Ο ΤΡΟΠΟΣ – Ο ΧΡΟΝΟΣ – Η ΣΗΜΕΙΟΛΟΓΙΑ -ΤΟ ΜΕΤΡΟΝ -ΤΗΣ ΚΑΤΑΣΚΕΥΗΣ ΤΟΥ ΜΝΗΜΕΙΟΥ

**Δημήτριος Σ Δενδρινός**

Ομότιμος Καθηγητής, Σχολή Αρχιτεκτονικής και Αστικού Σχεδιασμού, του Πανεπιστημίου του Κάνσας, Lawrence, Κάνσας, Ηνωμένες Πολιτείες της Αμερικής. Στην κατοικία, στο City του Ormond Beach, Φλόριντα, ΗΠΑ.

Περίληψη

Σε αυτή την εργασία εξετάζεται μια απροσδόκητη ανακάλυψη, σχετικά με την αρθρωτή δομή του οικοδομήματος που αποκαλύφθηκε στον λόφο Καστά στην Αμφίπολη .

Όπως όλα τα μεγάλα κτίρια της αρχαιότητας, αποδεικνύεται εδώ με τη χρήση σχετικά απλών αλγεβρικών σχέσεων, ότι αυτός ο ναός / μνημείο / τάφος χτίστηκε με μια υποκείμενη σπονδυλωτή δομή που συνδέει το περιεχόμενο (τους ενταφιασμένους) στους Ουρανούς, και συγκεκριμένα στο νυχτερινό ουρανό ,όπου μπορεί να γίνει η (τότε) γήινη παρατήρηση

Έχει ανιχνευθεί η χρήση ενός κανόνα ,μια αστρονομική σταθερά, ενσωματωμένη στο μέτρο του σχεδιασμού του μνημείου αυτού . Αυτό το μέτρο εμφανίζεται τόσο στον εξωτερικό χώρο όσο και στην με μάρμαρο κάλυψη των εσωτερικών τοίχων. Το γεγονός αυτό δείχνει ότι η θέση της εισόδου του ναού / μνημείου / τάφου προσδιορίζει μια συγκεκριμένη ημέρα του έτους και ακόμη μεγαλύτερη ακρίβεια, ένα ιδιαίτερο, 6-ωρο εκείνη την συγκεκριμένη ημέρα.

Επιπλέον, η βορειοανατολική (ΒΑ) είσοδος ως προς τον άξονα Νότου-Δύσης μπορεί να καταδεικνύει μια αστρονομική ευθυγράμμιση. Από αυτή την άποψη, αυτό το οικοδόμημα είναι ένα μοναδικό αρχιτεκτονικό δημιούργημα. Στον αρθρωτό σχεδιασμό του, το σπονδυλωτό συγκρότημα, μεταξύ των πολυάριθμων άλλων ,με τις ίδιες αρχές που βρέθηκαν σε αυτό, ο κανόνας «χρυσή τομή» φαίνεται να ρυθμίζει το ύψος στις Καρυάτιδες και στη βάση τους.

Ωστόσο, η μονάδα αυτή του ναού / μνημείου / τάφου στον λόφο Καστά , το οποίο απασχολείται ο έμπειρος αρχιτέκτονας ,ήταν πολύ πιο περίτεχνο και πολύπλοκο από μια αρχή συμμετρίας με βάση έναν απλό κανόνα.

Η εργασία αναρτάται εδώ αυτούσια στην Αγγλική γλώσσα στο τέλος μπορέιτε να την διαβάσετε σε μετάφραση.

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: [email protected]

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Abstract

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.

Introduction.

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.

Discussion

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:

http://www.theamphipolistomb.com/wall

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:

http://www.theamphipolistomb.com/caryatids

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* IS THE INTERIOR LENGTH MODULUS IN TWO DIMENSIONS.

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* ≈ 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 (.44×1.36=.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 ᴧ, involving N* slits, each slit obviously extending for a length of λ=ᴧ/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 ᴧ=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 4.50-2×1.38=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 ᴧ 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: http://www.antiquityofman.com/Fairall_Orion_precession.html

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 reader.

Conclusions.

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.

ΣΤΗΝ ΣΥΝΕΧΕΙΑ ΕΙΝΑΙ Η ΕΡΓΑΣΙΑ ΜΕΤΑΦΡΑΣΜΕΝΗ ΣΤΑ ΕΛΛΗΝΙΚΑ ΜΕ ΤΗΝ ΒΟΗΘΕΙΑ ΔΙΑΔΙΚΤΥΑΚΗΣ ΜΕΤΑΦΡΑΣΗΣ ΚΑΙ ΣΧΕΤΙΚΗ ΕΚΦΡΑΣΤΙΚΗ ΔΙΟΡΘΩΣΗ

ΟΙ ΦΩΤΟΓΡΑΦΙΕΣ ΤΟΠΟΘΕΤΗΘΗΚΑΝ ΑΠΟ ΤΟΝ ΑΡΧΑΙΟΓΝΩΜΩΝΑ -Η ΜΕΤΑΦΡΑΣΗ ΠΙΘΑΝΟΝ ΝΑ ΠΕΡΙΕΧΕΙ ΕΚΦΡΑΣΤΙΚΑ ΛΑΘΗ ΚΑΙ ΤΕΧΝΙΚΕΣ ΑΣΑΦΕΙΕΣ ΑΠΟ ΤΟΝ ΤΡΟΠΟ ΤΗΣ ΔΙΑΔΙΚΤΥΑΚΗΣ ΜΕΤΑΦΡΑΣΗΣ- ΓΙΑ ΑΚΡΙΒΗ ΚΑΤΑΝΟΗΣΗ ΤΩΝ ΕΝΝΟΙΩΝ ΜΠΟΡΕΙΤΕ ΝΑ ΤΙΣ ΔΙΑΒΑΣΕΤΕ ΣΤΗΝ ΑΓΓΛΙΚΗ ΣΤΟ ΑΝΩΘΕΝ ΚΕΙΜΕΝΟ

Εισαγωγή.

Το έγγραφο αυτό αναφέρει μια ανακάλυψη που βασίζεται σε μία αστρονομική σύνδεση ενσωματωμένη στην αρθρωτή δομή του πρόσφατα αποκαλυφθέντος τάφου-μνημείου στην Αμφίπολη, στο βόρειο τμήμα της Μακεδονίας, στην Ελλάδα. Το μνημείο στον λόφο Καστά έχει μια συμμετρία σε αυτό, πολύ πιο περίπλοκη από ό, τι μια απλή αρχή με τον «χρυσό κανόνα». Ενσωματωμένο σε διαρθρωτικές κατασκευές ,ο «κωδικός» του μνημείου αυτού είναι πολύ περίπλοκος, και αυτή η εργασία είναι μια πρώτη προσπάθεια να την αποκωδικοποιήσει.

Με ακόμη και απλή ανάλυση των σπονδυλωτή δομή του, πολλά θέματα αναδύονται και χρήζουν περαιτέρω μελέτης. Όλοι επισημαίνουν το επίπεδο των μαθηματικών και της μηχανικής πολυπλοκότητας που χαρακτηρίζει τον αρχιτέκτονα που έχτισε αυτό το τάφο. Θα πρέπει να διέθετε προηγμένη γνώση της άλγεβρας, τις δεξιότητες μηχανικής, ευφάνταστη αρχιτεκτονική σε αρχές και αστρονομικές γνώσεις. Τέτοια επιδέξια συγχώνευση της σύλληψης και της τεχνικής μηχανικής σπάνια συναντώνται στη μνημειακή αρχιτεκτονική.

Η ενότητα στο συγκρότημα αυτό που χρησιμοποιείται από τον αρχιτέκτονα του μνημείου, τόσο στο εσωτερικό όσο και έξω από αυτό, το θαύμα της αρχιτεκτονικής και της μηχανικής χρειάζεται προσεκτική ανάλυση, καθώς και μερικά από τα ενσωματωμένα στοιχεία και τις σχέσεις τους είναι ακόμη κάτι που πρέπει να ανακαλυφθεί . Η εργασία αυτή εξετάζει και προσδιορίζει ακριβώς μόνο ορισμένα συστατικά του συμπλέγματος τριών διαστάσεων της μονάδας, η KANABOS (ή πλέγμα) οι μαθηματικές και όχι μόνον σχέσεις που χρησιμοποιούνται από τον αρχιτέκτονα, δηλαδή η ενότητα «μήκος – πλάτος της, αλλά όχι το ύψος του (εντελώς) – λόγω έλλειψης στοιχείων. Έτσι, για τα συμπεράσματα θα χρειαστεί περαιτέρω επαλήθευση όταν θα είναι διαθέσιμα περισσότερα δεδομένα.

Οι αναμφισβήτητες σημαντικές συμβολικές πτυχές του συνόλου της μονάδας (το μέτρον ) δεν απευθύνονται σε συγκεκριμένη μία και μόνον έννοια εδώ, αλλά αφήνονται στον ενδιαφερόμενο αναγνώστη.

Μερικές σημειώσεις που πρέπει να γίνουν κατά την έναρξη.Το μνημείο έχει μία πολύπλοκη σπονδυλωτή (δίκτυο) δομή. Η έννοια της «μονάδας » είναι το κλειδί για τον προσδιορισμό της τρισδιάστατης δομής τόσο αυτής καθαυτής της «μονάδας» αλλά και του κτίσματος . Μας επιτρέπει να διεισδύσει κάποιος σε προσέγγιση για τους στόχους του αρχιτέκτονα στο σχεδιασμό αυτού του οικοδομήματος.

Έχουμε αρκετά στοιχεία από την αρχαιολογική ομάδα προς το παρόν που μπορεί να καθορίσει κάποιος με ακρίβεια το μήκος της μονάδας , αρκετά δεδομένα για να προσεγγίσει το πλάτος του, αλλά όχι αρκετά για να υπολογίσει την ακριβή εφαρμογή της για όλα τα ύψη μέσα στο μνημείο, με το υλικό που έχει δημοσιοποιηθεί. Είναι προφανές πλέον, ως αποτέλεσμα αυτής της μελέτης, ότι η μονάδα του μήκους χρησιμοποιείται τόσο στο εσωτερικό του τάφου καθώς και έξω, και ενεργεί ως ο ακρογωνιαίος λίθος στην κατασκευή του μνημείου. Η ακριβής εκτίμηση στο μήκος του δικτύου μας επιτρέπει να εκτιμήσουμε την αισθητική του αρχιτέκτονα, το δυναμικό συμβολισμό, καθώς και τους γενικούς στόχους κατασκευής του.

Μια επίμονη ερώτηση, δεδομένου ότι η ανακάλυψη του τάφου, ήταν γιατί ο αρχιτέκτονας κατασκεύασε έτσι ένα τεράστιο ταφικό ανάχωμα με περίμετρο 497 μέτρων, μόνο για να στεγάσει ένα οικοδόμημα μόλις 4,5 μέτρα πλάτος και περίπου 25 μέτρα; Γίνεται εμφανές από αυτή τη μελέτη ότι ο κύριος στόχος του αρχιτέκτονα δεν ήταν τόσο πολύ να επιτύχει το μεγαλείο με την οικοδόμηση με μια τέτοια γιγαντιαία από κάθε άποψη μνημείο. Αντ “αυτού στόχος του ήταν να σχεδιάσει μια δομή που θα αναπαράγουν τον ετήσιο κύκλο και μέσα σε αυτό να εντοπίσει ένα συγκεκριμένο ημερολόγιο ημερομηνία και την ώρα της ημέρας , πιθανόν να σχετίζονται με μια σημαντική εκδήλωση στο χρόνο ,για το άτομο αυτό που κατασκευάστηκε το μνημείο-τάφος και αυτό είχε ως στόχο. Το πιο πιθανό, το τελετουργικό στοιχείο μιας επαναλαμβάνεται ετησίως ως ιεροτελεστία ήταν βασικός στόχος του αρχιτέκτονα.Υπάρχει ένα εντυπωσιακό στοιχείο που είναι η συμμετρία τόσο στο εσωτερικό όσο και έξω από το μνημείο. Ωστόσο, συμμετρία δεν ήταν ο απώτερος στόχος του αρχιτέκτονα για την οικοδόμηση αυτής της δομής. Ήταν κάτι πολύ πιο σύνθετο και περίτεχνο από μια συμμετρία. Ήταν μια προσπάθεια να δημιουργηθεί ένα φόρουμ για την ετήσια τήρηση ενός γεγονότος, ατενίζοντας στο συγκεκριμένο σημείο στον ουρανό σε μια συγκεκριμένη ημερομηνία και ώρα.

*ellinondiktyo*