On the Parthenon’s Mathematics, Astronomy and its Embedded Harmony: its Skeletal Geometry, Modulus, three Key Triangles and Core Angles Dimitrios S. Dendrinos Emeritus Professor, School of Architecture and Urban Design, University of Kansas, Lawrence, Kansas, USA. In Residence at Ormond Beach, Florida, USA. Contact:
[email protected] June 29, 2017
The Parthenon restored according to reference [2.1]
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Table of Contents Abstract Prolegomena Brief Summary Introduction: Towards an Efficient Statement of Parthenon’s Geometry The Parthenon’s Modulus: its Mathematics and Astronomy The Parthenon’s Skeleton: its Three Key Right Triangles and Core Angles Stating the Parthenon’s Optimization Problem and its Heuristics Primitive Pythagorean Triples and the Parthenon’s Triangles Conclusions and Suggestions for Further Research Epilogue References Acknowledgements Legal Notice
The Parthenon restored. Source: [2.2]
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The Parthenon at present, Eastern view. Source of photo: [2.16]
Abstract A novel view of the Parthenon’s structure is taken in this paper. Instead of analyzing the Parthenon’s final configuration, either in its various reconstructions or in its current condition, the study draws the Temple’s 3-d skeletal structure. Based on that sketch plan, the Parthenon’s modulus and its grid pattern are derived. In closely examining this skeletal morphology, a number of findings emerge. The Parthenon was built on the basis of a critical ratio and a set of interconnected angles, generated by following a set of instructions. Utilizing the sketch plan (the Parthenon’s skeleton), the paper provides a mathematical optimization formulation, involving an objective function and a set of constraints. On the basis of that problem, one can derive, through the associated efficiency conditions, the entire Parthenon structure. Along the way, some topics of mathematical interest are presented and partially elaborated. Suggestions for further research are also provided.
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Prolegomena There are many claims that have been made about various topics surrounding the Parthenon, by many individuals and agencies over the course of time, since its construction in the middle of the 5th Century BC. In contrast to the monument’s widespread popularity, its gleaming aura and glamor, its Architecture is clouded in mystery and uncertainty. Even who are the builders and designers, architects and engineers of this extraordinary structure, is a question that still has a fuzzy response. Time has been rough on the monument, as humans and Nature have taken their toll on it. Yet, much of the structure still stands, in defiance and in spite of the ravages of time. A few things, among the many over which we still seem to be plagued by a fair amount of doubt, are somewhat clear. Architects ΚΑΛΛΙΚΡΑΤΗΣ and ΙΚΤΙΝΟΣ were the, known now, architects of the Temple. ΙΚΤΙΝΟΣ, who also built the Temple of ΕΠΙΚΟΥΡΙΟΣ ΑΠΟΛΛΩΝ at ΒΑΣΣΑΙ contemporaneously with the Parthenon, wrote a book about the structure, in which he laid out the “proportions” he implanted on it. But, unfortunately, the book didn’t survive and only parts of the final structure did. Hence, the contemporary Architecture students, scholars and detectives are confronted with a Herculean task. Putting the pieces of the complex Parthenon puzzle together still remains a formidable task. The careful analyst still has his/her work cut out for him/her. Within this mindset then, one must approach the study reported here and its findings. The debate still rages on, as to what were these “proportions” ΙΚΤΙΝΟΣ sought to implant on the Parthenon and exactly where and how. Hopefully here, one might find some answers, not contemplated by previous studies. These answers might be surprising, as they come from an unexpected source. They are not found at the Parthenon’s final structure, but at its skeleton – the sketches that probably the architects drew when contemplating and planning the Temple’s construction and designing its morphology. In that skeletal structure, one discerns significant mathematical sophistication, in both Geometry and Calculus, incorporated into the structure’s form. This study finds that the “proportions” ΙΚΤΙΝΟΣ strived to attain and embed on the Parthenon involve lengths, angles and instructions of how to derive them. The length part of the proportions is hidden deep in the Parthenon’s structure, close to its sacred spot where the sculpture by ΦΕΙΔΙΑΣ of goddess ΑΘΗΝΑ ΠΑΡΘΕΝΟΣ was set to stand, inside the ΝΑΟΣ. The angles and the instructions on how to generate them on the other hand, figure prominently and are based on the very orientation and the pediments of the Temple on the ΑΚΡΟΠΟΛΙΣ of Athens for all to see. We do not possess evidence that the architects of the Parthenon were familiar with either Operations Research methods, algorithmic Computing, or modern-day Genetics and DNA. Yet, they apparently had a very strong intuitive understanding of the basics involved in these mathematical methods and biological processes. It is the purpose of this paper to make this case.
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Brief Summary The thesis is advanced in this paper that one could potentially view the design of the Parthenon as a solution to an optimization problem. For the first time, to the author’s knowledge, an attempt is made to derive the Parthenon’s exact site plan (floor plan) and elevations morphology, including the Temple’s precise orientation, as the optimal solution to a problem drawn from the Mathematical Theory of Optimization. This Optimization problem’s objective function has a specific (albeit abstract) interpretation: Harmony. The paper argues the thesis that the architects of the Parthenon ΚΑΛΛΙΚΡΑΤΗΣ and ΙΚΤΙΝΟΣ derived the 3-d skeletal form of the jewel of Classical Greek Temples, as the outcome of a general (potentially applicable to all Classical Greek Temples) optimizing process. Two major sets of findings are reported. A third set involves findings that potentially open up new landscapes for further future research in Mathematics. First, and foremost, is the set of findings associated with the efficiency conditions for deriving the Parthenon’s optimal 3-d skeletal Geometry. Parthenon’s structure efficiency conditions operate under maximization of an objective function with a concrete interpretation: to maximize harmonious links among the basic elements of the entire Temple. To state the efficiency conditions and the essence of the objective function they obey, the paper discusses at some length the specific variables defining the modular three-dimensional structure of the Parthenon’s skeletal form and its average cubic grid pattern. It presents the precise manner in which the absolutely necessary and sufficient mathematical (geometric and algebraic) design specifications are set, expressed as variables and conditions they obey. These conditions are responsible for the Parthenon’s 3-d grid. Further, the general form of the problem’s objective function is stated. It is concluded that a far more advanced and complex set of instructions can be discerned in the design of the Parthenon’s 3-d skeletal structure, than merely the seemingly a priori and largely ad hoc determination of its major elements including the total number of columns at the peristyle (46), their 8x17 split in the Parthenon’s narrow and long sides, various inter-columnia, the Temple’s 3-d measurements, and the Temple’s orientation. It is proposed (and proved in the paper) that the fundamental Geometry of the Parthenon and the edifice’s 3-d modular grid were based on three key right triangles, their three core angles, and their spatial interconnections. Of special note is a core angle of 22.5 in a right triangle associated with the Parthenon’s orientation, representing exactly one sixteenth of 360; and an exact ratio of 1.50 associated with the Parthenon’s barycenter and omphalos – the location of ΦΕΙΔΙΑΣ sculpture of ΑΘΗΝΑ ΠΑΡΘΕΝΟΣ. The second set of findings identify the Astronomy related objective held by the Parthenon’s architects. It is demonstrated that the orientation of a key Parthenon axis’ front end is towards the sunrise rays of the Summer Solstice; and the concomitant orientation on the other end of this key axis is towards the last glim rays during the Winter Solstice. A third set of findings is linked to the Parthenon’s floor plan design and its 2-d modular grid skeletal structure. It involves some Number Theory aspects of the Temple’s 46 (in a split of 8x17) columns, comprising its peristyle. Finally, whether the key right triangles of the Parthenon belong to the class of right triangles called “Primitive Pythagorean triples” is explored, and some preliminary findings reported. 5
Introduction: Towards an Efficient Statement of Parthenon’s Geometry The Parthenon’s skeletal Geometry and its modulus Why does the Parthenon have 46 columns at its peristyle of about 1.90 meters in diameter (the four corner columns being slightly, by 1/40th, larger than the other 42)? Why are there eight columns in the frontal (Eastern) and back (Western) sides of the Temple, and seventeen columns on each of its North and South sides? Why do the columns stand at about 10.40 meters high? Why is the Parthenon’s Eastern/Western side (at the ground level of the crepidoma) about 31 meters (actually the Eastern side is a bit shorter than the Western side – 30.875/30.884), while the Northern/Southern is about 70 meters long (again, there is some minor differences, as the Northern side is slightly longer – 69.5151 – than the Southern side - 69.5115)? Why does the Parthenon’s main (long, East – West) axis is at about an 80 azimuth and not at a due East – West axis (like almost all Doric Classical Greek Temples – the Temple of ΕΠΙΚΟΥΡΙΟΣ ΑΠΟΛΛΩΝ at ΒΑΣΣΑΙ notwithstanding)? These are the kind of questions we are set to explore in this paper. A framework one may be able to adopt or wish to employ in seeking answers to such basic questions is none other than the field referred to as “Operations Research” or more formally, the “Mathematical Theory of Optimization”. Of course, Engineering related concerns (primarily having to do with designing a structure that could support the Temple’s roof) must have been involved in such decisions by the two architects of the Parthenon, ΚΑΛΛΙΚΡΑΤΗΣ and ΙΚΤΙΝΟΣ, and the Temple’s engineers in building a structure with the specifications we currently observe imprinted on it. Moreover, setting aside for the moment the Engineering and Topography related issues facing the Parthenon’s builders, Architecture and Art related factors, we do know, did play a major role and did enter a calculus in the design of a structure in which attention to detail was of paramount importance, and where Harmony was sought in its proportions at all levels of scale, from the miniscule to the grand. The enormous literature on the Parthenon is replete with references to these types of analyses. This paper will attempt to neither review nor supplant this plethora of studies and their wide scope. However, no matter the voluminous work on the multiple subjects concerning the Parthenon, perhaps the best studied monument of the ancient World, no study does, or even attempts to, answer the basic questions posed at the beginning of this Introduction. In addition, a careful and systematic consideration of the numerous ancient (used in Athens during its Golden Age) units of measurement (in length, area and volume) fail to exactly reproduce key elements of the Parthenon. Measurements of (for example) the Parthenon’s 3-d sizes do not come up to be exact fractions or multiples of the abundance of units employed back then, as it will be demonstrated. Why is it so, when such meticulous attention to detail was a hallmark of the Temple’s Architecture? Is it possible, that we are not measuring the real quantities that determined them? Hence, one is left pondering these questions. So, here an attempt is made to systematically address the issue of Parthenon’s construction from a different angle, as the solution to a problem from an Optimum Design perspective. Hence, the hypothesis is suggested that there must have 6
been, in short, certain variables – such as inter-columnia and columns’ diameters - that on their average (and possibly on an individual level) somehow optimized a function subject to constraints, the nature of which (objective function and constraints) can be and are subject of this work, drawn from shown and manifest evidence. By looking at the Parthenon (as is the case with any architectonic structure) one must ask a number of existential questions (and then attempt to find some answers). This genre of questions, introductory to an extent, may be the following. Why was the Temple built there, at that specific site and location with that specific orientation and not at a marginally neighboring one with a slightly different orientation? Why do the Temple’s elements have these exact (whatever that “exact” may entail) sizes and not marginally different (smaller or greater) ones? The same set of questions but with the “marginally neighboring” expression substituted by “significantly different”, so that the question becomes ‘why was it not built at a significantly different location, with a significantly different orientation’, or ‘why built it at all’, or “why not having elements with significantly different sizes” etc., clearly fall outside the purview of this study. But to the extent that they are posed ‘at the margins’, they remain valid (and potentially insightful) questions to ask and ponder. A similar (and again never posed by anyone to this date) question is this: What is the skeletal structure of the Parthenon? Is it possible that the architects of the Parthenon were precise and exact at that architectonic preliminary drawings (sketch) phase? Was it there the opportunity to obey the then current units of measurement in lengths, area and volume, and then (once the building acquired its final form and the masonry construction was dressed and all construction adjustments finished), the ensued measurements disappeared under the dressing, carving, imperfections and all other factors forcing differentiation and breakup of the homogeneity and conformity to a standard, and consequently what was left were approximations at the margins? Could that be a statement with social underpinnings? Is in the diversity of the Parthenon’s columns and their inter-columnia a declaration of all individuals being marginally different although in unison and collectively performing one function, supporting the Parthenon’s roof? Larger in scope questions may also be posed. Why does the Parthenon have these exact length, width, and height specifications? The Temple could possibly be constructed maintaining the exact same ratios it does possess at this scale, but at a marginally different size (either slightly smaller or bigger). Why at the size it currently stands? Is the answer to this question of a socio-economiccultural type, say for instance an optimum in total expenses due to finance related constraints or maybe resource related constraints? Or is the answer purely architectonic and based on Aesthetics? One will obviously say, it was both. But exactly how? Continuing along these lines, why does the Parthenon have a system of 8x17 columns at its peristyle, with a ratio R=17/8=2.125 (where the four corner columns are counted twice), a ratio critical in the discussion of Greek Classical (early, middle or late) Temples, and not any other? Some say, that this is simply the outcome of 8x2+1=17. But this doesn’t really answer the questions: why 8? And why 8x2+1? These, largely vacuous, answers describe the obvious, without explaining it. 7
Numerous similar questions one may ask, along the lines suggested. But those posed certainly must figure prominently in one’s architectural and Engineering approach to the Parthenon (or any other Temple of Classical Greece). It turns out that a number (if not all) of the above questions can be answered adequately, if one examines the appropriate modulus and a 7x16 average grid pattern under which the Parthenon’s skeleton was put together, and what lies underneath that modulus in the Parthenon’s skeletal structure. By so doing, one can approach the Temple’s construction on the basis of a set of optimality subject to constraints conditions involving the skeleton and its modulus. In turn, the associated efficiency conditions can show how that modulus was derived. It turns out that it was made on the basis of three key constituent right triangles and their three core angles, and a ratio involving two key lengths.
Number Theory and the Efficiency conditions of an Optimization Problem for CGTs Subjects related to the Parthenon’s modulus, along with other quantitative features of this Temple and other Classical Greek Temples (CGTs) have been addressed by this author in a number of contexts, see for example [1.1]. There, the role shadows played in the design of the Parthenon and a number of other CGTs was explored and identified. Also analyzed there was a number of key ratios embedded into the design of the Parthenon and four other CGTs, as well as discussion and documentation was supplied on the modular structure of these Temples. As a result of this analysis, the basis was set for a statistical approach to their design specifications, and towards the derivation of a typology in the CGTs architectonic morphology. This paper is a continuation of reference [1.1], where references to prior and existing literature on the Parthenon are found. Material from that paper will not be duplicated here, but simply extended and to an extent revised. Such modifications will be accomplished by a more in depth look into the pinnacle of Classical Greek Architecture, and by analyzing from a novel perspective this great monumental structure, designed by architects Callikratis (ΚΑΛΛΙΚΡΑΤΗΣ) and Ictinos (ΙΚΤΙΝΟΣ), the Parthenon. Much has been written about the Architecture of the Parthenon since Vitruvius, and especially during the past two centuries. Not much of that literature will be repeated here. Some of that has been reviewed in [1.1]. The interested reader may consult for informative introduction references [2.9], [2.10], [2.15] and references therein. That literature however, to the extent that it delves on the basics of the Parthenon (like for instance its modular structure), doesn’t approach the subject on a proper basis. In most cases the erroneous Vitruvius assertion about the modular structure of CGTs is adopted. The vast majority of the existing treatises tend to look at the Parthenon by examining the actual dimensions (length, width, height) of the structure. In them, ratios of different types (the Golden Ratio for instance) have been sought as intentionally having been implanted in the structure. This work does not intend to refute the validity of some of these assertions. This author is neutral on their validity, as questions about measurements and approximations can’t be ascertained in toto. Instead of placing emphasis on the simple and obvious quantities, 8
like for instance the Parthenon’s external 3-d sizes, a look into the deeper skeletal design of the monument will be taken. It is suggested (and documented) that such underlying skeletal structure was drafted with a great deal of care and attention to detail, and at its core it obeys exact quantities (angles and ratios) not approximations on lengths. Underlying skeletal morphology coupled with a set of instructions give rise to these surface ratios and to the overall Parthenon form. It is demonstrated here, that by looking into the skeleton of the Parthenon, its modulus and average grid pattern involving its constituent elements (three key triangles and their core angles plus a key ratio involving length), a set of complex instructions are uncovered and a huge amount of information is stored. This depository of instructions is uncovered here, as it has been overlooked by conventional approaches to the Architecture of the Parthenon. Central in this quest is the uncovering of the efficiency conditions found to govern the basic Geometry of the Parthenon’s skeletal structure. This angle, on how one can derive the Temple’s design specifications by analyzing its skeleton (a structure’s DNA to use a convenient analogy), leads towards stating the Mathematical Optimization Theory based problem of Classical Greek Temples’ morphology. In effect, the CGTs can be approached as the optimum solution to a comprehensive (entailing most quantities of a CGT’s architectonic form) optimization problem. It has been suggested by the author (and in contrast to the generally adopted Vitruvius view of a Greek Temple’s modular structure – namely that the modulus of a Greek Temple is the distance between two columns at their base on the stylobate) that the proper definition of a CGT’s 2-d modulus is the average distance between two columns’ very center. The 3-d manifestation of the modulus is then derived from the height of a column (including the capital), i.e., the distance between the crepidoma’s stylobate from the entablature’s lowest level (the bottom of the Temple’s frieze). It is reminded that the Parthenon’s columns do not have a base. They also sport different (some minute, some considerable) distances between any pair of them on any side. By closely analyzing the modular structure of the Parthenon under this angle, some Number Theory related issues immediately emerge. Further, a better understanding is obtained as to why the architects of the Parthenon chose this particular 8x17 split of the peristyle’s four colonnade segments (East, West, North, South) of 46 columns in toto. In approaching the subject of a 3-d modular structure of the Parthenon in accordance with the suggestions first put forward by this author in reference [1.1], one can derive an efficient mathematical method to define and arrive at the Parthenon’s skeletal geometric structure including the columns’ spatial arrangement (46, 8x17). It will be shown how the unfolding of the structure is pegged to efficiency conditions. Obviously, to derive such geometric efficiency conditions, aspects of sufficient approximation to actual measurements of the Parthenon’s structure are of critical importance. See for instance reference [2.10], [2.15] in which the almost miniscule differences in the Parthenon’s crepidoma East/West and North/South lengths are listed. In [1.1] the author argued that it is not possible for a Classical Greek Temple (any Temple, not just the Parthenon) to have identical (at a set level of approximation of, say, the millimeter) E/W or N/S length sizes, or column widths and heights, 9
or any quantity which is to be reproduced. The demon of construction imperfections doesn’t allow it. These differences were unavoidable back then, as they are to a lesser extent today. However, this is not the sole reason why differences, as those recorded in the Parthenon’s intercolumnia, are observed see [2.15]. Engineering related considerations dictated very likely the inter-columnium differences with columns next to the corner columns significantly differing with respect to inter-columnia. On top of that, and remaining on this example, with regards to same side inter-columnia differences, one can’t fail but to notice the discrepancies characterizing measurements of the same distances derived by different researchers. As the author has argued in the past, Quantum Mechanics theoretic reasons might underlie such differentials, see [1.5]. This is a fortiori a reason why, statistical averages must be considered. It is on statistical averages that the narrative here proceeds. Using such averages, an aggregate approach can be undertaken (whereby some averages are computed and their average effect ascertained). In the optimum design framework adopted here, however, some disaggregation can be accommodated (where some classes of variables, like for instance various types of inter-columnia) can be considered and their sizes and effects endogenously computed. In any case, the author has limited access to detailed and precise measurements, and no means to independently derive them. Hence, this paper can be construed as a first attempt towards achieving this goal, of stating the architectural structure design specifications as an optimal design problem and supply, to the extent possible and based on available evidence, convincing (although not beyond reasonable doubt) evidence to justify the claim and the findings. Subsequent attempts by other researchers, with access to more accurate measurements and computing capabilities, are of course needed and scientifically necessary in replicating the findings reported here and thus provide support to the claims made. Having supplied this caveat, it must also be noted that due to the nature of the enterprise (the search for exact ratios and exact counts on angles) and the finding of these building blocks, the work reported here seems very promising. The paper presents the formal mathematical statement of the (aggregate and disaggregated) optimum design and orientation of the Parthenon in the final section of the paper. Due to the Mathematics required to follow the exposition, it can be construed as an Appendix and available to those readers familiar with the basic elements of Optimization Theory and with the Operations Research literature. Related to the inherent difficulties involving accuracy in measurements, is the subject of entasis. In Doric Temples’ entasis (ΕΝΤΑΣΙΣ, tension) related effects were structurally reflected by: a slight inward leaning of the columns; by the frontal and sideways slight inward concavity and upwardly directed convexity of the crepidoma); by the slightly uneven diameter at the four corner columns from the rest of the peristyle’s columns; by a smaller inter-columnium between the corner columns and their neighboring ones than the average; and by the slightly uneven diameter on all columns along their vertical unfolding.
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All these variations are not incorporated into the analysis that follows. Such architectonic and engineering detail can only be incorporated into the present mathematical (geometric and algebraic) analysis when the precision by which the various relevant Temple dimensions (such as the total frontal and side lengths, as well as the columns’ height) enjoy equal confidence in measurement approximations. At the level of approximation in measurements contained in this paper, these fine details built into the Parthenon’s structure (for both, Art/Aesthetic as well as Engineering/Architecture, structural/material support involving weights and loads, lateral and vertical forces related factors) can be with ease bypassed at this stage. These aspects of a CGT’s Architecture and Engineering are in part presented and discussed in [1.1] and widely elaborated in the existing literature on the Parthenon. Again, here the emphasis and the focus is on the skeletal (sketch drawn) structure of the Temple.
Towards a Further Understanding of the Parthenon’s Astronomy Building on the Astronomy related considerations identified in reference [1.1] by the author, a closer look is taken into the orientation of the key triangles identified as being at the basis of the Temple’s structure and appearing in its skeletal configuration. It is concluded that these triangles, and especially the one responsible for a key diagonal in the rectangular part of the Temple’s superstructure formed by the peristyle’s 46 columns, are very informative in providing some evidence as to why the architects of the Parthenon’s decided to orient the edifice the way they did – which is not exactly in accordance with any major Solar alignment of any other Doric CGT. Further room for refinement is left in this angle of the exposition, by future work involving more accurate approximations to the Temple’s three key triangles’ precise orientation.
Primitive Pythagorean Triples and the Parthenon In the last section of the paper, a search is undertaken to examine whether some key right triangles this study identifies as constituent elements in the Parthenon’s Architecture are in fact so-called Primitive Pythagorean Triples (PPTs). This is a set of triangles that exhibit certain properties of pure mathematical (and thus possibly Aesthetic) interest. The core mathematical property of the set of PPTs (they comprise a Tree) is that the sides of these right triangles (expressed in some modular unit) are coprime. The right triangle with sides (3,4,5) is the root of that Tree, and the best-known PPT. Here, a more specific set of PPTs is examined as to possible connections to the study’s key right triangles – those with a long side close in length to their hypotenuse. The outcome of this search at this stage is inconclusive, albeit promising.
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The Parthenon’s Modulus: its Mathematics and Astronomy In accordance with arguments put forward in [1.1], the 2-d (floor plan) modulus of the Parthenon (and all CGTs) is found by drawing lines through the very center of the peristyle’s columns, along an approximately North-South, as well as an approximately East-West direction given that both the narrow (usually East and West) sides of the Temple, as well as the long (usually North and South) sides are so oriented. In the specific case of the Parthenon, its 2-d floor plan modulus forms a grid (the precise dimensions of which will be estimated in a bit) with its border lines going through the very center of the seventeen Northern and Southern most sides’ columns, and the eight Eastern and Western most sides’ columns.
Notation and some nomenclature. For the purpose of this paper, the Parthenon’s peristyle 46 columns (split according to a ratio R=17/8=2.125, with 17 columns along the North and South sides, and 8 columns along the East and West sides, corner columns counted twice) will be numbered as follows: the corner column at the South-Western corner will be designated as column #1; the next columns as #2 and so on, till the last (North-Western) column which will be designed as column #8. Moving Easterly, the second column at the Northern side of the Parthenon (and to the East of column #8) will be column # 9, and so forth till the last column of the Northern side (at the North-Eastern corner of the Temple) which will be designated as column #24. Consequently, the South-Eastern corner column will carry the number #31; and the last column of the Parthenon’s peristyle (at the Eastern side of column #1, and the penultimate Southern column) will carry the number #46. The very center of the Southwestern corner column #1 will be designated as point A, see diagram in Figure 1. Moving North, the Western border line of the grid pattern intersects the Parthenon’s East-West axis of symmetry (designated by the blue line in Figure 1), at point E. The very center of the South-Eastern corner column #31 will be designated as point B (see Figure 2). At the Eastern frontal (narrow) side of the Temple, the center of the NE corner column (column #24) will be designated as point C (see both Figures 1, and 2). The intersection of the modular grid’s Eastern border line intersects the (approximately) East-West axis of symmetry at point F (see both Figures 1 and 2). It is noted that this axis of symmetry does not have a due East-West orientation, as the diagram of Figure 3 demonstrates. The main orientation of the Parthenon is towards the East North-East at an azimuth of about 80. Inside the cella is the Parthenon’s Naos (ΝΑΟΣ), which contains the Eastern and main Entrance to the Temple, and the Sekos (ΣΗΚΟΣ), that has a Western entrance, see Figure 3. Slightly to the West of where the location of the statue of Athena, sculpted by Phidias, used to stand inside the Naos, the barycenter of the Parthenon’s floor plan is found, to be designated as point H (see Figures 1, and 2). It is in effect the Temple’s omphalos, as we shall see that a number of critical for the Temple’s structure intersections occur, and a key ratio, responsible to a large extent for the Parthenon’s overall morphology, is located on a line passing through this point H. 12
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Figure 1. The Parthenon’s floor plan, its 2-d modular and average grid structure. The basic floor plan diagram is from source [2.3]. Drawing of the modulus and its grid is by the author. North is almost straight towards the right side of the Figure. The scale of the floor plan is given at the top of the diagram. At bottom, is the Eastern – and main – Entrance of the Temple. The 2-d sizes of the modulus, that is the grid pattern (which is not exactly a square), are such that the distances among East-West running lines are slightly smaller than the distances among North-South running lines of the grid. The red diagonal line AC shown above has an orientation such that on the one end it points towards the sunrise during Summer Solstice at Athens; and on the other end, it aligns with the sunset during Winter Solstice. Blue line EF is the Parthenon’s 2-d axis of symmetry (approximately in an East-West direction at an azimuth of about 80). Point H is the Parthenon’s barycenter and omphalos, the point where the two axes intersect. The rectangle close to point H and to the East is where the statue of Athena by Phidias was located. Source of Figure 1: the author.
The Temple’s modular structure and Number Theory Parthenon’s modulus, its grid, and the Temple’s cella. Noticeable are the following (at least) seven specific connections between the suggested modulus proposed by the author and the key elements of the Parthenon’s floor plan: (1) the step of the cella’s platform coincides exactly with the grid line linking the centers of two sets of columns, set #46 and #9, and set of columns #23 and #32; (2) the cella’s Southern wall’s outer surface coincides exactly with the grid line joining the centers of columns #2 and #30, whereas that of the Northern wall coincides with the grid line joining the centers of columns #7 and #25; (3) the edges of the two cella’s exterior East-West (long) walls – the antae - start exactly at the grid lines joining the centers of columns #45 and #10, and columns #22 and #33; (4) the pair of ten-column sets running on an East-West direction inside the main space of the cella, the Naos, as well as the six internal columns running on a North-South direction, all these three sets of columns, align with three grid lines of the modulus, lines joining the centers of columns #6 with #26, #3 with #29, and #15 with #40; (5) the pair of 6column sets at the East and West sides of the Temple on the cella’s stepped up platform and in front of the antae at the Eastern and Western entrances into the Naos and sekos correspondingly (at the two porches that is), line up with the grid lines joining the centers of the pair of columns #9 and #46, and the pair #23 and #32; (6) the front of the base on which the statue of Athena was set at the interior of Parthenon’s Naos lines up exactly with the grid line joining the centers of columns #17 and #38; (7) the diaphragmatic wall separating the cella’ Naos from the sekos aligns precisely with the grid North-South running line joining the centers of columns #14 and #41. As it can be seen in Figure 1, the two diagonals drawn on the Parthenon’s floor plan go precisely through the four exterior corners of the cella’s antae. The above evidence leaves little doubt that this is in fact the Parthenon’s modular structure and grid. In the diagram of Figure 1 the units of the modular grid pattern are equal along the E-W axis, and also equal in the N-S axis. However, this does not imply that they are necessarily forming a square grid pattern (they don’t). 14
The Parthenon’s modular grid and Number Theory. It must not go unnoticed the fact that, numbering of columns has some Number Theory related properties. For example, due to the choice that involved the total number of peristyle columns (46), and their 8x17 rectangular arrangement, the grid alignments involve two pairs of columns with reversed numbering: #14 and #41, #23 and #32. And, of course, their sum is 55 in both cases. Another key matchup is that between columns #22 and #33, with a sum of 55, as well. Out of these examples, some general Number Theory related conclusions can be drawn. Column numbering of the 8x17 peristyle is such that when the first column is either at the Southwestern or the Northeastern corner, the sum of all pairs at the East and West sides of the grid is 32; for example: 1 + 31 = 32; 8 + 24 = 32, and it is remarked that in the number 32, 3+2=5. Whereas the sum at the North and South sides for all pairs, with the exception of the two (South-Western and North-Western) columns at the West end of the Temple, is 55 (so that 5+5=10). The totals 32 and 55 will be designated as N1=32, and N2=55. If the starting column is either at the Northwestern or the Southeastern corner, their sums are either 64 (on the East-West) or 41 (on the North-South) direction. Both of these sums are such that (as in the previous case), 6+4=10, and 4+1=5. Noted is the fact that the properties just outlined (the 5 and 10 sums) are invariant with respect to the assigned location of the #1 column, as long as #1 column designation is a corner column. Commutation of the numbering system among columns on this rectangular grid pattern, with a starting column numbering system at any off-corner point, is of interest. Such commuting schemes do not produce quantitatively and qualitatively similar sums to those postulated for the corner-column starting positions. This variance is meaningful, and with interesting mathematical Number Theory related extension, when Temples with different peristyles colonnades are so analyzed. For example, in the case of Temples with a 6x15 (38 total columns, as is the case of Selinunte’s Temple E, analyzed in [1,1]), or a 6x14 (36 total columns, as is the Temple of Hera II at Paestum, also analyzed in [1.1]) system of peristyle colonnades. The smaller the total number of columns, the smaller the East-West, and North-South totals. And the manner these totals are connected, could supply the basis for a Number Theory approach to the issue. The import of this Number Theory aspect of the columns’ rectangular configuration will be seen in the penultimate section of the paper where the Optimization Problem for CGTs will be addressed. The manner in which these totals, and the columns’ split between the narrow and long sides of a Temple, could potentially affect the structure and value of the optimization problem’s objective function. How this is possible it will be pointed out then. It is hence suggested that the choice of 46 columns at the Parthenon’s peristyle was not totally independent of these occurrences in and properties of Number Theory. This is a broader topic, involving formal mathematical analysis in Number Theory, which will not be further elaborated. It is left to future work and to the interested reader. 15
The Astronomy embedded into the Parthenon’s Design. The Astronomy built into the Parthenon’s design is depicted by the red line drawn in Figures 1, 2 and 3. Line AC, which joins the centers of columns #1 and #24, is a key feature in the design of the Parthenon. It is quite close to a 62 azimuth – very close to the azimuth of the Summer Solstice sunrise at the Athens’ location. The Parthenon is located at Earth’s latitude of about 38 (to be exact at 37.9714N). At that latitude, the sunrise at Summer Solstice occurs at about a 60 azimuth angle or 30 North of the due East-West orientation. The AC axis has an angle (as it will be shown below, this being one of the three core angles of the Parthenon’s skeletal structure) of about 22.3 to the Parthenon’s axis of symmetry EF, Figure 1. Given that the inclination of the EF axis is about 10, see Figure 3, it follows that the AC diagonal axis points to the Summer sunrise rays at Solstice. The one or two degrees of possible difference is negligible. Given the uncertainty surrounding the change in the declination of the ecliptic (as the Earth’s axis of rotation traces an approximately 26K-year cycle) over the intervening 25 centuries; and in view of any other competing and compelling rationale for such a deviation from a pure and due East-West orientation of the Parthenon’s main symmetry axis; it can be asserted with some degree of confidence that the objective of the Parthenon’s architects Callicratis and Ictinos in sketching out the blueprint of the Parthenon was to align this key diagonal of the Temple dedicated to goddess Athena to the sunrise during Summer Solstice on the one end, and to the sunset during Winter Solstice on the other end. It could be an orientation that the prior to Parthenon Temple had. Of course, the orientation of the key diagonal line AC leaves one wondering what was the function (or intent) of the other diagonal’s orientation. Parthenon’s EF main axis of symmetry (at about 10 North of the due East-West orientation) is understood now (as it will be demonstrated in more detail in a bit) to have been decided in the pursue (and the result) of a desired AC orientation. This reason still leaves as an open question the other diagonal’s orientation. Further aspects regarding the Astronomy built into the Parthenon are discussed and analyzed in [1.1], whereas the basic issues regarding the Astronomy related shadows of monuments are found in work by this author in [1.3] and [1.4].
The Parthenon’s 2-d floor plan modulus sizes and some Ancient Greek units of measurement. From the manner in which the modulus (and associated grid structure) of the Parthenon’s skeletal structure is now set, one can derive a basic ratio: the number of narrow sides’ modular units (7) over the number of long sides’ modular units (16), or R’=16/7. This ratio (an important ratio in the exposition that follows) will be further refined in a bit, by weighing these units employing their actual average lengths. This ratio must be viewed in conjunction with the partitioning of columns ratio R=17/8=2.125 (counting corner columns twice) as discussed in an
16
earlier section. Of course, R’ is a function of R. These two ratios, R and R’ are critical in the analysis that follows. Ratios, R and R’ are potentially of some mathematical interest as well. They belong to a sequence containing factors constructed by the following rule: the starting value is the ratio R; the next term in the sequence be a ratio of a numerator less than a unit from the numerator of the prior term; the denominator also be less by a unit from that of the prior term’s denominator. It must be noted that all CGTs’ partitioning of their peristyle’s colonnade and modular units occupy two consecutive terms in a series within the class of sequences they belong. Each Temple has a fingerprint type sequence, belonging to a class of Mathematical Sequences where all CGTs with identical peristyles belong. Each class depends on the peristyle’s narrow/long side number of columns. The arithmetic values of these sequences are (in discrete steps) exponentially increasing functions of the order in the sequence. There are upper and lower bound limits in these sequences. These limits are functions of the narrow side’s number of columns. Other properties apply as well. For example, bulkier Temples (more square looking – column-wise than rectangular ones, where T1 is closer to T2) experience lower upper bounds. In the case of the Parthenon, T2-T1=9; in the case of the Temple of Epicurius Apollo, built by Ictinos as well, the T2-T1=15-6=9 also; similar to Epicurius Apollo is the difference and colonnade at Selinunte Temple E (see [1.1]); at Paestum, the Hera I Temple has a difference of T2-T1=18-9=9, and at Hera II has T2-T1=14-6=8. Further analysis on this topic is left to the interested reader. Upper bounds of such class of sequences enter the objective function underlying the efficiency conditions, to be addressed at the last section of the paper. Inter-columnia. A major finding of this paper, confirming prior work by the author on this subject, is the existence of different on the average inter-column distances (the distance is referred to as “intercolumnium” in the literature) between columns at the narrow (East/West) sides of the Parthenon, and the long (North/South) sides. Moreover, these inter-columnia are such that the narrow sides’ average inter-columnium is shorter than that along the longer sides. This subsection’s objective is to document this claim. Estimation of this difference will lead to a precise estimation of the Parthenon’s real modulus and the sizes of its rectangular grid pattern. To estimate the sizes of the two-dimensional modulus and associated grid of the Parthenon’s floor plan, one must create a system of equations on two unknowns. It will be shown that the average distance between columns at the peristyle (ΠΕΡΙΣΤΥΛΙΟΝ) of the Parthenon (and in almost all CGTs) along the long (usually their Northern and Southern) sides and along the narrow (usually the Eastern and Western) sides are not identical and neither are the two sizes (East/West and North/South) of the modulus’ grid. To do so, and following the algebraic system of equations first presented in [1.1], one need compute the values of the two variables at both sides of the Temple, having made an estimate of the sizes of the Temple at the stylobate (ΣΤΥΛΟΒΑΤΗΣ) level: 17
8x + 7y’ = L1 and
17x + 16y” = L2
(1)
where variable x designates the columns’ average diameter at its base on the stylobate (a known quantity or parameter here); y’ is the value of the average inter-columnium along the narrow (Eastern and Western) sides; y” is the value of the average inter-columnium along the long (Northern and Southern) sides; L1 is the length of the Temple at the stylobate along the narrow sides; and L2 is the length of the Temple at the stylobate along the long sides. This is the simple linear version of a far more complicated non-linear system on three variable (x, y’, y”) assuming uniform in diameter columns at all spots of the peristyle and average inter-columnia. The nonlinear comprehensive version of this system of equations in (1) will be presented in the next section of the paper. The pair of numbers (7, 16) are the number of spaces among columns at the narrow and long sides of the Temple correspondingly. If by T1 and T2 we designate the number of columns along the narrow and long sides of the Parthenon respectively, then: R=T2/T1=2.125, and R’=(T2-1)/(T1-1).
(1.1)
For the Parthenon, and using the current metric system, one obtains the following measurements, x=1.9 meters; L1=28.50 meters; and L2=66.80 meters. The last two counts have been estimated by the author and they are based on the diagrams of Figures 1 and 3. The first count, on the average diameter of the columns’ base (which is measured at the edge of the flutes of the columns’ shafts), is a widely used count, encountered for instance in reference [2.5] and its sources. However, the architects of Classical Greece did not use a metric system. Instead they used an assortment of units to measure lengths, areas, and volumes, see [2.6] for an introduction to them. Here a few will be mentioned, of assistance in deriving the measurements of key components of the Parthenon’s structure, as perceived by the architects and engineers then. They will also be used as evidence to support a major thesis of this paper: namely that no elements of the Parthenon’s structure are either exact fractions or multiples of any of these units. This realization forces the analyst to seek more basic components and processes as being the constituent elements in the design of the Parthenon, and some of these are identified here. Or, alternatively to assume that the construction of the Parthenon was not carried out according to a preestablished plan or blueprint, but it was instead the product of an ongoing improvisation and ad hoc decision making. Such a hypothesis has been fueled by the considerable (to some) variations in columns’ diameter and inter-columnia. However, this author rejects the latter hypothesis. A number of units of measurement used in Ancient Greece, seemingly of import here, are the following: the pous (ΠΟΥΣ) or “foot”, a unit of about 30.8 centimeters; the lichas (ΛΙΧΑΣ) or “ten fingers”, a unit of about 19.26 centimeters; the cubit or “pechys” (ΠΗΧΥΣ) or “24 fingers”, a unit of length measuring about 46.2 centimeters; the single bema (ΒΗΜΑ) or “step”, a unit measuring about 77 centimeters; the “pace” or a double step, about 154 centimeters; the “fathom” 18
(ΟΡΓΥΙΑ) about 1.85 meters; and the plethron (ΠΛΕΘΡΟΝ) a unit of longer distances, measuring about 30.80 meters in length. These units of measuring length are of some use within our scope to interpret and put into perspective the measurements we obtain in meters or centimeters, along with making the case for the major thesis to be advanced, just stated. Specifically, we shall see that none of the Parthenon’s basic structural components (the columns’ diameter or height, the stylobate’ width and length, etc.) are exact fractions or multiples of (but only close approximations to) any and all of these units of measuring lengths. Why it is so is a fundamental question in the Architecture of the Parthenon, and it will be addressed and explained in the next section, where and when the derivation of the Parthenon’s skeletal structure is presented and analyzed. On the basis of the above estimates and known lengths, it is suggested that the columns’ diameter is close to (but not exactly as long as) a fathom. The width of the Temple at the top of the crepidoma (the stylobate) is close to (but not exactly equal to) a plethron; and the length of the stylobate is close to (but not exactly equal to) 36 fathoms. These quantities result in the following average inter-columnia (measured in the metric system): narrow side, y’=1.90 meters and equal to the columns’ average diameter at their base, x being an exogenously determined quantity in the system of equations (1); while in the long side, the average inter-columnium is y”=2.16 meters. As presented first in the work by the author in [1.1], the narrow side average inter-columnium y’ is smaller than the long side average intercolumnium y”. This relationship, attributed mainly to Engineering Statics but also to Aesthetics as discussed in [1.1], is fundamental in CGTs’ design and construction: (y’ < y”).
(2)
This is indeed a major aspect of CGTs Architecture, and a topic that has gone largely unnoticed in the Literature, which focuses on measuring individual inter-columnia. We now turn to a closely related subject, equally neglected and (due to the errors by Vitruvius) largely misrepresented in the current literature on CGTs and in the Parthenon’s case as well. The modulus and its grid. Hence, we derive the average sizes M1 and M2 of the two-dimensional grid pattern’s unit of the Parthenon’s floor plan modulus. The total (narrow sides) extent of the grid is seven times the grid’s unit length, M1, along the East and West sides. Thus, we obtain at the narrow side: M1 = (x + y’) = (1.90 + 1.90) = 3.80 meters, so that 7x3.80=26.60 meters (this length being .87 of a ΠΛΕΘΡΟΝ). While, the equivalent dimension of the grid’s unit along the long (North and South) sides is: M2 = (x + y”) = 4.06 meters, so that 16x4.06=64.98 meters. This comes close to 2.1 times the length of a ΠΛΕΘΡΟΝ but not exactly twice it. Consequently, here we have clear confirmation of our hypothesis, that these sides of the modulus (and by extension the sides of the stylobate) were not constructed according to this or any other unit of length. They were rather derivatives of other considerations. 19
In [1.1] the author, due to their closeness, rounded up these two counts to an almost 4-meter square grid pattern modulus, which was an approximation to their estimated average sizes M1 and M2. This study, in that regard, slightly adds to the previous work by the author. In [1.1] the author estimated the vertical dimension of the Parthenon’s modulus M3 to be also about 4 meters, where it actuality is close to 4.16 meters, so that the Parthenon’s columns average height of 10.40 meters is approximately 2.6 times a four-meter M3. Hence, the 3-d modulus of the Parthenon is approximately a cube with side lengths of about four meters, and applied in ratios M1:M2:M3 as 1:1:2.5 in average modular units, a result roughly confirmed by this analysis although slightly refined, where we found that the modulus dimensional ratios are exactly (in meters) 3.80:4.06:4.16, and the relationship to hold: M1 < M2 < M3.
(3)
The reasons why these (otherwise unexplained) discrepancies in the 3-d modular grid average unit sizes exist will become clear in a bit. In addition, the 4-meter 2-d modulus (grid average unit size) of the Parthenon is close but again not identical to two fathoms (3.70 meters versus an actual 3.80-meter modulus narrow side). On the long side of the 2-d modulus (4.06 meters) the closest one comes to an ancient Greek unit of measurement (multiple or fraction) is the pace (2.5 paces to be exact). However, it is highly unlikely that two different units of measuring lengths were used to form the 2-d horizontal surface part of the Parthenon’s modulus. Thus, again, the rationale of a different way these Temple average sizes (x, y’ and y”) were arrived at (as it will be presented in the penultimate section of this paper) looms as a far more likely scenario. Earlier we encountered the modular units’ ratio R’, where the unweighted by their average actual lengths modular units, 7 (along the narrow sides) and 16 (along the long sides), were employed to derive it. Now we shall weigh these variables by their actual (estimated) average lengths, M1=3.80 meters and M2=4.06 meters respectively. The modified R” is now given by the expression: R” = (16x4.06)/(7x3.80) = 2.4421 This of course is the cotangent of the angle (BAC) from Figure 2, its tangent being .3914. An angle ’ with tan=.3914 is (from trigonometric Tables) equal to ’=21.375. Angle ’ is the core angle used in the previous subsection to derive the orientation of axis AC towards the Summer Solstice sunrise. As the estimation of this angle is based on an estimation of the 2-d modulus’ average grid it involves approximations which might be too rough. A more direct (and thus slightly more accurate) estimation of this core angle will be supplied later in the text (next sub-section). Thus, to derive the rectangle containing the floor plan of the Parthenon and its 46 columns at its peristyle one needs just (the located at some point in space) point A; the direction towards the sunrise at Summer Solstice; the length of the line AC; and the core angle ’. These two quantities (given point A) are necessary and sufficient to generate the Parthenon’s rectangular base, the stylobate, on which the columns of the peristyle are set. One may argue that two items of
20
information are also needed to draw a rectangle the conventional way: length and width, plus a predetermined (corner) point and orientation. However, here we were able to also obtain a specific reason as to why the particular orientation of the Parthenon (off by an otherwise unexplained 10 to the North of the due East-West axis) was selected. Location of the corner point could be related to other (exogenous) factors, possibly having to do with the physical morphology of the site at the top of the Athens Acropolis; the possible presence of a prior Temple at the site; and the relationships sought to govern the location of the Parthenon in reference to other Acropolis structures. One of these factors apparently had something to do with shadows cast by the Parthenon and other edifices at the Acropolis, as presented and discussed in reference [1.1]). Thus, the reader must note that here the information regarding the direction of the monument was endogenous to the analysis, and not exogenous. By solving the system of equations in (1), if the columns’ diameter at their base x is known, then the inter-columnia (for both the narrow, y’, and long, y”, sides) are directly derived. In a more general formulation, where x is also an unknown, one can endogenously also obtain the columns’ diameter at their base. But for that to be accomplished, one needs a broader framework – the subject of a later section of this paper.
North-Western view of the Parthenon. Shadows and the various roles shadows played in the location, as well as the functions of the Temple was discussed in [1.1]. ΕΝΤΑΣΙΣ and its connection to the shadows cast by the entablature upon the Parthenon’s columns was discussed, offering a novel explanation and rationale for its purpose and effects. Source of photo: [2.14]. 21
The Parthenon’s Skeleton: Its Three Key Triangles and Core Angles
Figure 2. The Parthenon’s skeleton in a 45 axonometric diagram showing the Temple’s basic Geometry, and the three key triangles in its structure (schematically, not in scale). Point H is the Parthenon’s barycenter and omphalos. Line HH1 is the intersection of two of the skeleton’s three key triangles, (CAD) and (FEG). Point H2 is the two triangles’ intersection line extended on the surface of triangle (FEG) till it crosses line EG. The three key right triangles and their core angles are: key triangle (BAC) and its core angle BAC (angle ); key triangle (CAD) and its core angle CAD (angle z); and key triangle (FEG) with core angle FEG (angle ω). Source of diagram, the author.
The Parthenon’s key triangle (BAC) and its core angle . We now return to the system of equations in (1). Before we provide the non-linear version of this complex system of equations involving three variables (x, y’, and y”), an independent check will be provided on the magnitude of angle ’. Previously, an estimate of ’ was derived on the basis of the 3-d modulus. Now it will be derived directly from the sizes of lines AB and BC of the Parthenon’s skeleton. Obviously, and given the L1 and L2 as given in the previous section, AB=66.8-1.90=64.9 meters; and BC=28.5-1.90=26.60 meters (the columns’ diameter x being 1.90 meters). The above triangular specifications of the right angle (ABC), as shown in the 45 axonometric diagram of Figure 2, result in a hypotenuse AC=70.23 meters and an angle BAC = = 22.3. Difference in and ’ is due to estimation errors, and can be disregarded. In the analysis that follows the 22.3 angle size will be employed. Its tangent is tan = 26.60/64.9=.40986. 22
23
Figure 3. The Parthenon’s floor plan, source [2.4]. The orientation of the floor plan’s axis of symmetry EF and the axis AC (a key axis in this paper’s analysis) are shown, with the direction of the North at the upper left-hand side of the diagram. The orientation of the AC axis is towards the rising sunrays at Summer Solstice to the East -Northeast, and towards the last glim rays of the Winter Solstice at the West-Southwest. Length and width of the crepidoma (ΚΡΗΠΙΔΩΜΑ) in meters are also indicated. Source of the final diagram: the author.
This is a core angle in the triangle (ABC), since the length of AC and precisely specify it. Moreover, a 22.5 angle is exactly one sixteenth of 360, and the angle (22.3) is extremely close to this angle size. This relationship is of extreme import (hint: the 22.5 angle can be easily drawn with just a compass and a ruler, and is a good starting value in the search for an optimum, the 22.3 value) in the formation of the skeletal structure of the Parthenon. Its import will be further analyzed and highlighted in the last section of the paper, where the heuristics involved in finding the optimal configuration of the Temple will be exposed. What is of course of import is the convergence to AC=70 meters. This represents a ratio of about 1:12.5 as the hypotenuse of the 2-d modular structure is: d=5.56 meters. Before we look at the other two key right triangles in the Parthenon’s skeletal structure which involve the Parthenon’s columns height and its height including the pediment at its entablature, a few remarks are necessary to address at this point the problem of the design specifications of the Classical Greek Temples in 2-d. These are introductory remarks to a fuller exposition of this central topic to this paper, to be offered in the paper’s penultimate section.
The Parthenon’s skeletal specifications problem. Let us now review the system of equations in (1). It was assumed there that the total number of columns (46 in an 8x17 arrangement), the columns’ average diameters (x), and the total length of the stylobate’s 2-d dimensions L1 and L2 are known (exogenously supplied) parameters. However, for the designer of the Temple, these are not exogenously supplied parameters, but variables the values of which the designer and engineer of the Temple is supposed to derive. It is precisely this type of questions we are set to address now more in depth, to the extent that the Parthenon is concerned. Are there any clues or any systematic procedure that can be set up to arrive at and answer the question of “how did the architects, Callicratis and Ictinos, settle on the configurations we currently record” on this classical of all Classical Greek Temples. Of course, Engineering and Statics had a lot to do with these variables; however, here on the architectural and purely mathematical (Geometry and Algebra related) aspects of these variables will be the focus. The original system of equations in (1) must be re-written in full as follows and in a nonlinear form:
24
T1x + (T1 - 1)y’ = L1; T2x + (T2 – 1)y” = L2
(4)
where T1 is the number of columns at the narrow sides, in the Parthenon’s case, 8, and T2 is the number of columns along the long side, in this case 17. Variables x, y’ and y” as well as L1 and L2 are as defined in the analysis surrounding conditions (1) – (3). The first three variables are expressed in their averages. System in (4) is a second-degree nonlinear system of two equations (plus the seven positivity conditions) with seven unknowns. It is noted, that among the seven unknown variables, T1 and T2 must be in addition to positive also integers. Obviously, the system is undetermined, which means that there must be other conditions to be supplied for it to have a set of finite solutions. Due to its nonlinearity, these solutions may or may not be unique. Mathematical Optimization Theory and the CGT problem. It is obvious then, that other considerations must be introduced, for the system to obtain endogenously derived solutions, and not be forced to have solutions by decreasing its degrees of freedom, by arbitrarily (and exogenously) specifying some of its variables. How to do this efficiently, is what this effort is all about. Moreover, we kept the design specifications of the cella and its interior configuration (interior columns and walls, and their specific morphology) out of this analysis, to simplify it. Adding the cella specifications (and following up from the analysis involving the seven ratios in reference [1.1]) could provide avenues to address this set of issues. What is pursued here now, in an effort to address the question of determining a Temple’s morphological specifications as the outcome of an endogenous calculus, is to detect the embedded instructions linking the three key right triangles and their core angle, as shown in the design of the Parthenon’s skeleton of Figure 2. It is reminded that for a right angle to be specified the length of any side and a core (one of the two non-90 angle) or one of its trigonometric specifications (sine, cosine, tangent or cotangent) are needed. In the mathematical Theory of Optimization, the information needed to obtain values of a system of variables where the number of variables exceeds the number of equations available (in effect, the number of constraints) is obtained through the system’s objective function. In that case, the optimization process (either by maximizing or minimizing the value of the objective function subject to the available constraints, hence optimally deriving values for the variables involved) provides the means to solving the system of variables by the first (and appropriate second) order conditions, in accordance with standard Theory of Optimization, see for example [2.11]. Standard Theory of Optimization (which will not be reviewed here, the interested reader can consult any reference from the very rich literature on this branch of Mathematics) affords the derivation of the dual formulation of a primal objective function. This platform and problem structure offer rich insights, as it points to the inherent dual nature of Optimization (involving both maxima and minima). This is a very promising avenue of research in the statement of the CGT problem of condition (4). This effort, towards an efficient approach to solving the Classical Greek Temple optimization problem, will be more formally addressed in the following section. 25
The Parthenon’s key right triangle (CAD) and its core angle z. From the previous section, the value of diagonal line AC is used (AC=70 meters) along with the height of the Parthenon’s columns (10.40 meters) for key right triangle (CAD). This combination of the main components of this key triangle results in the core angle CAD = z = 8.45 and a hypotenuse, see Figure 2, of length AD=70.77 meters. Its tangent is: tan z=10.40/70=.14857.
The Parthenon’s key right triangle (FEG) and its core angle ω. Now we examine the other key right triangle of Parthenon’s skeletal structure, seen in Figure 2, triangle (FEG). The dimensions of this right triangle are, long side EF=64.90 meters, and FG=17.3 meters, see [1.1]. On that basis, one derives the length of the hypotenuse EG= 67.14 meters, and the core angle FEG = ω = 14.85. The tangent of this angle is: tan ω=17.21/64.90=.26518.
Figure 4. Parthenon’s Southern corner of the entablature’s Eastern pediment. The estimated about 13 (12.7 to be exact) basis angle of the Parthenon’s pediment (formed by the ΣΙΜΑ framing the ΤΥΜΠΑΝΟΝ) is shown above. It is the place where the architects ΚΑΛΛΙΚΡΑΤΗΣ and ΙΚΤΙΝΟΣ embedded one of the constituent angles of the Temple. Source of photo: public domain. 26
The Parthenon’s pediment angle It is noted that the angle found on the Parthenon’s pediment, see Figure 4, is close to 12.7. It is the angle formed by the sima (ΣΙΜΑ) at the two bottom corners of the tympanum (ΤΥΜΠΑΝΟΝ), at the pediment’s structure. The angle at the top of the pediment is about 154.6. This result is directly derived from the Parthenon’s façade reconstruction, shown in Figure 3.4, p. 35, in reference [1.1], among a number of other reconstructions available in the literature, see for example that outlined in [2.7] and [2.8] the reconstruction of the West pediment shown in Figure 5. In this reconstruction, the top angle of the pediment is about 155, directly implying that the two base angles of the tympanum triangle are about 12.5. Furthermore, this result is confirmed by the line CD of Figure 2, which coincides to a great extent in the 45 axonometric with the righthand side line of the Temple’s pediment skeletal structure (at the point of the Parthenon’s North-East corner column’s center). The geometric proof of this is left to the interested reader to confirm. The pediment angle is about 10 less than the angle (22.3). By examining a number of reconstructions (some shown in this paper) the author estimates the tangent of the pediment’s angle to be about .22564, corresponding to an angle of about 12.71.
Figure 5. Parthenon, West pediment reconstruction from reference [2.8].
It is apparent that these three core angles of the three key right angles (8.45, 14.85, 22.3) and the 12.7 pediment angle are critically linked and intrinsically related to the instructions resulting in the Parthenon’ skeletal morphology. How is that set of instructions embedded into the Temple’s overall form, resulting in the pediment angles is the subject of the section that follows. 27
Three Lemmas and two Comments: three angles as the building blocks of the Parthenon Lemma 1: the key ratio. As a consequence of analysis in [1.1] it was estimated that the total height of the Parthenon (including the entablature) is 17.20 meters; see the relevant estimates at p. 35 in [1.1]. With this analysis, one derives now that the ratio of lengths HH1 to H1H2 is exactly 1.5. This is trivially obtained by the simple geometric property that: since HH1=10.40/2, and HH2=17.30/2, then HH1/(HH2-HH1) = 5.20/3.465 = 1.5. Point H of course is the omphalos of the monument as already pointed out. The 1.5 ratio is central, as it will be shown in the last subsection of the penultimate section of the paper. It deals with a heuristic based search for the optimum solution of the optimization problem underlying the Parthenon’s skeletal morphology. Lemma 2: the efficiency conditions. The three key triangles with their three core angles, three variables in all, plus the information that they are of the “right triangle” type and that two of these key triangles intersect at the barycenter H, plus the location of point A and the astronomical orientation based direction of axis AC, are the necessary and sufficient conditions to uniquely define the Parthenon’s skeletal structure in situ. Additional information concerning the Parthenon modulus, is in an optimization process that can generate the number of columns on each side of the peristyle, and the length, width and height measurements of the Temple. Comment 1. In effect, the above algorithmic process is a holistic approach to precisely specify both the location and orientation of the Temple, and optimally define the Parthenon’s skeletal morphology, as it will be further shown by Lemma 3 and the following section of the paper. Lemma 3: the instructions. The instruction on angles is derived by directly comparing the three core ones, 8.45, 14.85 and 22.3 and connect them to the pediment angle of about 12.7. A number of possible sequences (scenarios) can be drawn to generate these three-angle measurements. A possible sequence (although by no means unique) is the following: use the core CAB angle, , of 22.3, which is approximately one sixteenth of 360, as the base core angle; subtract from the base core angle 8 (1/32 of 360) to obtain a (gross) approximation to the core FEG angle, ω, of 14.85; subtract from the base core angle 10 to obtain an approximation to the pediment’s angle (12.7); and finally subtract 14 from the base core angle to derive an approximation to the core CAD angle, z, of 8.45. Thus, all four angles are linked to a base core angle (=22.3) and to differences in even steps (14, 10, 8). Comment 2. Hence, the proposition is put forward and the thesis is advanced in this paper that the Parthenon’s building blocks were three core angles that can be generated as a sequence (subtracting 14 and 8 correspondingly) from a base core angle – which is the sixteenth of 360. At the very center of this sequence of the three core angles (and the result of their meshing in forming the Parthenon’s skeletal frame) is the pediment’s angle 12.7. it is thus concluded that a set of instructions on angles is responsible for the Parthenon’s skeleton. Of course, numerous other sequences can be found to generate these angles. Integrating the three key right triangles (containing the three core angles) frames the specific morphology of the Parthenon’s skeleton of Figure 2. How this is done critically hinges on the location of point H, the barycenter of the Parthenon’s floor plan. The heuristic process involved is discussed next. 28
Stating the Parthenon’s Optimization Problem and its Heuristics
Introductory remarks: the Parthenon “genetic code” as a set of instructions The exposition that follows can take two avenues to unfold. One is the geometrically based avenue, in which the optimum configuration is attained by following a set of geometric instructions. The other avenue is a Calculus based procedure. It produces the (same) optimum values of the variables included in the Parthenon’s skeletal structure, except it does so analytically, employing differential Calculus. To those familiar with the Mathematical Theory of Optimization, this double approach to finding an optimum solution to an optimization problem is also present in the (simple) case of Linear Programming. There, a set of (linear) constraints bound the feasibility set; then the slope of the (linear) objective function is used to locate on (one of) the feasibility frontier’s corners a (unique) solution, that is a point which maximizes the value of the objective function subject to the set of constraints. This double in nature (but of course identical in essence) procedure will be employed here to derive the optimal values of the (geometrically in the form of a set of instructions and, at the same time, algebraically in the form of a Calculus) stated variables (components) giving form to the Temple’s skeletal structure. Of course, we have absolutely no evidence that 5th century BC Athens (even at its Golden Age) and its mathematicians, astronomers, architects and engineers were in possession of the types of Mathematics found in the contemporary mathematical Theory of Optimization. They were also not familiar with modern Genetics, where in the DNA structure (the genetic code of living organisms) as we now understand it, a set of instructions are embedded that are responsible for generating the living organism’s morphology and internal structure. To them, the “instructions” embedded in the mathematical specifications of the Parthenon’s modulus and overall morphology were in all likelihood intuitively derived. The geometric procedures they followed in effect were a substitute for searching for the optima of an optimization process regarding Parthenon’s form. These geometric procedures, that involved the modulus and the key right triangles and their core angles, will be exposed now, alongside the Calculus based procedures. It is justified to assert that the emphasis by the Parthenon’s architects was on the geometric instructions for creating a structure with angles (and right triangles) of the type discussed in the previous section going through point H, the barycenter of the Temple and its omphalos, while the Temple was anchored at point A with an orientation (line AC) towards the sunrise rays at Summer Solstice. These angles present a mathematical (geometric in this case) sequence. That sequence resulted in the unique solution which is manifested by the Temple’s specific morphological configuration. This configuration is translated into a specific number of columns (46), arranged in a configuration of 8x17 column peristyle, with specific (and differing) intercolumnia, and the Temple’s length, width and height. All that information is contained in the Parthenon’s skeletal structure morphology as shown in Figure 2.
29
Angles , z, and ω are by themselves and in combination of the existence and geometric properties of point H, sufficient to geometrically define the (unique) 3-d skeletal structure of the Parthenon. The ensuing specifications involving its 46 columns and crepidoma’s length and width as well as the columns’ height and the pediment’s height are algebraically derived once the geometric structure of the Parthenon’s skeleton is set. To do so, one need translate the aforementioned angles (by computing their tangents) into quantities involving lengths, of course. In other words, we need to state the necessary and sufficient conditions in terms of these variables by the way of core angles of key right triangles, for one to derive from the skeletal geometric structure, schematically shown in Figure 2, the algebraic (and differential Calculus) conditions of the system in (4). Doing so requires that the angles , z, and ω be derived and be drawn by means of a compass and a ruler (the only means available back then to doing Geometry). It turns out that these three angles are produced by a set of instructions, with a starting value as their base an easily defined and drawn angle, the core angle of triangle (BAC) – angle . This angle is very close to one sixteenth of 360 (22.5). The other two angles (z=8.45 and ω=14.85) are a bit more complicated. Their derivation, and the architects’ ability to draw them, requires an iterative method to be discussed shortly. In that iterative algorithmic procedure, the embedded instructions are followed in a step-by-step schema. Before we do so, however, a few notes are needed.
Two Notes Point G (see Figure 2) is now (for simplicity) the top point of the Parthenon’s pediments without accounting for the height of the top – middle – acroterion (ΑΚΡΩΤΗΡΙΟΝ) showing above the pediment’s sima (ΣΙΜΑ) framing its tympanum (ΤΥΜΠΑΝΟΝ). According to this analysis of the Parthenon, and in effect of all CGTs, there are some variables (elements of the Parthenon or any other CGT’s Architecture) that are included in the skeletal structure. They weigh more than others in a Temple’s construction. Here, the suggested hierarchical import of the architectonic elements discussed (having to do with a Temple’s location, orientation, floor plan and elevation) are superior in importance than the detailed sizes of the various sub-elements of the structure, such as the steps of the crepidoma, or the various sizes of the different components of the columns’ capitals, or the entablature’s elements (frieze, metopes, triglyphs, etc.) This does not necessarily imply that these second-tier components of a Classical Greek Temple are not important – in fact some of these elements and their dimensions are in effect time markers assisting us now on how to date the approximate construction phase of the Temple and even identify its architect(s) and sculptors. Moreover, their advanced mathematical sophistication in their detailed configuration is also a time marker in the Evolution of Form, Architecture, Engineering and Mathematics. Meticulous attention to detail is a hallmark of Advanced and sophisticated Architecture, and certainly this is the case involving the Parthenon. 30
The translation of angles into concrete variables of the CGT optimization problem In this sub-section, the basic (skeletal) Geometry of the Parthenon, and its embedded geometric instructions will be presented. The three core angles of the three key right triangles can be translated as follows, so that the full optimization problem’s constraints (or a priori conditions) are fully deployed in the form of a differential Calculus, as it will be subsequently shown. The geometric instructions must correspond to the set of conditions found in the 2-equation system (4), plus the positivity and integer conditions as stated earlier. On top of these positivity and integer conditions, one must assume (exogenous) upper and lower bounds, since the (integer) number of columns can’t be less or greater than certain limits. These limits can be statistically derived, by inspecting the currently available records of CGTs. It is reasonable to assume that a minimum of six and a maximum of nine are the columns encountered in the narrow sides of a CGT, while a minimum of ten and a maximum of twenty are encountered at its long side. Hence, in addition to the two conditions in (4), one must also assume the following conditions also to hold on the variable T1 and T2. Upper and lower bounds are not necessarily hard bounds (either for the case of the Parthenon or any other CGT) as other upper and lower limits can be tested in the geometric solution, and the differential Calculus based search for an optimum solution to be discussed following the Geometry based solution. Hence one has: 6 < T1 < 9 ,
10 < T2 < 20.
(5)
Or stated differently, in an Integer Programming form, these variables can only obtain one of the values in the vectors: T1 = [6, 7, 8, 9]’
T2 = [10, 11,...,19, 20]’.
(5.1)
Of course, these are suggestions (indeed, weak assumptions) as already stated. In the case of the Parthenon, how the solution regarding the peristyle’s colonnades (that is the optima) involved: T1* = 8, and T2* = 17.
(6)
will be demonstrated next. Optimum values for variables T1 and T2 (designated as T1* and T2*) and the optimal average values of variables x, y’ and y’’ (designated as x*, y’*, y”*) given the conditions in system (4) and the constraints in (5.1) emerge as the geometric solution to the problem of “fitting” or “meshing” the three key right triangles from the three core angles CAB, that is angle (the base angle of the entire system, and about one sixteenth of 360, that is 22.5, and with tan = BF/AB), CAD, that is angle z (8.45, with tan z=CD/AC) and GEF, that is angle ω (14.9, with tan ω=FG/EF). The columns’ ratio R (17/8=2.125), with the four corner columns being counted twice, and the (unweighted) modulus grid ratio R’ (16/7=2.2857…) are values derived from this geometric solution, a consequence of the two lines AC and EF intersecting at point H and the two key 31
triangles (CAD) and (FEG) intersecting at line HH1 under the relationship (see Lemma 1 of the previous section) linking line segment HH1 to H1H2 (responsible for the exact ratio of 1.5), as shown in Figure 2 and proved in Lemma 1. The ensuing modulus is the outcome of this (heuristically derived) geometric structure of the Parthenon’s skeletal structure. The step-by-step procedure is as follows.
The step-by-step heuristic derivation of the Parthenon’s skeletal structure and orientation We now present the systematic search for the optimum configuration of the Parthenon’s skeletal structure, shown in Figure 2. It follows a heuristic method in its Geometry (the Calculus from the nonlinear optimization process will be outlined later in this section). This process is necessary, as the exact drawing of the basic angles z, and ω can’t be done employing just a compass and a ruler, due to the fractions they contain. The keys in this procedure is the closeness to the angle 22.5 by (22.3, and how it is approached by successive approximations) and the stopping rule of attaining the exact ratio of 1.5 between the HH1 to H1H2 line segments. Step One: Set point A on the Acropolis site plan, an exogenous decision. Step Two: Set the line AC, oriented towards the sunrise of the Summer Solstice on the one hand, and towards the sunset of the Winter Solstice on the other end. Note: point C will be derived at a following step of this search process. Step Three: At point A draw a line AB at an angle approximately equal to 22.3. For example, an initial value of 22.5, an angle BAC which is exactly one sixteenth of 360 and can be easily drawn, can be a good proxy (starting value). Note: location of point B will be derived in a later step. Step Four: At point A and on a plane perpendicular to the ground intersecting it at line AC, draw a line forming an angle CAD equal to 8.45 (angle z). Starting at a 9 is a good starting value, as a 9 angle is precisely one fortieth of 360. Note: point D will be located by a later step. Step Five: At point A draw a line AE perpendicular to AB towards the North-Northwest. Note: The decision to form a rectangular Temple skeletal 2-d structure is exogenous to the above described process. Note: point E will be derived from an iterative procedure (and this constitutes part of the heuristic aspect of the optimization process). Step Six: At line AE, on the plane perpendicular to the ground, and from any point E draw a line forming an angle FEG equal to 14.85 (angle ω). Starting at a 15 (1/24 of 360) is a good starting value, as it was the case for step four and angle z. The two planes containing angles ω and z intersect (designate that line as HH1). Point H on the ground is the barycenter of the 2-d schema of the Parthenon’s skeletal structure and its 2-d modular grid. Points F and G will be derived from a later step.
32
Step Seven: At a distance from point E on the AE line, mark point E’ (note: E’ is not designated on the diagram of Figure 2) such that AE=EE’. Step Eight: At point E’ draw a line E’C perpendicular to AE (hence, perpendicular to AE’ at E’), a line which is also parallel to line AB. This line intersects the AC line at point C. Note: this is how point C is obtained on line AC drawn earlier at step 2; furthermore, the intersection of line AC and the vertical to the ground line CD at point C is how point D on the plane containing triangle (CAD) is arrived at from step 4, and this key right triangle (CAD) is formed. Finally, point B from step 3 is also defined, thus forming the key right triangle (ABC). Step Nine: On the plane vertical to the ground and containing line CD, draw a plane parallel to line AE’ (which is also then vertical to the line AB at point B). This defines points F and G from step six, and thus the key right triangle (EFG). It also identifies point H2 on the HH1 line, this being the intersection of the two key right triangles (ACD) and (EFG). Step Ten: Compute the ratio HH1/H1H2. If different than 1.5 (a critical value obtained from the previous section) then repeat search process from step 3.
Heuristics, the Parthenon and the CGTs The step-by-step process presented above applies to the specific case of the Parthenon’s skeletal structure. However, under appropriate modifications it might apply to other CGTs as well. It contains in toto four loops: the first and longest loop is between steps 3 and 10 (it adjusts the size of angle with a very good starting value at 22.5); a second loop is between steps 4 and 10 (it adjusts the angle z); a third loop is between steps 6 and 10 (it adjusts the angle ω); a fourth loop is between steps 7 and 10 (it adjusts for the magnitude of line segment AE). Step 8 makes simply an accounting of whether the three key right triangles are formed; and step nine computes the magnitude of the resulting critical ratio and its deviation from the exact value (1.5). In this heuristic process, the stopping rule involves the formation of the key right triangles and the arrival to the ratio of 1.5. Within this ratio, the conditions set (integer total number of columns and their partition into T1=8 and T2=17) and the quantities of x, y’ and y” are obtained. These quantities are obtained thus, because numbers 17 and 8 are the only integer numbers which precisely and perfectly subdivide the line segments AC and BC. Steps 1 to 10 listed above outline a qualitatively equivalent heuristic process of locating an optimum within the constrained set (and on the feasibility frontier) of a Linear Programming problem. In this case, things are a bit more complicated, because it is a Nonlinear Programming problem we are confronting. Magnitudes of changes in the heuristic procedure sketched, and the direction of change are fine points in need of further analysis, not to be supplied here. Hence, we arrive at a conclusion demonstrating how a set of instructions on angles involving a sequence of steps in the meshing of three key right triangles generate the modular structure (and grid) of this Classical Greek Temple’s skeletal morphology. 33
But this optimal values can be obtained as well by appropriately solving a Calculus based optimization problem, the possible nature of which and the meaning of its objective function are topics to be pondered next.
The CGTs Objective Function: Primal (Harmony) and its Dual (Costs) We are now set to head-on confront the core argument of this paper: that the Parthenon, as an architectonic structure, was the outcome of an optimization process, the morphological result of a heuristic algorithm partly embedded in a sequence built into the Temple’s skeletal structure (its key right triangles, their core angles and their links as discussed earlier in the text of the previous sections and especially the prior sub-section), that produced the various components of the structure as optimal solutions to a problem in optimization. We suggest an interpretation of this objective function: the maximizing of a harmonious coexistence among the Parthenon’s various sizes of all the elements (number of columns, their average thickness, and their average inter-columnia) of the Parthenon’s sophisticated skeletal structure and modulus. This syllogism implies that the objective function is an Aesthetics driven function. Employing averages in the optimization function, aggregate analysis can be carried out. However, various classes of these variables can be explicitly recognized. In which case, the optimization problem acquires a disaggregate form. For purposes of illustration, the aggregate analysis case will be shown below. Parametric specifications of the objective function contain within them trade-offs. These parameters weigh the importance of the various state variables (reflecting the concrete morphology of the Temple) as to their relative contribution to the value of the objective function. The objective function is optimized subject to constraints. Next, we analyze the form of these constraints, conditions reflecting the Parthenon’s overall (albeit skeletal) morphology. In conditions (4) and (6), where (4) is a generalization of (1), the constraints of the optimization problem were set, plus the positivity conditions on x, y’ and y” and the integer conditions in (5): 0 < x, y’, y”
(7).
Again, the location of the anchor point A (an exogenous decision, possibly related to a prior Temple on this site), and the orientation of the axis AC (an endogenous decision in the overall process outlined here, but possibly associated with the prior Temple as well, and where its angle to the due East-West orientation is an endogenous variable to the specific problem of optimization) are given as specified earlier. Now the focus is exclusively on the optimization problem’s objective function involving the following set of variables: x, y’, y”, T1, T2, L1, L2. These are the seven state primal variables involved in a primal objective function, where all seven are linked so as to maximize the harmonious relationship simultaneously among them all. In abstract, the general primal objective function stated as to maximize Harmony can be written as: 34
Maximize = F{x, y’, y”, T1, T2, L1, L2}
(8)
subject to the constraints in (4), (5), (6) and (7). By substituting the values of L1 and L2 from conditions (4) into , one has the transformed objective function ’: Maximize ’ = F’{x, y’, y”, T1, T2}.
(8’)
This is an objective function on five unknowns (state primal variables), subject only to positivity conditions in (7), regarding the state variables x, y’, and y” and the integer conditions regarding T1 and T2 found in (5). Of course, through T1 and T2, the R and R’ enter the objective function, and thus, all considerations brought up in prior sections about Number Theory and Mathematical Sequences become of essence. The specific, nonlinear, parametric and algebraic form of is of course subject to differing formulations, which will not be further elaborated here, as they involve mathematical aspects far beyond the objectives of this paper. But speculation as to the possible (and quite likely) interpretation of this -function is appropriate at this point. It is of little doubt that Harmony among the Parthenon’s components, and through them the projection of Harmony in human relationships, was a critical issue of import at that time, manifesting itself in both Art and Architecture, as well as in Athenian society and its broader culture during Pericles’ Golden Age. Harmony can be argued to be present in the face of structures embedding relationships such as the Golden Ratio, or the Ratios found in other irrational numbers (such as the Silver Ratio and ). Such Ratios have played a role in the overall design specifications as well as in the details of edifices (Temples and other private as well as public buildings). They have been detected in the design of artifacts of the time. Is it possible that some of these Ratios were involved in the function? This remains an open question, and a subject for future research. Given the known Parthenon skeletal form, one can work backwards, deriving the Parthenon’s original design objective function, by analyzing the Optimization Problem’s first order conditions. These conditions require the first partial derivative of the Lagrangian function L, a function which includes and a set of five Lagrange multipliers (the vector ’, associated with the primal formulation’s set of five constraints), with respect to each primal state variable to be set to zero: L = - 1C1 - 2C2 - iDi
(9)
L/x = 0, L/y’ = 0, L/y” = 0, L/T1 = 0, and L/T2 = 0
(10)
where L is the Lagrangian objective function; and the vector ’ are the five Lagrange multipliers (and dual variables) associated with constraints C1 and C2, these two being the two integer requirements associate with the number of the peristyle’s columns in the narrow and long sides correspondingly; and iDi are the Lagrange multipliers i(i=1,2,3) associated with the three positivity conditions of variable x, y’ and y” respectively. The vector of the multipliers ’ is in effect the vector of the Optimization Problem’s dual variables. 35
Equations in (10) provide the necessary conditions for optimality. Sufficiency is met by the second order conditions associated with the optimization problem. Since the primal objective function is maximized, the second order conditions (the second derivatives) must be negative. For a standard introductory reference to this and associated issues in the Theory of Optimization, the reader is directed to reference [2.11], which has a nice exposition of the economic aspects involved here as well. Economics are of essence, since architectural construction is the subject. In deriving the full form of the primal objective function , and thus the form of its Lagrangian L, we need to also revisit the Number Theory aspects of the numbering involved with the 46 columns of the Parthenon’s peristyle discussed earlier, and we will also revisit the number N1=32, and N2=55. We shall also touch upon the nature of the ratios R and R’. In so doing we will revisit the discussion on the embedded sequence. One might be willing to speculate that the optimization problem, i.e., to maximize subject to the stated constraints, entails a quantity and a function in which the product of these numbers (or a nonlinear parametric function of them), N1 and N2, R and R’, must be involved as well. In this case, part of the objective is to create a set of column numbers (along the narrow and long sides of a Temple) such that their product is maximized, always subject to the constraints stated. These constraints guarantee that the three resulting main right triangles produce core angles as specified. Of course, the solution to this integer (due to T1 and T2 being integers) optimization problem is also heuristically derived as indicated previously, and being reflected in the geometrical configuration of the Parthenon’s skeletal structure, provided in Figure 2. It is widely accepted that Classical Greece was the place and time that Harmony entered the Arts and Architecture. Interpreting in detail and elaborating on the notion of “harmony” isn’t of course the subject of this paper. This is a complex notion, subjected to multiple interpretations and culturally determined formulations over millennia. But in essence, what is offered here is a concrete method in which this complex, abstract and largely qualitative notion acquires a quantitative representation. The primal objective function (which incorporates all the aforementioned variables, comprising the Parthenon’s skeletal structure and modulus) can be aptly interpreted as an expression of Harmony. The constituent elements are nothing but the components of this Harmonious coexistence at the optimal value of the composite objective function. However, this is by no means the only interpretation one might be motivated to attach to this primal objective function ; others may suggest different interpretations, which might be at least equally appealing interpretations. This task is left to the interested reader to ponder. Before concluding this exposition, let us briefly elaborate on the dual formulation of this Theory of Nonlinear Integer Optimization Problem, since this aspect of the Theory of Optimization is critical in fully understanding what is involved in the optimization process. The dual objective function, to be designated here as is of course to be minimized subject to the equivalent dual constraints. As is well known in the Theory of Optimization, at optimum, * = * (that is, the maximum of the primal equals the minimum of the dual – if the primal objective function is to be 36
maximized, conversely if the primal objective function is to be minimized, then the dual objective function is to be maximized). Moreover, the Lagrange multipliers that are in effect the dual variables of the dual formulation (associated with the dual constraints) are the primal variables. Another critical condition linking the primal and dual formulations at optimum is that the product of the primal variable at optimum times the corresponding dual constraint be zero; and the product of the dual variable at optimum times the corresponding primal constraint be zero as well. Let’s briefly address what might be the interpretation of the dual objective function . In the field of Economics, see [2.11], the standard interpretation of the dual of a primal maximizing objective function dealing with tangible quantities, is that they (the dual variables, in this case the array of ’), represent the real ‘economic cost” or “opportunity cost” or “real economic rent” of the primal quantities entering the primal objective function. In the case of stating the dual of a primal objective function in the optimization process involving CGTs, the likely interpretation of the dual variables (the Lagrange multipliers corresponding to the primal set of constraints) is that they represent bundled (or “comprehensive”) engineering and economics related costs embedded in a dual function that the designer wishes to minimize. This is a suggested interpretation, but other interpretations are also plausible.
Instructions, the skeletal structure and the modulus: some thoughts. A clue as to the presence of an optimization process at work, what in contemporary parlance is referred to as a heuristic algorithm in search of an optimum solution to a Theory of Optimization type problem, is a dep seated paradox that surrounds the Parthenon. Given that the architects of the Parthenon paid extraordinary attention to detail, one would expect to record measurements of its structural components that are exact fractions or multiples of known to exist at the time units of measuring lengths, areas, volumes etc. Yet the actual sizes of certain basic elements of the Parthenon like its façade’s length or the length of the Southern side, or the columns’ height, do not seem to be either exact multiple or exact fractions of known units of measurement used back then. The fact that none of the actual (as estimated at present) features-components of the Parthenon exactly fit any of the known ancient units of measuring length, area or volume, as indicated in an earlier section of the paper, or at least they do not do so in a consistent manner, might be an indication of two things: first, that these components’ lengths were the result of the optimization process (or instructions sequence resulting in some configuration) which produced the geometric form of the Parthenon’s skeleton as shown in Figure 2; and second, that deep inside the Parthenon’s structure, there are some key components (found in its skeletal morphology) which are exact and beyond any approximation and accuracy of measurements doubts in their derivations. In fact, we located two of them. They were related to angle - an angle linked to the 37
Parthenon’s orientation; and to the ratio 1.5, linked to the Temple’s barycenter in its 2-d floor plan, where the featured attraction of the Parthenon was located, the statue of Athena. Basic features of the Parthenon, namely elements (T1, T2, R and R’) of the Parthenon structure, as well as x, y’, and y”, were not the initial elements of the design process, but the consequence of some other more primordial and fundamental design considerations. They were in effect the outcome of a heuristic process involving a set of fundamental instructions embedded in the formation of the Parthenon’s skeletal morphology. Put differently, the Parthenon’s 7x16 2-d modulus and also the modulus in its unfolding in 3-d sizes, were the outcome of the more basic instructions involved in meshing the three key right triangles and their three core angles to form the skeletal outline of the Temple as shown in the author’s diagram of Figure 2. One step in these instructions was the derivation of the angle shown at the Parthenon’s two pediments, the Southern corner of the Eastern pediment shown in Figure 4. The various discrepancies observed between the actual components and the fractions or multiples of various ancient Greek units measuring lengths are due to the formation of the structure obeying these instructions. The exact replication of the three core angles and the pediment’s angle of Figure 4 are evidence that these triangles, angles and instructions were the starting steps of the design process, whereby the final skeletal form was heuristically derived. To this heuristic one may also attribute the differentials among columns’ diameters, and intercolumnia. An exact replication of the currently observed individual column dimensions and intercolumnium would require a powerful computing, based on parallel processing procedures to use. Whether this parallel processing would be able to precisely replicate the current Parthenon morphology at a level of approximation leaving no reasonable doubt is itself doubtful. It could be that the architectonic production process that resulted in the Parthenon as it now stands (in all the minute details) is not possible to reproduce, being a unique type of experiment (route towards solution, in a nonlinear optimization problem) in the search for an optimum. Going back to the point, regarding our inability (or failure) to replicate specific elements of the Parthenon’s Architecture by using Ancient Greece type units of length, as evidence that deeper quantities (associated with the skeletal structure of the Temple, angles and ratios and on objective function largely responsible for these observed lengths) one may also reasonably counter-argue that the differences we record between the exact fractions or multiples of the various units of measuring lengths available to the architects and engineers at the time of Classical Greece and its Golden Age, i.e., to the architects ΚΑΛΛΙΚΡΑΤΗΣ and ΙΚΤΙΝΟΣ, and the actual components of the Parthenon as we have them today and measure them, may also be the result of the weathering over the millennia of these Pentelic marble elements of the Temple; as well as the humanly caused destruction the Parthenon experienced over time. However, it is suggested here that the main factor is not the latter, but rather the former. Earlier, and by this analysis, the author rejected the competing hypothesis that the cause of all these discrepancies was due to the possibility that the architects built the Parthenon not on the 38
basis of a blueprint but on an improvising, unplanned mode. To the contrary, the findings here supply evidence that a very careful and sophisticated sketch plan, involving a set of instructions on angles plus a core and important ratio, were behind the construction of this Temple.
Primitive Pythagorean Triples and the Parthenon’s Triangles Before concluding this study, it would had been imprudent if one were not to check whether the three key right triangles of the Parthenon’s skeletal structure, as well as half of the triangle embedded at the Temple’s pediments (shown in part in Figure 4) are in fact Primitive Pythagorean Triples (PPTs). The author demonstrated in his paper on Le Grand Menec monument of standing stones at Carnac, Brittany (see reference [1.4]), that a tradition in monumental Architecture may have existed, even prior to Pythagoras, that could possibly entail at least some knowledge and potentially some minimum understanding of PPTs since the 5th millennium BC. Hence, quite likely in a post Pythagoras world of Mathematics in 5th century BC Classical Greece, the implanting in monumental structures of some key triangles from the hierarchical list comprising the Tree of PPTs may had been underway, and thus it might be possible to detect some of them, implanted in the structures of CGTs and quite likely in the case of the Parthenon, at present. It is worthy thus asking the research question of whether any (or all) of these four important right triangles (the three key ones and that at the pediments) of the Parthenon, that were identified here and possibly other triangles found on the Parthenon’s various components, are part of that list. In [1.4] links to some introductory references on the topic of PPTs are found, for the interested reader. Their mathematical property is that the three sides’ length (measured in modular units) of these right triangles are coprime. Such a search entails a systematic (possibly computerized) examination of the candidate PPTs as to how well they match these four key triangles of the Parthenon. Here, a rather rough approach will be taken, whereby three possible PPTs will be tested as to how close they replicate these four important Parthenon right triangles. In a, more or less, random search approach, two attributes of the four Parthenon right triangles were compared to three PPTs chosen to be tested: core angle, and length of the hypotenuse (expressed in modular units). It is recognized that the search requires confirmed accuracy and excellent approximations to be available on the Parthenon’s right triangles sides’ lengths and angles, for the conclusions to enjoy a high confidence level. Hence, the same caveats apply as those stated earlier. The three PPTs examined were: (7,24,25), (11,60,61), and (13,84,85). They were selected because their hypotenuse is close to their long side – an attribute exhibited by the four Parthenon right triangles, but especially triangles (CAB) and (FEG). In the three PPTs chosen to investigate, their corresponding core angles are: 16.3, 10.4, and 8.8 respectively. It would seem then that the only candidates for PPT is the right triangles (CAB) – possibly corresponding to PPT (13,84,85). However, this topic needs further study, and of course better approximations to the actual sizes of the three key triangles in the Parthenon’ skeletal structure and the pediments’ triangle. 39
Conclusions and Suggestions for Further Research A number of conclusions can be drawn from the above analysis. First and foremost is the conclusion that the Parthenon was built not on the basis of setting a priori the number of columns and their split along narrow and long sides; or by pre-setting length and width measurements of the Temple’s rectangular floor plan. Instead, it is established by the analysis provided that these components of the Parthenon’s overall morphology were the result of a far more complex algorithm (or a set of instructions) involving a process of optimization. This is not to imply that the Parthenon’s architects ΚΑΛΛΙΚΡΑΤΗΣ and ΙΚΤΙΝΟΣ were familiar with either Genetics (the DNA embedded sort of instructions) or the mathematical Theory of Optimization and its various branches (including Integer Programming and Computing). Apparently, they had a strong intuitive sense of the Heuristic Geometry based algorithmic process involved in the search for optima (or satisficing conditions) in an optimization process. Their intuition is to be appreciated in conjunction with their search for harmonic relationships among the elements of the Classical Greek Temple. Their reliance on the angle (one sixteenth of 360) and the exact ratio of 1.5 in the H1/H1H2 line segments at the barycenter (omphalos) of the monument, where ΦΕΙΔΙΑΣ located his statue of ΑΘΗΝΑ ΠΑΡΘΕΝΟΣ, are architectonic components one must marvel at and at least deeply appreciate. These two quantities were the building blocks of the Parthenon’s skeletal (and possibly entire) structure’s unfolding in 3-d. A major conclusion of this work is that the Parthenon (and possibly all Classical Greek Temples) were built according to a heuristic mode in the search for an optimum solution. The optimality condition was based on an objective function that was nothing else but a comprehensive harmonious relationship linking all basic elements of the Parthenon’s (and possibly in the case of all CGTs) skeletal structure. Maximizing this harmonic relationship embedding objective function, subject to certain realism containing constraints on the number of columns in the Temple’s peristyle, formed the framework within which the Parthenon (and quite possibly all CGTs) were made. The search involved central components of the Temple’s architectonic elements beyond the total number of columns at its peristyle, like for example their ratio of splitting them into narrow versus long sides’ colonnade, the Temple’s length, width and the columns’ height, etc. It could be that the Parthenon, in this genre of architectonic construction of that Era (Athens’ Golden Age), was an optimum optimorum type of solution. This will require a more elaborate research effort involving a number of CGTs to confirm. In this regard, this paper can be viewed as having setting up a program of research activity. It was found that the basis of the Parthenon was three key right triangles, and their three core angles. The building blocks of this basis were angle and the ratio 1.5, plus a set of instructions governing the generation of the other two core angles in the Parthenon’s skeletal structure. In this interplay among the three key triangles, the central role of the Temple’s barycenter and omphalos of the monument was highlighted. A fourth angle, related to the core angles’ 40
underlying instructions to build the Temple, was imprinted onto the Parthenon’s Eastern and Western pediments. Another conclusion derived here is that the split of a Temple’s 46 columns at its peristyle between the narrow (8) and long (17) sides was a derivative decision tied (among other factors) to the Number Theory aspects of the CGT’s colonnade. This is an area which requires also further research. Moreover, in the paper’s conclusions it must be noted that, the finding (first reported and documented in a previous paper by the author) that the inter-columnium along the narrow side (usually, although not always, this being the East and West sides of the Temple) is smaller than the inter-columnium along the long side (usually, but not always, this being the North and South sides) of the CGT was reconfirmed. The average inter-columnium along the narrow side y’ was found to be equal to the average diameter x of the typical column at its base (y’=1.90 meters); whereas the average inter-columnium along the long side (y”) was found to be 2.06 meters. Finally, it was reconfirmed that the modulus of a CGT is the distance between the very center of two neighboring columns; and that the Temple’s modulus length is the sum of an average column’s diameter at its base on the stylobate plus the side’s average inter-columnium (x+y’, and x+y”). On the average, it was also re-affirmed that the Parthenon’s 3-d modular grid was a cube of approximately four meters in length on each side. The closest to this modular size (four meters) any ancient Greek unit of measurement comes to is two fathoms (about 3.70 meters). Using the Greek cubit, it would require about 8.79 of them to come close to the 4.06 meters modulus’ sides length. It is abundantly clear that these Parthenon components were not derived using any of these units. All estimated measurements reported here, and the resulting relationships need to be confirmed by other researchers. Before being admitted as evidence of the suggested optimization theory based propositions put forward here as underlying the Parthenon’s design. This is usual scientific practice, and it applies to all empirical studies. In this regard, this study is no exception. A more refined exposition of the method proposed, involving closer approximations to the measurements of the various Parthenon components, and especially those incorporating the entasis effects of the structure, is a way to also extend and elaborate on the topics of this paper. Two subjects of potential mathematical interest were brought up, as a result of analytically approaching the Parthenon’s (and in fact all CGTs’) morphology and in particular its (and their) colonnades. One had to do with Number Theory, the other with Mathematical Sequences. It is expected that these two areas of investigation, sprung from the detailed analysis of these Temples’ colonnades, might open up new areas of mathematical research.
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Epilogue There are numerous aspects of Architecture, Art and Engineering; of History and Culture; of Economics, Politics and Religion; and of Mathematics of course, that this marvel of monumental construction has stimulated writers and analysts, students and researchers in all these fields of human endeavor to record, contemplate and upon which to reflect. A good summary is found in [2.12]. It is neither astonishing nor surprising then, that this detailed view of the Parthenon’s skeletal morphology evokes work in at least five additional fields of Mathematics, with interesting findings added to its already well stocked arsenal of wonders: Mathematical Theory of Optimization; Algorithmic Computing; Number Theory; Theory of Sequences; and the Theory of Primitive Pythagorean Triples. To those, one might also add Statistics, a subject alluded to here but not covered in any detail. In simply addressing the Theory of Optimization part of this list of fields in Mathematics, one is led to the suggestion that the Parthenon’s construction possibly attained the maximum maximorum in its objective function, indicative of the accomplishment that was achieved, when compared to that of any other Classical Greek Temple. This attainment might reveal the reason why the Parthenon has been singled out as the pinnacle of Classical Greek Architecture within this category of monumental construction, and possibly beyond it. Undoubtedly, a number of additional areas of Mathematics are still awaiting work, to be fully employed in an ongoing effort to fully gauge and grasp the Temple’s inner workings as well as its impact on the Human Heritage.
The Parthenon from a South-West view. Source: [2.13]. 42
References Author’s work [1.1] Dimitrios S. Dendrinos, 10 April 2017, “Moving Shadows and the Temples of Classical Greece”, academia.edu. The paper is found here: https://www.academia.edu/32383784/Moving_Shadows_and_the_Temples_of_Classical_Gree ce [1.2] Dimitrios S. Dendrinos, 24 January 2017, “The Mathematics of Monoliths’ Shadows”, academia.edu. The paper is found here: https://www.academia.edu/31101997/The_Mathematics_of_Monoliths_Shadows [1.3] Dimitrios S. Dendrinos, 7 April 2017, “The Dynamics of Shadows at and below the Tropic of Cancer in the Northern Hemisphere”, academia.edu. The paper is found here: https://www.academia.edu/32426535/The_Dynamics_of_Shadows_at_and_below_the_Tropic _of_Cancer_in_the_Northern_Hemisphere_update_1 [1.4] Dimitrios S. Dendrinos, 21 November 2016, “A Carnac Conjecture: Neolithic Experimentation with Primitive Pythagorean Triples?”, academia.edu. The paper is found here: https://www.academia.edu/30163918/A_Carnac_Conjecture_Neolithic_experimentation_with _primitive_Pythagorean_triples [1.5] Dimitrios S. Dendrinos, December 1991, “Methods in Quantum Mechanics and the SocioSpatial World”, Socio-Spatial Dynamics, Vol.2, No.2: pp 81-108.
Work by others [2.1] http://www.goddessathena.org/Museum/Temples/Parthenon/Parthenon_NW_from_NW_rec.html [2.2] http://www.dvdtalk.com/reviews/33095/nova-secrets-of-the-parthenon/ [2.3] https://en.wikipedia.org/w/index.php?title=File%3AParthenon-top-view.svg [2.4] By Io Herodotus - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=56293415 [2.5] http://www.ancient.eu/parthenon/ [2.6] For an Introduction to Classical Greece units of measurement, see for example reference: https://en.wikipedia.org/wiki/Ancient_Greek_units_of_measurement
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[2.7] http://www.ancientgreece.com/s/Parthenon/ [2.8] http://www.newworldencyclopedia.org/entry/Parthenon [2.9] https://www.britannica.com/topic/Parthenon [2.10] http://ancient-greece.org/architecture/parthenon2.html [2.11] Michael D. Intriligator, 1971, Mathematical Optimization and Economic Theory, Prentice Hall, Englewood Cliffs, New Jersey. [2.12] http://ancient-greece.org/architecture/parthenon.html [2.13] http://www.ancient.eu/image/943/ [2.14] http://imgarcade.com/greek-architecture-parthenon.html [2.15] http://www.metrum.org/key/athens/dimensions.htm [2.16] https://oedipustheking.wordpress.com/what-oedipus-rex-tells-us-about-ancient-greece/
Acknowledgements The author wishes to acknowledge the contributions made to his work by all of his Facebook friends, and especially by the members of his twelve groups the author has created and is currently administering. Their posts and comments have inspired him in his research over the past two and three-quarter years. But most important and dear to this author has been the almost 22 years of encouragement and support he has received from his wife Catherine and their daughters Daphne-Iris and AlexiaArtemis. For their continuing support, assistance, encouragement and understanding for all those long hours he allotted doing research, when he could have shared his time with them, this author will always be deeply appreciative and grateful. Special thanks, in addition, are due to my daughter Daphne-Iris who assisted me in the trigonometric analysis of this paper.
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