Μια εξαιρετική μελέτη του αρχιτέκτονα, ομότιμου καθηγητή στο πανεπιστήμιο του Κάνσας των ΗΠΑ, Δημητρίου Δενδρινού, σχετικά με τα κρυμμένα μυστικά του ταφικού μνημείου
Η εργασία στα αγγλικά.
The Mosaic of Kasta Hill’s Tomb
The algebraic wonders of its MAIANDROS, including an Exact Ratio embedded in it and its Magnificent Cycles
Dimitrios S. Dendrinos
Emeritus Professor School of Architecture and Urban Design, University of Kansas, USA. In residence at Ormond Beach, Florida, USA.
July 17, 2015
Abstract
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 international public to an extent fascinated and stunned by these discoveries. A magnificent mosaic was found to decorate C2’s floor. The relatively small in size (4.5 meters wide, 3 meters long and about 6.5 meters from floor to the top of the arched ceiling) chamber proved to be huge in importance. It also proved to be the penultimate chamber in a tomb full of unexpected findings, in a corridor of history packed with unexpected spaces and objects. Further deep laid the space everyone among the hundreds of millions of people closely following the excavation was eager to mentally enter – the funerary chamber and the dead person(s) buried there.
The stir C2 created mostly centered upon its spectacular at first view mosaic floor, a floor designed for not many to step on or see. On that mosaic floor at its center, a magnificent rendition of the abduction of Persephone by Hades, was offered. Under Hermes’ watchful eyes, while riding on a chariot pulled by two galloping horses in the direction from East to West, the soul was to embark on a ride to eternity.
Naturally, like all things associated with the findings at Kasta Hill’s tomb, this mosaic produced a flurry of interpretations and had its own share of controversy. Who are the real two persons on the chariot was the question that attracted most of the attention. Left alone in that discussion was the Hermes figure. The view that indeed Hermes was the witness to whatever was taking place and by whomever was on that chariot wentunchallenged. Very few elements about this tomb in general, and this mosaic in specific are beyond dispute, and the Hermes figure was one of them. It was and has been accepted like a mathematical or physics related “constant” by almost everyone who ventured into the public eye and expressed a viewpoint on the subject.
Hermes may have been a point of agreement among analysts, but little else was. As the sealing soil was being removed, a round hole in the middle of the mosaic was found, void of all pebbles used by the artist that created this beautiful Persephone-Hades-Hermes rendition. The archeological team informed the public that a large number of these pebbles had been recovered, and that the mosaic could be almost fully restored. Of course the presence of such a gap just in front of another riddle, abroken down, double-leaf heavy marble door leading into what proved to be the final (funerary) chamber inside the Kasta tumulus caught a fair amount of the public’s wonder and bewilderment.
But this was not all. A bunch of all kinds of other topics preoccupied the public’s attention, as this mosaic was revealed in C2. From the artwork per se of the mosaic (pebble based not done with tesserae), to the female figure’s attire, to the apparent “blue left eye” of the horse in the foreground, C2’sfloor had its field day in the sun. More than two and a half millennia after its construction, and at least two millennia in oblivion and “eternal” peace, the archeologists’ axe suddenly and permanently put an end to that “non-existent” for us till then “existence”.
By the time C2 was being gradually and slowly cleared up from the sealing soil that the whole tomb was filled, speculation as to what lay beyond the broken marble door was reaching a crescendo. The World’s interest was at its peak as the excavation phase entered its apparent final stage, C3. Amphipolis, Kasta Hill and its contents were already in the headlines; but as the archeologists advanced from C2 to C3 the movement made TV prime time news not only in Greece, but as far as the US and China, and in fact the World over. The bubble of public interest was inflated to the tilt. But that bubble was about to burst, and its echo to reverberate a year later, down to this day. The bad news about C3’s condition and perplexity surrounding the bones found there, caused international interest to dramatically wane.
Major questions and minor insights.
Who built this monument there in Amphipolis, for whom, when, and how are of course the primary running questions in search of an answer. Most likely, they will not be settled for decades to come. But there are also secondary running questions. Like for example, what is the perimeter wall all about? What about “the tomb’s sealing with soil”? Who did it, how, for what purpose and intent? What was exactly the real effective (immediate and surrounding) environment of this tomb? What were the specific socio-economic conditions that surrounded its existence from birth to death? What is the real symbolic meaning of all these various items in the tomb, like the Sphinxes, the KORES, the mosaic? What is the interior and exterior architectural code used to build this monument or tomb or temple? All these are questions which will need possibly centuries to be fully answered.
And then there are fine and detailed type questions, one might classify them as “tertiary”, associated specifically (by no means exclusively) with C2. They consist of details such as: what’s the exact nature of the three fragments of marble found in C2, with faded remnants of some painting? How do the faded painted representations on the fragmented marble pieces relate to the rest of the items in C2, and especially its mosaic floor? The initial location of this marble coverage was on top of the beam supporting the arch transitioning from C1, with the two KORES, to C2 with the mosaic floor. Does this have any meaning? Since apparently little was left to chance in the construction of this monument, these become sources of more pondering and derivative questions.
How exactly did the marble door shatter the way it did, when and by what agency? How are the door’s dimensions related to the dimensions not only of C2, but to the whole tomb? Numerous more questions arise as one moves along the 25 or so meters of corridor-space, or finds him/herself inside the 3-meter long C2. Focus on the major ones is of course justified and expected. However, the devil is always in the detail; so, by addressing and answering some minor ones, insights into the major questions will undoubtedly be gained. And as it will be seen here, that may be exactly the case with the mathematics and construction details of the complex meander at the floor of C2.
So, from the potentially thousands of questions that cross anyone’s mind in reference to this tumulus, some obviously grand in scale, some minor at first glance, in this paper just a few will be addressed of the second type. They will cover just a small band in the spectrum of questions potentially to be asked, even about C2. For sure, the questions posed here haven’t attracted nowhere near the attention other questions have so far. In C2, the public’s inclination has been to speculate as to the persons depicted on the mosaic’s central representation, the shattered marble door, the huge almost perfectly circular gap in the mosaic, and the faded paintings on the marble fragments. But none of these questions will preoccupy the analysis here. Instead, attention will be focused on the frame this representation was enclosed.
The double meander frame and the waves that frame it.
Little did any one notice or publicly comment about the frame of that mosaic. However, just a glance on it would impress anyone with its beauty, complexity and size in reference to the total area taken up by the floor’s whole mosaic. It was by all means an astonishing and unique double MAIANDROS structure. But for a variety of reasonsmostly having to do with politics, and little to do with Art, Archeology, History, Architecture, Mathematics, or the fine points of mosaic representation of that era, next to nothing was said about it from the archeological team. The end result was, that it seemed to attract very little attention, if any at all. Just like the Hermes in the rendition itself.
Figure 1. The mosaic floor, in chamber #2 at Kasta Hill’s Tomb, with the almost circular gap with the missing pebbles. Source: Greek Ministry of Culture.
This double (to be exact, deployed in two levels) meander structure, consists of a set of individual counter-clockwise spinning meanders. Each meander has linkages (through its four tentacles) to four neighboring, at two levels, meanders. In the frames hown in Figure 1, these meanders move collectively in interlocking sets of waves. That wave motion is further aesthetically enhanced by a set of two (inside and outside) secondary frames on the meanders’ main frame, which depict actual waves moving simultaneously in both directions, by a play on color tone. After the analysis finishes dealing with the main frame, it will turn its focus on these waves as well.
Searching for the implied sophistication in mathematics of the double meander frame, may guide the analyst into its inner code. Thecode (shown here to exist in the form of two key sequences) embeds in it an eternal continuous 2-way (directionally) as well as up-down (2-level) motion of the meander in the frame. The code has a story to tell. In it, both the strengths of this extraordinary meander and frame are included; but in its design, also its inherent shortcomings, the cause of its impurities, are also revealed.
The double meander.
Next, the untangling of the mathematical complexity of this meander frame will be attempted, recognizing that there are many ways one could possibly approach the algebra of this representation. The one chosen here is simply a matter of personal preference.
Some preliminary comments and algebraic designations are in order. In the comments below, the term “pebble” will be used. A more mathematically appropriate term would be “tile”; but I avoided using this special term, because the expression “tiling of a plane” has a very specific meaning in mathematics – which as it will be demonstrated here – doesn’t seem to hold. Thus, I prefer the term “pebble” to “tile” for the time being, knowing that the mathematics to be explored further down might generate a branch of the term “tiling” specifically addressing the complex mathematics of “framing” any given mathematical configuration, into a specific shape, in this case that of a frame in the shape of a rectangle. In the mathematics of the theory of tiling, the plane “tiled” is unlimited, with no impurities in the repetition of the tiling pattern. Here, the type of “pebbling” the artist presented has impurities, which will be extensively addresses in this paper; these impurities do not allow for a boundless or arbitrarily limited tiling given the pattern chosen.
The actual pebbles used are obviously not all of the exact same size Pebbles form lanes (thick lines in effect) that can be broken down as having squares in their constitution. All these (approximately equal in size) squares have a dimension to be designated as variable X of some unit length (L), a length in centimeters to be determined by the available information from the record. The pebble based constituent square is the primary meander’s unit (with an area A=LxL). That square will from now on be referred to as a “pebble”, although in actuality not a single pebble produces this square, but rather a number of them. It could be considered as a “tile”. As already mentioned though, in view of the fact that the tiling of a plane is a specific term used in mathematics, and here we are not directly dealing with tiling of a whole surface, the term “pebble” is usedinstead.
These “pebbles” are either dark blue or white, both intended to have equal size, although in real they may randomly differ in size by a slight variance, which for all practical purposes can be assumed to approach zero. It remains as a statistical hypothesis to be tested by field work. The specific patterns they form, as well as the overall pattern they create are unambiguously clear.
Even though they actually slightly differ in size, all these “pebbles” in toto produce a coherent overall pattern, consisting of dark blue and white thick lines or lanes. White lines are supposed to provide the foreground, while blue lanes or simple squares in some instances form the background in the overall pattern.
Finally, in the presentation which follows, the mathematical formulations have been kept to a minimum. No formal mathematical statements and proofs are given, since the primary audience of this paper is not mathematicians but the general public with some knowledge of algebra.
On the mathematical aspects of this extraordinary meander, it is emphasized that the analysis presented below shows the close connection between algebra and geometry, in its full extent. The reader must be careful in distinguishing between “rows” and “columns” in their geometric representation on a 2-dimensional plane, from the algebra of a 2-dimensional graph.
The basic pattern (BP) the basic square pattern (BSP), the two constituent patterns (CP) involved in the BSP, and their embedded algebraic functions on the plane.
Let’s designate as Basic Pattern (BP) the overall pattern of the mosaic frame that includes meanders and chess pieces, in two levels, the constituent elements (CE) of this pattern. This CE can be analyzed either as the sequence S1: top [meander – chess piece], bottom [chess piece – meander]. Or sequence S2: top [chess piece – meander], bottom [meander – chess piece].
Proposition 1. The frame of the mosaic floor at Kasta Hill consists of whole patterns.
Proposition 2. The sequence in effect in this specific frame is the S2. It will be argued that S1 doesn’t “turn” either up or down, to continue the pattern on the sides of the frame.
Within BP we can distinguish two overlapping to an extent forms, the basic square pattern (BSP) and the Basic Repeated Pattern (BRP). Both are shown in Figure 2. BRP is the pattern of the meander frame, which contains within it the continuous flow so characteristic of a meander. It will be shown here later how this flow takes place, and generates the cycle embedded in the particular meanderused here.
The size of this BSP is 17 lengths (L) horizontally by 17 lengths vertically on the plane, forming a perfect square of 289 dark blue and white “pebbles” within it. The two colors form two distinct constituent patterns (CPs) one being the elementary meander formed by the white pebbles, whereas the other is formed by both white and dark blue pebbles. Mathematical details on these two types of CPs will be supplied in a bit.In Figure 2 the 17 pebble lengths along the x and y axisare shown. The frame’s BSP contains two pairs of two CPs – a pair of GAMMADIA, to be designated as CP1, and a pair of a chess board parts each consisting of a three by three set of squares, to be designated as CP2. A GAMMADION is a classical Greek pattern of four (capital) letters Gamma, joined at their heads and forming a counter-clockwise continuous flow (motion) pattern (a symbol of eternity, in the form of continuous thus eternal flow).
Each CP constitutes a square itself, of dimensions 8.5Lx8.5L, and it takes up exactly a quarter of the area of the BSP. That is exactly 72.25 expressed in unit length (L) squared. Note that these four squares, each containing a CP, share in area (half) some pebble squares (those at the four borders of the four CPs). There are 4×8=32 such neighboring pebbles. The pebble sitting at the very center of the BSP has a unique property among all 289 pebbles; it is shared equally by all four CPs. These special neighboring (border) pebbles will be designated as N(i) [where i=1,2,…I (I=32)]; whereas the very central one as K. A breakdown of these special pebbles by color (white and dark blue) will be given and examined later.
Figure 2. The 17×17 square of an S2 type BSP and its key axes on a 2-d plane. The x’ and y’ axes are at right angle; the x* and y* axes are symmetry axes of the BSP. The BRP in its horizontal motion, omits either the first [or the last (17th)] column of the BSP. Equivalently, the BRP in its vertical motion omits either the first or the 17th row. Both center C, and the y* axis are key elements of this double meander Basic Pattern (BP), which contains a pair of sections from a chess board , and a pair of meanders. Source: the author.
Theorem 1,Rotations: (a) The CP of Figure 2 contains within it a (plus or minus) 180-degrees rotation of all its parts (CPs and pebbles) about the origin C;(b) each CP1containing the full area of all its neighbors N(i)and K, (that is, a square of 9×9=81 pebbles) can be independently rotated by (plus or minus) 90 degrees about an origin located at the very sub-center of each of the pair of CP1s;The four sub-centers in question are found at coordinate pairs {CP1: (4.5L, 4.5L), (12.5L, 12.5L); CP2: (12.5L, 4.5L), and (4.5L, 12.5L)}; (c) Each of the pair of CP2s containing the full area of all its neighbors N(i) and K (i.e., again 9×9=81 pebbles) can be independently rotated (plus or minus) 180 degrees. The fact that CP2s can only rotate 180 degrees, and not 90 degrees as the CP1 can, is the single factor that contributes to their static nature. The proof for all these rotations (a, b, c)is left to the interested reader (and it’s not that difficult to obtain). The subject of ‘rotations” is of some import here, because it shows that none of the BRPs were rotated, implying that all BRPs in the frame have the “orientation” of Figure 2.
Theorem 2, Symmetries: The clean (i.e., not inclusive of neighboring N(i) pebble shares) pairs of both CPs(i.e., areas of 72.25 L squared) are symmetric with reference to point C.
Now the focus will turn onto the flow, or to be more exact, to the cycle found in the CP, as well as the CP1. It is noted again that from the two CPs within the BSP, the chess pattern CP2 is static. The dynamics in the frame giving rise to the flow within itself and in conjunction with CP2 (the double flow, or the double meander of the frame’s description) is of course the meander of CP1.Each CP1 is here flowing directly into two neighboring CP1s and two distant CP1s around a couple of neighboring CP2s with all its four tentacles. In unison with its own flow within each GAMMADION, inengulfing the chess piece, shapes the continuous flow pattern of the overall meander frame of this exquisite and unique mosaic floor. The CP1 in effect has within its motion a complex general (overall) movement along the x-axis, as well as the y-axis. This pattern is encountered for the first time in any representation (in mosaic or other contexts) that contains a meander. Although the pattern of alternating CP1 and CP2 has appeared in a Pella mosaic, the Kasta Hill mosaic frame takes up two levels, whereas that in Pella is a single level mosaic frame. Usually a single meander is shown, moving along a single overall general direction. It should be here noted that this complexity in flow movement came at a cost. It produces obvious impurities in the overall meander pattern of the frame. Some of these impurities will be discussed here.
Before we turn into the detained study of the two patterns, the BSP and the BRP, a note on the S1 vs S2 sequencing types in BP is needed. Whereas S2 does move (with impurities), making a turn up or down, S1 doesn’t afford to do so. So, we can state as a Lemma 1 the following: S1 doesn’t turn, S2 does. Again, the proof is left to the interested reader. It has something to do with the location of one of the Gammadion’s branches in the CP. We can also state as a Lemma 2, that within the frame, there are no BSPs but only BRPs.
The cycle’s dynamics and the “pebbling” of the overall meander space.
No matter how informative (and spectacular in its internal structure) the BSP of 17×17 (289 pebbles in all) squares (each of a unit length L) is, it is not the key here to unlocking the mystery code of this design pattern. The key is(BRP) that part of the BSP which is repeated horizontally and/or vertically along the frame.Only the first (or last – it is not of import) 16 (out of the 17 in total) columns or rows(as shown in Figure 2) do repeat. In its vertical (up or down) motion BSP’seither top row, or the bottom one (it isn’t of mathematical import), is omitted in the BRP. In its horizontal (right or left) motion BSP’s either first or last column are omitted.
BRP thus is a rectangle, 16×17=272 pebbles (or squares of unit length L). Of course, it doesn’t enjoy the properties of the BSP. Nonetheless, it has some interesting properties of its own. To analyze however its properties, and discover the cycle embedded in it, some accounting of the pebbles it contains is necessary. But before we take on the tedious task of accounting for BRP, a closer look at the accounts of the BSP is necessary. One may ask here, why did the artist settle on a pattern unfolding on a (16×17) rectangle (which doesn’t afford the property of “tiling” that a square pattern does at least for a form of a tile of some interest to the subjects here in Kasta Hill) is an interesting question to which we will attempt to offer an answer.
The accounting of pebbles in the BSP, the sequences and the Code.
Moving horizontally in the x, y plane of Figure 2’s BSP, along the x-axis, and counting the number of white (W) and deep blue (B) “pebbles” encountered in each of the 17 rows, from pebble row 1 to pebble row 17 (moving vertically along the y-axis) one has:
Table 1. Accounting of the White (W) and blue (B) pebbles in the BSP.
Summing up along columns. Summing up along rows.
_________________________ ________________________
R1 W: 16 B: 1 C1 W: 15 B: 2
R2 W: 3 B: 14 C2 W: 3 B: 14
R3 W: 13 B: 4 C3 W: 13 B: 4
R4 W: 8 B: 9 C4 W: 8 B: 9
R5 W: 12 B: 5 C5 W: 12 B: 5
R6 W: 8 B: 9 C6 W: 8 B: 9
R7 W: 13 B: 4 C7 W: 13 B: 4
R8 W: 2 B: 15 C8 W: 3 B: 14
R9 W: 15 B: 2 C9 W: 15 B: 2
R10 W: 2 B: 15 C10 W: 3 B: 14
R11 W: 13 B: 4 C11 W: 13 B: 4
R12 W: 8 B: 9 C12 W: 8 B: 9
R13 W: 12 B: 5 C13 W: 12 B: 5
R14 W: 8 B: 9 C14 W: 8 B: 9
R15 W: 13 B: 4 C15 W: 13 B: 4
R16 W: 3 B: 14 C16 W: 3 B: 14
R17 W: 16 B: 1 C17 W: 15 B: 2
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From the above Table 1, a few things become immediately quite clear. There is a symmetry in these counts. A look first in the row totals, summed up by columns. With center at row #9’s counts (W-15, B-2) the counts on both whites and blues repeat in two complementary sequences, a “white rows sequence”(Wrs) and a “blue row sequence” (Brs) as follows (keeping of course in mind the conservation condition, that for all rows: Wrs + Brs = 17).
The Wrs is [moving away from the central (#9) row’s counts, either up or down]: (2, 13, 8, 12, 8, 13, 3, 16). The corresponding Brs is of course its complementary (to 17) sequence: (15, 4, 9, 5, 9, 4, 14, 1).It is recalled that the central (#9) row with the totals over all 17 columns contains in its middle the horizontal line (y*=8.5L) of Figure 2. This is a line (y*=8.5L) with a special interest, as it will be seen later in the text.
Now, a look is taken into the equivalent column totals over all rows, to be designated as Wcs and Bcs (as obtained by summing up along 17 rows). With center at column #9’s counts (W-15, B-2) – identical in counts as those in row #9, and depicted by the vertical line at x*=8.5L in Figure 2 – we have the corresponding Wcs: (3, 13, 8, 12, 8, 13, 3, 15), while the corresponding Bcs is of course a sequence with each cell the complement of the corresponding Wcs: (14, 4, 9, 5, 9, 4, 14, 2).
Comparing the two sets of two sequences each, [Wrs with Wcs, and Brs with Bcs] one sees immediately the small in number but key for the final result overall differences: the first and last numbers in these two sets differ by just a single square (2 versus 3 and 16 versus 15), while the rest six pairs are identical (13, 8, 12, 8, 13, 3). This is the “core sequence” that links the two basic square patterns (BSP). This is the FRAME’S one part of its CODE. One might ponder the nature of these numbers, to obtain an insight as to the artist’s motivation to settle on this specific pattern.
The motion in the whole pattern is attributed to just the minor (by a single square) difference in the marginal (beginning and ending) row and column totals. This small variation is the cause of the impurities in the overall Basic Pattern (BP), preventing it from being able to perfectly “tile” the (unlimited or arbitrarily bounded) plane.
In reference to Table 1, it is noted that for the mathematically inclined reader, the entries of Table 1 can also be represented in the form of two matriceswith 17×2 cells each. Designating as Wr(m), [where m=1,2…M, with M=17, and Br(m), where again m=1,2,…M and M=17] one has the cells with column totals for the white and blue pebbles correspondingly. The other matrix consists also of 17 cells with row totals for these two types of pebbles. Specifically, Wc(n), [where n=1,2,…N, with N=17] are the entries for the white pebbles summed up over N (i.e., 17 rows); whereas, Bc(n) [where n=1,2,…N, with N=17] are the entries for the blue pebbles. Notice here, in the BSP, M=N=17.
In terms of the symmetry conditions identified earlier, one notes that:
Wr(m=9) = 15 Br(m=9) = 2
Wr(m=8) = Wr(m=10) = 2 Br(m=8) = Br(m=10) = 15
Wr(m=7) = Wr(m=11) = 13 Br(m=7) = Br(m=11) = 4
Wr(m=6) = Wr(m=12) = 8 Br(m=6) = Br(m=12) = 9
Wr(m=5) = Wr(m=13) = 12 Br(m=5) = Br(m=13) = 5
Wr(m=4) = Wr(m=14) = 8 Br(m=4) = Br(m=14) = 9
Wr(m=3) = Wr(m=15) = 13 Br(m=3) = Br(m=15) = 4
Wr(m=2) = Wr(m=16) = 3 Br(m=2) = Br(m=16) = 14
Wr(m=1) = Wr(m=17) = 16 Br(m=1) = Br(m=17) = 1.
It is also noted, that a further but more limited symmetry exists in so far as:
Wr(m=4) = Wr(m=6) = Wr(m=12) = Wr(m=14) = 8
And, of course (due to the conservation condition W + B = 17):
Br(m=4) = Br(m=6) = Br(m=12) = Br (m=14) = 9.
Similarly for the pair:
Wr(m=3) = Wr(m=7) = Wr(m=11) = Wr(m=15) = 13,
Br(m=3) = Br(m=7) = Br(m=11) = Br(m=15) = 4.
The accounting of pebbles in the BRP.
Analyzing now the equivalent table for the basic repeated pattern (BRP), we include only the repeated 16×17 = 272 squares of the double meander. This table contains only 16 columns of pebble squares (one less than the BSP), and 17 rows (as does the BSP). The conservation conditions now differ, they are still whites+blues=17 on the 16 column totals from all rows (since we still have 17 rows), but it’s now whites+blues=16 on the row totals, as we now have only 16 columns. One can obtain a transformed table, such that its section dealing with summed over all rows column totals, up till column 16 (again, there isn’t a column 17 in the BRP) are identical to those of Table 1. Where Tables 1 and the transformed table will differ, is the column totals for each of the 17 rows it contains. For simplicity and economy of space, that table isn’t provided; instead the difference between the two is given in Table 2. Also for reasons of avoiding significant duplication the case of a vertical motion in the BRPs will be omitted.
In matrix notation, one has the following matrices and cells, where the cells and sequences will be designated with lower case w and b, for simplicity, as (wr, brand wc, bc) for the cells, and (wrs, brs) and (wcs, bcs) for their sequences. In matrix form, this BRP table has entries which could be represented by a pair of matrices with16x2 cells (as there are now only 16 repeated columns with their row totals). Specifically, [wr(m) where m=1,2,…M and M=17] are the row cells of the new matrix for the white pebbles; whereas, for the blue pebbles we have cells given by in the case of the 17 rows (which contain sums over columns) [br(m) m=1,2….M and M=17], but for the 16 columns (which have sums over 17 rows) we now have [bc(n’) where now n’=1,2…N’ with N’=16]. Thus one can obtain the equations linking the two sets of paired matrices, Table 2.
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Table 2. The Basic Repeated Pattern (BRP) pair of matrices, and their comparison to those of the Basic Square Pattern (BSP).
Wr(m) = wr(m) – 1 for all m=1,2,….17 except when m=8, 10;
Wr(m=8,10) = wr(m).
Br(m) = br(m) for all m=1,2,…17 except when m=8,10;
Br(m=8,10) = br(m) – 1.
Wc(n) = wc(n’) for all n=n’ where n’=1,2,…N’=16;
Bc(n) = bc(n’) as above stated.
______________________________________________________
The transformed sequences from the BRP matrices are then:
The center row is again that on the y*-axis containing 14 white pebbles and 2 blue. Moving up or down from that axis one has the wrs of {2, 12, 7, 11, 7, 12, 2, 15} and its complimentary (with 16 total) brs of {14, 4, 9, 5, 9, 4, 14, 1}. These sequences are the second part of the frame’s CODE.
Both sequences demonstrate as expected the same symmetry types of Wrs and Brs that were discussed earlier. On the column side, and for the 16 columns we now have in the BRP, one has the identical sequences of the BSP, with the exception that these sequences extend to only 16 counts (not the 17 counts of the BSP). Specifically, one has moving to the left of the x*-axis[where wc(9)=15 and its compliment (with a total of 17) bc(9)=2] the full eight count sequence wcs1is {3, 13, 8, 12, 8, 13, 3, 15}.Moving to the right of the x*-axis one obtains the same exact sequence, truncated however, since it only has 16 cells. Namely, one obtains the sequence wcs2 {3, 13, 8, 12, 8, 13, 3}. Easily one obtains now the compliments of both wcs1 and wcs2 (with 16 being the conservation condition) deriving the bcs1 and bcs2. The differences between the two sets of sequences (of two each for both white and blue), in the BSP and the BRP are in combination the force which produces the meander’s flow.
The cycle.
By looking at Figure 3, one can estimate the total number of BRPs in the mosaic’s frame that is the cycles and the cyclical feature of the pattern. Moreover, one obtains the sequencing of the two constituent designs [GAMMADION (G) and chess piece (C)] at both the top and bottom sections, and at each side of the frame. These cycles’ nature will be the focus now.
The photo although not complete, seems to supply enough evidence so that these sequences can be inferred. In toto, there are horizontally (top and bottom) ten whole pairs of each basic pattern (chess piece and Gammadion).Vertically, right and left sides sections there are seven of these pairs (or at least that’s what the artist intended, if the photos are accurate). One may be rather confident that this is the case, in spite of the mosaic construction faults, unavoidable approximations (not all “pebbles” could be exactly squares, and of the same exact size), and design flaws (an issue which also involves the impurities examined here). In Table 3 the sequences are shown for the top and bottom sections of the frame.
Figure 3. The mosaic floor, the representation of the abduction, and the meander as the frame. From this photo the estimation of the BRPs along the bottom/top, and the right/left side sections of the frame is made. There are seven at each side and ten at the top and bottom parts of the frame. Source: Greek Ministry of Culture.
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Table 3.a. The ten Basic Pattern (BP) sequences of the two constituent patters (CP) at the top and bottom sections of the frame: G stands for “GAMMADION”, and C for “chess piece”.
C G C G C G C G C G C G C G C G C G C G
G C G C G C G C G C G C G C G C G C G C
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A few things are clear from Table 3.a: the top left corner of the mosaic’s frame and the bottom right corners are of a C-type; whereas the right top corner and the left bottom are of a G-type. There are no BSPs at any point in the frame, and all BRPs in the frame have the orientation shown in my Figure 2. At the top right cornerstone of each BRP there is a G-type pattern. A G-type pattern is also found at the very top right and the very left at the bottom corners. Regarding the four BRP type corners, the right top side BRPis identical to the left bottom, and the left bottom to the right top. This difference in the two pairs of the four corner types is another source of impurities directly emanating from flaws in the overall design.
Some of all these flaws mentioned so far can be partially seen in the close ups of the four corners from photos made available to the public: Figures 4a, 4b, 4c, and 4b show these impurities.
Table 3.b. The seven Basic Pattern (BP) sequences of the two constituent patterns (CP) at the right and left sides of the frame’s BRPs.
Left side Right side
C G C G
G C G C
C G C G
G C G C
C G C G
G C G C
C G C G
G C G C
C G C G
G C G C
C G C G
G C G C
C G C G
G C G C
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Figure 4a. The upper right interior corner of the frame. The area obscured by dirt is supposed to contain a C-type basic component (a chess piece). It clearly is fit into inadequate space (as the size of a comparable C-type is shown to the left between two G-types.) Source: From photo made public from the Greek Ministry of Culture.
Figure 4b. The upper left interior corner of the frame. This the best corner fit, as all sections of the BRPs seem well aligned. It is possibly indicate of the fact that this is where the pattern was started being constructed. Source: From photo made public from the Greek Ministry of Culture.
Figure 4c. The lower left interior corner of the frame. The G-type shown is supposed to be a good 4/5 its size to the left. Notice the shape and size of the corner wave inside the meander frame: it’s severely suppressed. Source: Photo made public from the Greek Ministry of Culture.
Figure 4d. The lower right interior corner of the frame. Here the white lane up of the interior part of the frame hits the G-type part of the BRP off by a good 2/3 from where it is supposed to do. The corner wave in the interior part of the frame is again severely suppressed. Source: photo made public from the Greek Ministry of Culture.
A final note is due on this accounting of squares in BSPs and most importantly in the BRPs. With a 272 “pebbles” in each BRP, there are in toto 8,160 pebbles in the frame. To expect that all of them would be of equal size, would be to demand an extraordinary quality level of workmanship by the workers who did this piece of extremely rare and high quality Art. Thus, the impurities and the number of construction faults we see on this mosaic can be overlooked, in a final analysis although they are there.
In combination, such design and constructions failures we have seen in this and other parts of this tomb/monument, may have some interesting and basic things to tell, about the whole monument. It is here that answers to small questions may provide insights into questions of a grander scale. Especially, in reference to its Grand Renovation Phase, when the mosaic floor, the marble clad, the two Kores and some other parts of this extraordinary edifice were added.
Areas, lengths and measurements of the frame’s components.
The key photo from where all the measurements listed below are taken is the archeologists’ index shown in Figure 5.
Figure 5. The measurements basis. The BSP is a square of about 43.5 cm long and wide, as estimated from this photo. Source: Photo made public from the Greek Ministry of Culture.
There are 30 BRP’s in the frame, each frame taking up an area of about .18 square meters. Thus the whole frame (exclusive of the wave patterns in its inside and outside) occupies a total of about 5.37 m2, in an area of about 3mx4.5m=13.5m2 (the 3mx4.5m numbers were offered by the archeological team). Of course, at this level of analysis one would desire more accurate numbers. However, in absence of such accuracy we will have to satisfy ourselves with the numbers given us.
Given this .435m measurement in the length of each BRP, one can estimate the total size of the chamber 2 (C2). Without counting the space taken up by the outside wave pattern, ten BRP produce about a length of 4.35 meters (15 centimeters less than the official size of 4.5 meters; when the area taken up by the waves is added, then a number closer to the officially announced number can be obtained). Width wise, the seven BRPs on the frame’s each side result in a total C2 depth of about 3.045 meters (a number off by about 4.5 centimeters from the official number of 3 meters, without counting the waves.) As such, the numbers directly from the BRPs provide a good confirmation of the BRP’s sizes, as well as a close to perfect estimate of the number of BRPs horizontally and vertically found in the frame.
Given also this measurement (.435m) of a BRP, the X size of the “pebble” square can be directly now computed. Since the length of a horizontally running BRP is about .435m, and it takes 16 squares in it, the unit length of each square (the “pebble”) is about .027m. Thus X=.027m (and so we now have the area of each “pebble”, about 2.3 to the power of 10 to the minus 4, meters squared).
What will preoccupy the analysis now is the length of the y*-axis of my Figure 2. It is recalled that this is a line running at the very center of the largely white lane. This lane runs at the very center of the frame horizontally along its top and bottom sections, as well as vertically along its sides. It forms a rectangle, that will be designated as R, with horizontal dimension Rh and vertical dimension Rv. We now can estimate the approximate dimensions Rh and Rv of this rectangle. Its horizontal dimension Rh is exactly nine times the length of a BRP. A BRP length is as we saw just above .435m, thus the total length of R, is about Rh=9x.435=3.915m.
R’s Rv is six times the .435m of each BRP (running either vertically or horizontally the total length remains the same, only the specific rows/columns being repeated change) we obtain the estimated Rv:Rv=6x.435=2.61m.
The dimension 3.915 and 2.61 have a ratio r of exactly 1.5:
r=3.915/2.61=1.5
It is a ratio of some interest; its exactness can’t be random. I have argued in my “HFAISTION HYPOTHESIS” paper that the construction of the monument/tomb, during its Grand Phase renovation, was done inside out, and down up. It is clear that all dimensions in the tomb’s 25 meter corridor started in C2 and moved outwards. All was based on C2, which in turn was based on the floor’s mosaic. We do not have from either photos or descriptions by the archeological team a clear view, at this stage of the marble clad inside C2. Undoubtedly, the marble coverage and its related code (which I analyzed in my “The Modular Structure of the Tomb at Kasta Hill” paper) must be intertwined, as must be the dimensions of the marble door.
The waves.
We now turn our attention to the waves, found both on the inside and outside part of the mosaic’s frame. A photo provided above (Figure 1) shows that the artist placed five white and five deep blue color waves per BRP or per cycle. As already mentioned, the double play on the waves allow the white waves to move in a counter clockwise pattern, as does the G-type pattern, the Gammadion. Whereas the blue waves move along a clockwise pattern. No matter the specific imperfections in the size of the waves at certain points along the frame, the artistry of the pattern leaves the observer/analyst quite impressed.
The total number of the exterior (to the double meander section of the frame) waves is of course 50 (5×10) horizontally, and 35 (5×7) vertically, for a total of 170 waves. Whereas the total number of the interior to the frame wave pattern is 40 (5×8) horizontally, and 25 (5×5) vertically for a total of 65 waves. Thus in toto we have 235 waves on both sides of the frame (exterior and interior).
From the measurements obtained from Figure 5, it is deduced that the width of the waves pattern on the outside the meander frame must be about 10cm. It is not clear if this is indeed the width of this part of the frame all around the double meander part of it. If it is, then the whole size of the frame is about 4.35+.20=4.55m, over the officially announced size of 4.5m by 5 centimeters.
The interior wave pattern seems tomaintain an equal width, of about 10cm, horizontally and vertically. It is not clear though that this width is maintained consistently through the length of the frame. A close up, Figure 4c seems to indicate that this is not always the case. It could just be one of the overall construction and maybe design as well shortfalls.
An effort will be made to estimate the total area taken up by the wave pattern, aiming atderiving the total area taken up by this impressive mosaic frame. The area of two rectangles of length of 4.55m each and width .10m (.91m2), plus the area of two rectangles size .3045 and width .10m (if the width at the sides of the frame equal that running horizontally, and at this stage we simply don’t know, since no photos have been offered covering that area anywhere in the mosaic floor, so we just assume so) produce a total area of about .6m2. Thus the total area of the outside wave pattern amounts to about (.91+.6=1.5m2)
The interior section of the wave pattern is two rectangles with dimensions 3.4m (the length of eight BRPs) and width .1m for a total of about .68m2, plus two rectangles of length 2m (the length of five BRPs minus .2m) and width .10m for a total of .2m2. Thus in all, the waves pick about .88m2 of area on the inside of the double meander frame. By adding that to the total from the outside, we obtain a total wave area of about 2.38m2. When added to the total area picked up by the double meander part of the frame, the whole margin of the mosaic floor amounts to about 7.75m2 [5.37m2 + 2.38m2]. With a total chamber area of about 13.5m2, the area taken up (about 57%) by the frame alone is impressive. It exceeds the area picked up by the representation, and by a hefty margin.
The point is here, that we have a mosaic floor with a rendition of Persephone’s abduction, in which more than half its area is taken up by its overall frame. It clearly conveys the message that the information we obtain from the frame must be of equal to, if not greater than, import to that offered by the rendition itself.
Epilogue.
In concluding this analysis, one can return to Hermes. This figure in the pattern, didn’t commend as much controversy or analysis as anything else there in C2 did. It was a fixed point of reference, a figure that all can agree to and about, and immobile from any uncertainty in mind search and ubiquitous implied movement inscribed all over C2 – a static configuration in a highly volatile fluid environment. Waves and cycles, no matter how regular in their inner structure, rough nonetheless, are all around him in that small solitary room. Yet, in spite of this apparent gales of motion within this extraordinary representation, Hermes seems to convey a sense of stability over a cloud of uncertainty and winds of change spinning around the rendition itself and more broadly Chamber #2. The huge but amazingly regular in shape damage done to the mosaic, also escaped him. Somehow, it seems that all these man-made and nature-produced forces that shaped this magnificent Chamber, left us a message, to be conveyed to us by Hermes. Someday, humanity might be able to receive that message. Maybe the Code embedded in the sequences of the Basic Repeated Pattern (the BRP for short here) encrypts this message: 3, 13, 8, 12, 8, 13, 3 (the wcs2). May be.
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Acknowledgments: The author wishes to thank all members of the OMADA APOSYMBOLISMOY KASTA tumulus
Among those involved in the group, the author wishes to specifically mention Panagiotis Petropoulos, Photini-Effie Tsilibary, Vagelis Vagianos, and Theodoros Spanelis for having collaborated in some of his research and for having offered valuable assistance. Further, the author wishes to acknowledge very helpful interaction and stimulating discussion on a variety of subjects related to the Kasta tumulus by Athanassios Fourlis, Dimitris Savvidis, and Elena Vardakosta.
References:
- Dimitrios S. Dendrinos, 2014 “The Modular Structure of the Tomb at Kasta Hill” found in: http://www.academia.edu/10923712/The_modular_structure_of_the_tomb_at_Kasta_Hill_by_D_Dendrinos
- Dimitrios S. Dendrinos, 2014 “The HFAISTION hypothesis” found in:https://www.academia.edu/14138924/On_the_HFAISTION_at_Kasta_Hill_hypothesis
- More discussion on the subjects of the Kasta Hill tomb/monument is available by the author on his FB page:https://www.facebook.com/profile.php?id=100006919804554
Copyright. This work is copyrighted by Dimitrios S. Dendrinos. No parts of it are to be reproduced, without the author’s expressed written consent.
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