Modulus.
The Karyatides base. When the two Karyatides were revealed to us by the excavating team,
we were told that their rectangular prism base was i.36 meters in length, 1.40 meters in height,
and .72 meters in width:
In that reference other measures and photographic evidence released to the public is also
shown. Of particular importance here is the LENGTH, the 1.36 meters measurement. It should
be noted, however, that all these are measurements given to us from the archeological team.
We don’t know how accurate they are, and if they have a significant digit at the third decimal
level. In any case, there will be used here AS IF they are accurate. The base of these two
Karyatides is the grounds on which this analysis is itself fundamentally based. It provides very
clear evidence to set up the module used by the monument’s architect.
The Karyatides base. First of all, even a casual look at the above photo from the Karyatides’
base reveals that the architect of the monument did NOT use the “golden section” rule and its
ratio ,ϕ as a rule to design this base. It was certainly not done due to ignorance of the “golden
section” proportions. In fact, we know he must have known the golden rule quite well, since he
used it, as we’ll see a bit later. It will be claimed that the architect did not intend to use it here at
this base, because the architect wanted a more complex modular structure for his edifice.
Let’s take a close look at this base the two Karyatides stand on. Symmetry issues regarding the
overall arrangement inside the tomb will be explored very synoptically later, as the emphasis
here in this paper is the derivation of the module and not aspects of symmetry in the structure.
Each Karyatida’s base consists of four distinct layers. We’ll designate from the floor up these
layers as, Layer A, B, C and D. Layer A is made of a single marble stone, layer B consists of two
marble stones, layer C from a single marble piece, and as is also finally D’s quite thin layer on
which the Karyatides feet rest. The exact derivation of each layer and stone’s three-dimensional
measures requires that from the totals offered to the public, the various sections’ measurements
be derived.
From analysis of photographic evidence in the above reference, it is obtained that the following
proportional relations hold as far as the four layers’ height is concerned: let h be the unit of
measure for height, associated with the height of layer A; then it is observed that 2h is the
height measure for layer B, h/2 is the corresponding measure for layer C, and h/6 that of layer
D. Since we know that the total height is 1.40 meters, it follows that
h + 2h + h/2 + h/6 = 1.40 (1)
From the above we obtain that h=.38 meters. Thus, layer A has dimensions length (L) 1.40
meters, height (H) .38 m, width (depth) (W) .72 m. There’s a recess of about 4 centimeters
where layer B rests on A. Layer B’s height is 2x.38=.76 meters. The height of layer C, lying
directly on top of layer B, is .19 m; whereas finally layer D’s height is .06 m. We thus identify h is
the HEIGHT of the INTERIOR MODULE. All heights in the base are multiples of this measure
h. We don’t have a complete view of the interior clad, so whether this measure (h=.38 m) is the
base for all interior heights needs to be further checked once more evidence becomes available.
The Karyatides height (2.27 m) is a multiple of the height element of the internal module
(.38), since 2.27/.38=6.
Now, we turn our attention to the frontal length component (given to us by the archeological
team and estimated at exactly 1.36 meters) of the three dimensional grid found in the
monument. Estimating the frontal lengths L of these layers is straight forward as far as layers A,
C and D are concerned (1.40, 1.36, and 1.33 m correspondingly). Estimation of the two stones’
length in layer B requires finding their proportional relationship; by visual inspection, and if we
designate as x the frontal length of the stone to the right and by y that on the left, and z as their
sum, it is derived that:
x + y = 1.36 (2)
y / (x + y) = .68 (3)
From the above system of two algebraic equations we derive the dimensions of the two marble
stones of layer B, as:
x* = .44 m and y* = .92 m (4)
Clearly, these frontal lengths for the two marble stones of layer B do not obey a “golden section”
rule. The right hand side stone is slightly less than half in length than the one on the left – thus
clearly not obeying this particular rule. The reason why these two particular lengths were picked
by the designer of this edifice will be shown in a bit, below.
Golden section. It is recalled that the “golden rule” or “golden ratio” or “golden section” is such
that if length A and length B are such that: (A+B)/A = A/B = ϕ = 1.6180339887… where ϕ is an
irrational number, then A and B are said to be linked by a “golden rule”. Such a link has attracted
much attention in both mathematics and the arts and Architecture in particular. The question
though is, does it apply to this particular monument? And if so, what is the key length “module”
or “base” (for either A or B) here in this tomb? Furthermore, how accurately an architect
replicates this ratio is an indication of the architect’s both mathematical and construction
sophistication. In this case, the architect achieved this approximation at a remarkable for his era
level. Although the two rectangles at the base’s layer B do not obey this particular rule, the
Karyatides’ height (2.27 meters) to the total base height (1.40 meters) does obey the
golden section rule 2.27/1.40=1.6214 with an approximation of 3/10’s of one percent. Thus we
know the architect was very familiar with this rule. If the architect of the monument wanted to
use the golden rule at the Karyatides’ base, then the right rectangle’s base length (x) would be .
53 m, whereas the left rectangle’s base (y) would be y=1.36-.53=.83 m. But such clearly isn’t
the case. Instead, another relationship was in the designer’s mind when picking these two
lengths, x* and y* as we’ll see momentarily.
The MODULUS. To complete the size estimation of all marble stones of the Karyatides’ bases
we now have: layer A single stone (H: .38 m, L: 1.40 m, W: .72 m); layer B stone to the right
(H: .76 m, L: .44 m, W: .68 m) and stone to the left (H: .76 m, L: .92 m, W: .68 m); layer C’s
single stone (H: .19 m, L: 1.36 m, W: .68 m); finally, layer D’s thin stone dimensions are (H .06
m, L: 1.33 m, W: .68). Stones in layers B and C, on their exposed surfaces have a relief with a
frame of about 3 cm.
The real workhorse of the building, in terms of marble coverage is the stone of layer C.
This stone has been used extensively both on the interior, but most importantly the exterior
wall’s clad. It is the stone which links the interior with the exterior, as the building flows from the
inside onto the outside. The archeological team reported that the interior clad extends onto the
exterior. This can also be verified by the photographic material made public, and available in the
above given reference.
Let’s call z* the sum of x* and y* (z*=x*+y*= 1.36 meters) of the layer B’s stones frontal length; it
is the frontal length as well of the stone at layer C. Length z* is thus the modular frontal length
basis, with its depth wise (or width) length being about half of it (.68 m). Consequently, THIS z*
Now, the question as to why did the architect decide to have these two particular length sizes
(.44 m and .92 m) so prominently visible at the Karyatides’ base becomes apparent. The
engineer inside of the man probably dictated this split given the weight of these stones. But
there’s also the architect side of this man that offered the answer, since these two lengths
exactly depict the two sizes of the module. It is recalled that z*=x*+y*=.44+.92=1.36 meters;
and their difference y*-x*=.92-.44=.38 meters=h*. Thus, their sum depicts the frontal
horizontal module, z*, whereas their difference depicts the height of the module, h*. In these two
relationships the architect’s genius is shown.
One might be motivated to study the connection of the modulus length (1.36), width (.68) and
height (.38) to actual measures used back then. It is left for future research. Left to future
research is any connection of this modular structure to the double-leaf marble door and the
mosaic floor of chamber #2. When more accurate measurements become available from the
archeological team, this measure will need to be also more closely checked. Let’s now go back
to the outside of the tomb.
Back outside the tomb: the astronomical linkage.
The circle and the ellipse of the perimeter wall. We now know that the outside perimeter is
an actual ellipse; but on vertical projection (that is as a floor plan, or KATOPSH) it is a “perfect
circle” (according to the archeological team, that has not made available any accurate, digitally
obtained site plans, floor plans or anything equivalent to inform the public). Thus we will use the
“perfect circle with a 497 meters perimeter wall” phrase as a flat plane (perpendicular to its axis)
cutting a right circular cylinder with a radius, R, of 79.14 meters or with a diameter, (2R), of
158.8 meters (assuming that is, that’s what they meant, when they said the tomb is a perfect
circle with a perimeter of 497 meters). We’ll designate as P the actual elliptical perimeter length,
so that obviously: P*<P. It is noted that, since the ground’s plane represents a cross section of a
cylinder sufficiently off from being parallel to the cylinder’s axis, it is fine to consider the shape of
the cross section as an “ellipse”. It should also be made clear that given the Hill’s dimensions,
P* and P are very close indeed, and for all practical purposes one can safely assume that P* ≈
P. The angle to the horizontal plane (that is, a plane perpendicular to the cylinder’s axis) this
particular slope has is designated as ψ. Obviously, P is a function of both P* and ψ. The actual
perimeter length of an ellipse P is obtained by a complicated function, based on ψ, and R, which
isn’t needed here for the purposes of this paper.
As discussed already, all marble slabs visible on the surface of the wall have an equal length,
and this length must be equal to z* (that is exactly 1.36 meters.) This z* is in effect the tomb’s
EXTERIOR HORIZONTAL MODULUS. This module is directly linked to the length of the
Karyatides’ base and thus the interior horizontal modulus.
The Major finding. Since the monument is encircled in all its (circular projected perimeter
P*=497 meters) by marble slabs of this length, then the perimeter’s length (projected as a circle
by a flat plane perpendicular to the cylinder’s axis) must be (exactly) divided by this outside
modulus. And indeed, it is; and with a big surprise in store.
If one divides the total length of this perimeter P* by the length of the exterior modulus z* one
obtains the total number of marble slabs needed to cover the total perimeter wall in each row of
slabs N*:
N* = P* / z* = 497 / 1.36 = 365.44 (5)
In effect, this number N* is off the grid by a length of .5984 meters (.44x1.36=.5984 about 60
centimeters in an almost half a kilometer long perimeter). It could be that this is the difference
between the actual elliptical perimeter of the tomb P, and the projected circular perimeter P*
plus any space needed to attach and seal the slabs. However, there’s far more to this length
than simply that slack.
N* is a remarkable number with an astronomical connection. We now know from precise
astronomical measurements that there are 365.22 days in a year. The architect approximated
with N* through his module an astronomical number. The days-in-a-year number produced by
the monument’s architect is six hundredths of one percent (0.0006) different than the actual
astronomically derived number. One may comfortably say, it’s an exact estimate. This
astronomical link is quite informative, and in effect stunning, given the age of the monument.
Humans have constructed monumental structures obeying the daily cycle, the lunar cycle and
this monument was built obeying one significant element of the annual cycle, the whole
calendar year. Here, in effect, each marble stone on the periphery wall corresponds to about
one degree, in an approximate 360 degree circle.
Since this finding points directly towards a link involving an astronomical number of significant
import; and since the architect designed his module so that he was fully aware of this
connection (the chances that this N* is a random event are next to zero); one needs to ask
whether there is more to this finding. Obviously, it is now time to explore its significance and
symbolism further in reference to this particular monument. For example, is the location of the
entrance significant, given that the place it occupies corresponds to particular days of the year, if
one assumes that some point on the perimeter (say, the Northern most point) corresponds to
the first day of the year, if the year’s flow follows a clockwise motion? This interpretation would
point to an entrance at the June-July period, depending where exactly the entrance is. It would
point to a late May early June date if the flow is counterclockwise. This may offer some
additional clues as to the intent of the tomb’s architect, the purpose of the monument, and
possibly some extraordinary date in the life of its occupant. However, this symbolic aspect of the
tomb’s external structure will not be done here. The need to evaluate more the accuracy of this
extraordinary finding requires that we take a closer look as to the sizes of the circle and the
ellipse at hand, and their material differences.
The circle and the ellipse examined closely. A number of points must be kept in mind; first,
the above equation (5) is a calculation based on a vertical projection of a right circular cylinder
on a flat surface and its perimeter P*. It is not the actual perimeter P of the actual elliptical
shape surface, obtained by a flat plane cutting the axis of a right circular cylinder at an angle ψ.
So, the question is raised as to whether this difference is immaterial to our finding, that is in
other words, whether the finding about N* applies to P as well. In short, the answer is yes.
Of course, P is a function of the angle ψ, and put more formally:
P = f[ψ,R] (6)
Where, P is the length of the ellipse, and R is the cylinder’s radius; F[ ] stands for the algebraic
expression “function of”. As already mentioned, P* and P are about the same for all practical
purposes. This claim though should and can be demonstrated quite easily. Since the elliptical
plane (that is, the plane which contains the ellipse) intersects the cylinder it follows that the
diameter of the ellipse’s short axis is identical to the circle’s diameter. On the other hand, the
ellipse’s long axis is greater than the circle’s diameter (but by not much in this case). To prove
this, let’s designate by ϵ the difference between (P* - P). By the Pythagoras’ theorem, the
ellipse’s longest axis must be the positive square root of the square of the circle’s diameter plus
the square of ϵ. Given that the square of the 158.28 (the length of the circle’s diameter) is
25,052.36; to obtain an order of magnitude in the length of P equal to one meter (keeping in
mind that each stone is 1.36 m in length) one would need an ϵ of 17.8 meters. Such is clearly
not the case, as the height differential between the northernmost point of the tomb’s exterior
wall and the southernmost point is far less than that. Thus, the elliptic surface still remains
extremely close to a circle, and a circle for all practical purposes. This can be verified by a
casual look at Kasta Hill’s google earth map.
The archeological team hasn’t produced for us the angle ψ, the North to South slope they
announced on November 29th, 2014; thus no exact calculations of the actual length P in
equation (3) can be obtained). One however inevitably wonders if there’s a connection between
the angle ω discussed earlier and the angle ψ. A hint is that they are identical, another
speculative statement, a conjecture one might say, which needs research. Thus, conjecture
(c.1) is:
Ψ = ω ? (7)
The slab’s seamless attachment. The seal between the slabs (both horizontally and vertically)
is so thin (due to the excellent work done on smoothing the surfaces of the marble slabs) it’s
almost invisible. However, it’s there and does take some space, of course. Over a set of 365
stones, it becomes of some import. Let’s call this total slits’ length , involving N* slits, each slit
obviously extending for a length of λ= /N*. At this point one must remark at the masons’
precision in crafting each stone’s side, with an appropriate depth-wise angle by almost half of a
degree on each side of every slab, thus fitting 365 slabs into 360 degrees’ circle (a precision
which robots accomplish these days). By this construction, the architect of the monument
accomplished a unique feat; each of the stones in all Rings of the perimeter wall correspond
both (in a set of stunning approximations) to a day (of a year) and a degree (in the 360-degree
approximate) circle. In attaining this dual function for each stone we can detect the desire of the
architect to build such a huge perimeter to the tomb/temple/monument, not in seeking grandeur
due to size.
The total length taken up by the slits between slabs must have been insignificant given the
perimeter’s length (about 497 meters using 365 slabs) but not zero. How its likely magnitude
affected the tomb’s exterior design will be addressed a bit more in the next section. Although
extra length is required to be covered due to the actual elliptical shape of the perimeter, the area
required to attach and seal all of the 365 slabs must have been compensated enough by the
leverage the ellipse offered for all four rows-rings that are circling the perimeter wall. More
precisely, the actual difference between P and P* must not have been such that the calculation
in equation (5) was materially affected given a relatively small λ in sealing and attachment
length between slabs. However, an exact estimation of ψ would allow someone to verify this
conjecture. Thus conjecture (c.2) is:
N*[z** + λ] -> N*[z*] ? (8)
It implies that the N* is materially unaffected by the horizontal slit size among the clad of 365
marble stones. The magnitude =365λ offers an estimate of the total length taken up to seal
adjacent marble stones in each Ring.
The entrance
The modular size of the entrance. We are told that the net (or clear) width inside the
monument/tomb, that is the distance between the two parallel marble coverings of the side walls
(internal clad, or ORTHOMARMAROSH), is approximately 4.50 meters. The tomb doesn’t
narrow or widen as one moves inside it, the width remaining constant in all three chambers.
This width implies that the monument’s entrance, leaves a perimeter length equal to 497-
4.50=492.5 meters. This is an opening corresponding to three stone lengths plus a quarter of a
stone. In daily counts, it represents three days and six hours.
The margin of error. By looking at it from a slightly different angle, the opening between the
two Karyatides’ marble covered rectangular prisms (their two bases) is 4.50-2x1.38=1.74
meters. This represents a distance of one slab (1.36 meters) plus .38 meters. The extra 38
centimeters correspond to about a quarter of a slab’s length and a length approaching the x*
base of the internal modulus, off only by about a few centimeters. Given that the 4.5 meters
width is an approximate size given to the public by the archeological team, it is safe to argue
that indeed the width of the tomb inside obeys the interior length modular size z* as well.
Moreover, the extra centimeters not accounted for may provide an upper bound to the quantity
mentioned earlier. Adjusting for this quantity per marble stone, the entrance would obey
exactly the three days and six hours space, as discussed.
The astronomical linkage expanded. The entrance of the monument is located at a point
slightly towards the SW; the exact location has not been announced. It is not known where
exactly North was pointing back then, but astronomical research reveals that the North Pole (the
axis of Earth’s rotation) has been wobbling over the centuries. For sure, 23 centuries ago the
North Pole was at a slightly different position than today’s northern most direction. Precisely
computing the North back then, would more accurately identify the specific days of the year this
entrance meant to depict, assuming that North was the beginning of the year, depicting January
1st. Of course, different interpretations might exist, if for instance East (or West or South) were
taken as the annual commencement or anchor day. Moreover, one could consider that any of
the points corresponding to the ancients’ ritualistic observances (Winter and Summer Solstice,
or the Spring and Fall Equinox) were the corresponding by the marble slabs starting or anchor
points for the 365 days string of marble stones. So the orientation of the entrance on the current
NE-SW axis could correspond to such astronomical alignment. Finally, the entrance of the
monument could point towards specific Constellations or high in magnitude and brightness stars
of particular import to the religious customs of the space-time under consideration, as they
appear on the sky at particular dates of the year. The topic of astronomical alignments in major
monuments of antiquity is a major subject of research, particularly for the Egyptian Pyramids of
the Giza Plateau, see for example Anthony Fairall’s treatise:
However a more complete treatment of this topic is left for further research. Such research must
combine this entrance alignment with other symbolic components found inside the tomb, like for
example the motion by the chariot in the mosaic of chamber #2. One might be motivated to
imagine pivoting specific stones on the outside wall, or specific points in the cornice’ oscillatory
pattern, the architect planned to erect statues corresponding to specific persons/deities of
import at those particular days of the year. A public calendar of sorts.
The date and time depicted. Assuming a clockwise motion and the current North as the
beginning of the calendar depicted by the 365 marble stones of the exterior wall (something
which seems to be the most likely case and the simplest scenario here), the entrance does
depict three days and the first six hours of that day in late June and early July. These
days and hours may had been connected to some religious observances 23 centuries ago, or to
the time of the year the occupant of the tomb died (or was born, or something of significance
happened to his/her life). Such symbolic interpretations however, are left to the interested
A first and preliminary effort was undertaken to de-code this very sophisticated monumental
structure. First, a modulus was produced for the tomb, which replicates precisely both the
interior and the exterior marble clad of the tomb’s walls. This three-dimensional modulus was
shown to be 1.36 meters in length, .70 meters in depth, and .38 meters in height (although the
last measure is in need of further confirmation). The grid is found to apply in both the interior’s
four layers of stone found in the Karyatides’ bases, and the exterior wall’s six Rings of parallel
and overlaying stone coverage.
Second, it was found that the Karyatides’s height (2.27 meters) relative to their base’ height
(1.40 meters) obeys the “golden section” rule.
Third, the central finding of this research, was the discovery that the architect of the tomb had
devised an overall tomb modulus which embeds in it an astronomical constant. Through its
exterior lengthwise modular manifestation the architect links the monument to the annual cycle.
Specifically, four Rings of marble stone on the monument’s exterior wall, each containing exactly
365 stones of marble connect this monument to the Earth’s heliocentric motion. This connection
is achieved with an extraordinary approximation. Specifically, the architect’s estimate (365.44
days) differs from the actual (365.22) by a stunning for the era in question miniscule variance.
Whereas, the exterior module shown on the monument’s marble clad wall was found to depict
days, the interior module (through the monument’s entrance) was found to depict a quarter of a
day (that is, a six-hour interval). The associated symbolism in combination with the entrance
orientation must be further explored.
Finally, the paper produced a conjecture (c.2): is there is a connection between the leaning of
the exterior wall by an angle ω and the slope of the ground picked up by angle ψ? It is
suggested that such connection does exist; when more data become available, it is expected
that the conjecture will hold. As the tomb is positioned at an angle to the radius at the entry
point, ρ, one further wonders about the connection of that angle to the two angles, ω and ψ.
The paper’s third and final conjecture (c.3) is:
ρ = ω + ψ ? (9)
This monument is a treasure trove of researchable topics, like all great works of Art and
Architecture. Its sheer size should not be what must attract attention to it. Rather, it was shown
here, it is its sophistication that ought to matter when analyzing it. Within a broader framework,
this 4th century BC monument at Amphipolis should rank high in the World’s hierarchy of
monumental Architecture. Complexity in its underlying design structure is unparalleled in the
history of temples, monuments, or tombs, as this edifice seems to combine all three categories’
functions through a complex structural code: its module.
Acknowledgments. The author wishes to thank Panagiotis Petropoulos and Effie Tsilibari for
translating parts of this paper into Greek and for very helpful comments.
Latest revision: 7:15 PM US EST, December 12, 2014
This paper was written during the month of November 2014. It is based on information made
public as of November 29th 2014.
... The second, and possibly the most extraordinary feature of that monument at Kasta tumulus was the calendar depicted on the tumulus circular exterior wall. The study in [1] documents the modulus used to build the monument. Lengthwise, the size of that modulus was 1.36 meters. ...
... All these observations, propositions, suggestions, hypotheses, preliminary conclusions and proofs are fully documented in the series of papers found in their various versions in references [1] through [5]. Below, some more in detail and slightly more extensive review of the totality of findings found in these references is supplied. ...
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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.
... To this author, no matter any objections raised (the two are very close to being one half of the other to be two independent units of measurement, according to this author) these two measures are of some importance nonetheless. In a study undertaken by this author [15], regarding the modular structure of the tomb at Kasta Hill (as revealed on the marble clad of its interior and exterior walls) two key measurements of the three-dimensional module embedded in that monument were found to be 1.36 meters and .72 meters. The close proximity of the two sets of measures is noted, especially their ratios (1.6/1.36=1.1765, ...
... On the other hand, it is inconceivable (barring extraordinary evidence, forthcoming in the future, pointing to the contrary) that the designers and planners of the original end-of-4 th millennium BC henge at Stonehenge were cognizant or planning for the ultimate shape of Stonehenge (be that the 60-sarsens, 15 Trilithons, or the Bluestones Phases) although not necessarily the holes which designate their positions. It is though possible that those who set up the circle with the sarsens could envision the final Bluestones apse, and of course the Trilithons Phase. ...
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This is an updated version of an earlier paper titled "Stonehenge, Durrington Walls, Newgrange: Monuments to the Egyptian Bull and Cow Cults and Origins of Innovation" by the same author. However, in this paper new material has been included. Thus, this paper marginally amends (in view of the summer 2016 retraction of the summer 2015 announcement regarding Durrington Walls) and considerably extends the previous paper. On September 19th, 2016 I revised the view presented in this paper regarding the date of Gobekli Tepe's oldest layer's construction with this paper: A new version of this paper is forthcoming to account for this update.
... He suggested that the architect or designer of the sculpture's foundation had as a "unit" the size of a "foot" that had a length equal to a contemporary measurement of .333m. The reviewer here must mention that in his work in [6], he identified as the modulus of the Kasta Tumulus tomb 1.36m (length) and .72m width. ...
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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.
Full-text available
This is the edited version of the July 17th 2015 article by the author, titled "The Mosaic at Kasta Hill's Tomb: the Algebraic Wonders of its MAIANDROS, including an Exact Ratio Embedded in it, and its Magnificent Cycles.
This is the edited version of the July 17th 2015 article by the author, titled "The Mosaic at Kasta Hill's Tomb: the Algebraic Wonders of its MAIANDROS, including an Exact Ratio Embedded in it, and its Magnificent Cycles.
Here, work is presented by the author on the complex meander frame found in the mosaic of Kasta tumulus. Its detailed mathematical structure is analyzed, and an effort is made to derive its embedded code. It turns out that this code is a sequence derived from the implicit cyclical pattern it produces in the advanced by all means meander structure of this tomb/monument. Results are also presented regarding the wave pattern framing the meander part of the frame itself. Associated with these elements of the meander frame, certain measures are also obtained, pointing to some advanced geometric and algebraic sophistication by the artist. Impurities, as well as some minor construction failures found in the mosaic are put in their proper perspective. Some general remarks regarding Chamber #2 at Kasta Tomb. Uncovered during the recent excavation of the tumulus at Kasta, Amphipolis, Makedonia, Greece, the mosaic floor of chamber #2, created a sense of awe and wonder upon its discovery in October of 2014. It followed the discovery of the two impressive KORES in Chamber #1 of the tomb. The excitement these two statues created upon their emergence out of the sealing soil of the tomb and from the depths of about 25 centuries in oblivion was immense. As if not to be outdone, however, chamber #2 (C2 from now on) had its own ace in the hole to pull on an
Full-text available
The paper analyzes the Stonehenge Phase 3 II Architecture, modular structure, and their connection to the sun-induced cast-off shadows from its sarsens, and the motion of the shadows over the course of a day and throughout the year. It establishes that a direct link exists between the size of certain shadows and the design of the monument. It further documents that besides the summer solstice sunrise alignment, the vernal and autumnal equinox alignment was a major one in the monument's design. Moreover, the paper demonstrates that the Trilithons sarsens ensemble's quasi-elliptical form is a type belonging to an extension of the Alexander Thom's classification of stone enclosures.
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