Here are a few examples of how the "sacred" has entered into geometry in different eras and cultures:
1) The ancient Greeks assigned various attributes to the Platonic solids and to certain geometrically-derived ratios, investing them with "meaning." For example, the cube symbolized kingship and earthly foundations, while the Golden Section was seen as a dynamic principle embodying philosphy and wisdom. Thus a building dedicated to a god-king might bear traces of cubic geometry, while one dedicated to a heavenly god might have been constructed using Golden Section proportions.
2) When Hindus (ancient and modern) plan to erect any ediface for religious purposes, from a small wayside shrine to an elaborate temple, they first perform a simple geometric construction on the ground, establishing due East and West and constructing a square therefrom. (It's a simple, elegant piece of work, at about the level of high school geometry). Upon this dioagram they lay out the entire building. The making of this geometric construction is accomanied by prayers and other religious observances.
3) The Christian religion uses the cross as its major religious emblem, and in geometric terms this was elaborated during the Medieval period to the form of an unfolded cube (reminiscent of example #1 above, where the cube was equated with kingship). Many Gothic cathedrals were built using proportions derived from the geometry inherent in the cube and double-cube; this tradition continues in modern Christian churches to the present time.
4) The ancient Egyptians discovered that regular polygons can be increased while still maintaining the ratio of their sides by the addition of a strictly constructed area (which was later named the "gnomon" by the Greeks); the Egyptians assigned the concept of the ratio-retaining expansion of a rectangular area to the god Osiris, who was, therefore, often shown in ancient Egyptian frescoes seated on a square throne (square= kingship again) in which the original square and its L-shaped gnomon are clearly delineated, but the geometrical construction used to create the gnomon is not shown. It is, in fact, the absense of the attendent arcs and extension lines used in the creation of geometric forms that has led art historians and iconographers such a merry chase through history. It often takes the eye of a geomterician to spot the tell-tale signs of construction.
5) One of the best-known pieces of detective work in this regard was the discovery by Jay Hambidge, an art historian at Yale University during the 1920s, that the spirals on the Ionic column capitals of ancient Greek temples were laid out by the so-called "whirling rectangle" method for creation of a logarithmic spiral. He realized this by examining numerous Ionic capitals in art museums until he located some in which the holes made by the placement of compass points had not been obliterated over time. (One of these capitals was an unfinished, broken piece, dug up from a rubbish heap near a temple -- it had apparently been damaged during manufacture and was discarded; its burial preserved it from the elements, and the marks of the geometeric layout were remarkably clear upon it.) No "sacred meaning" for the log spiral form of the Ionic column capital has been determined from Greek writings, but the use of other log spirals in Greek temple architecture (for instance in floor-block proportions and their placement in relation to overall floor area) indicates that Greek architects, unlike the Romans who came after them, deliberately constructed their temples according to "whirling rectangle" geometeric ratios.
I could multiply examples almost endlessly.
Not everyone who catalogues and writes about sacred geometry considers geometry itself to be inherently spiritual; for some of us, sacred geometry is an adjunct to the study of archaeology, architecture, art history, comparative religion, anthropology, archaeoastronomy, or geometry itself.
Sadly, many books on the subject of sacred geometry are chock full of extraneous blather about UFOs and perpetual motion and Atlantean Science (whatever that is). Even worse, unexpected encounters with such drivel tends to discourage research into the subject by steely-eyed geometricians -- who are, of course, those most qualified to undertake the work.
Over the years i have come up with a theory that encompasses three interlocked reasons for the unfortunate state of affairs by which the once-honoured field of sacred geometry is now often perceived as something akin to pyramidology or spirit-channelling.
1) There was an actual loss of general geometric knowledge during the Dark Ages -- the old Egyptian and Greek geometry was no longer passed along as it had been; instead it became the secret of such trade guilds as made use of it. Thus geometry became "mysterious."
2) Although interest in geometry revived during the Renaissance, the adoption of the Arabic numbering system had already led many Europeans onto a different way of thinking when it came to numbers. Specificly, because the irrational numbers that are so common in geometrical proofs are difficult to handle arithmetically, they became the domain of academic mathematicians. Thus sacred geometry -- which allows one to rattle off irrational number formulae like ".618... : 1 :: 1 : 1.618..." before having one's morning coffee -- seemed rather hard to master. And, the joke of it all is that "1.618..." is but a rude approximation, anyway -- just something for folks with rulers to measure after the geometricians have put down their compasses.
3) During the 19th century the sizes of construction materials became quite standardized for the first time. A common brick was 2 x 4 x 8 inches; lumber came in 12 foot lengths that were 1 or 2 or 4 or 8 or 12 inches wide and a similar choice of numbers deep. Construction therefore took on a more arithmetic aspect than it had when geometric ratio was the prime mover behind design. In a wooden frame house of typical Victorian style, a vernacular builder could lay out the work using simple arithmetic. An "architect-designed" house of an earlier period might have included a spiral staircase -- a test of geometric knowledge -- but by the late 19th century tables of angles printed on steel framing squares obviated the need for carpenrters to study even the small amount of geometry used in figuring out the area of a roof gable.
I believe that the combination of these three factors led many 19th century scholars (especially those who were culturally bound up in colonialist feelings of superiority toward conquered races) to decide that it was inconceivable that ancient cultures could have known enough "math" to have used "irrational numbers" to construct architectural monuments. One thing these writers failed to consider was that a culture that relied on compass-and-straight-edge geometry rather than arithmetic to design structures would not give a fig that some of the lengths turned out to be irrational numbers. The numbers (or rather, the lengths they represented) would simply appear during the course of construction and that would be that.
Jay Hambidge pioneered the technique of searching for certain typically "sacred" geometric ratios among the arithmetic measurements of ancient articfacts. Like all sacred geometry detectives, he had to work backwards -- he took arithmetic measurements of Greek vases and temples and derived from them their geometric construction. This is not as simple as it sounds, because many times an arithmetical construction will duplicate the results of a geometric construction -- in fact, in order to derive even the faintest proof that geometric contruction underlay Greek arhcitecture, he had to perform calculations on dozens of items of differing size, establishing beyond doubt that it was RATIO, not measurement, that determined the relative lengths of crucial dimensions.
The hardest battle Hambidge fought was convincing academics of his day that the ancient Greeks were "sophisticated" enough to have used geometry to lay out their temples. For someone to double check his figures meant the work of years, and few wanted to devote the time to it. (Only recently, with the lightning-quick calculations offered to us by computers, has it become possible to "deconstruct" an ancient temple into its geometric basis with anything approaching efficient speed or to determine the intended astro-calendrical orientation of a temple constructed thousands of years in the past.)
Hambidge did fine work -- and one would think that he and his students would have proved the case for the legitimacy of sacred geometry beyond reasonable doubt -- but he wrote at a peculiar time in history. It was then, during the inter-War period -- when Hambidge's 1925 book "The Parthenon and Other Greek Temples: Their Dynamic Symmetry" was in print, when King Tut's tomb had been discovered, when Theosophy and Rosicruciansism were at the height of their popularity -- that the "mystery" of sacred geometry became bound up with the writings of people whose interests lay far afield from the use of the compass and the straight-edge.
It's hard to say where the art deco theories of Hambidge's students Edward B. Edwards (author of "Dynamarhythmic Design") and Walter Dorwin Teague (author of "Design This Day" and designer of Texaco Gas Stations) leave off and the metaphysical theories of fellow Hambidge student -- and Theosophist -- Claude Bragdon (author of "The Beautiful Necessity" and designer of railroad terminals) begin. Bragdon seems so...normal...as he writes about the Golden Mean and creates a new design style based on projective geometry, that it comes as quite a shock to find him edging into the theory that Man is a Cube, crucified in Time. Still, to his credit, Bragdon, despite dedicating one of his volumes to "The Delphic Sisterhood," was a practicing architect.
Beyond Bragdon, however, a line is crossed -- and one finds onself confounded by writers of that period such as Manley P. Hall, who lumps sacred geometry together with belief in Lemuria, spirit chanelling, Enochian magic, and Rosicrucianism. And he wasn't the worst, by far. The 1960s hippie interest in the occult and the 1990s New Age interest in spirituality have both given library shelf space to authors intent on inventing or perpetuatiing imputed connections between saced geometry, metaphysics, fringe archaeology, magic, and eccentric religions.
In maintaining the Sacred Landscape Bibliography, i list many books that present mystical and even other-worldly claims, but they have been clearly labelled as such. To ignore spiritually oriented or even scientifically lunatic texts would be to ignore a vast part of the writing on the subject of sacred geometry and sacred architecture. So drink deep, but do not swallow every draught uncritically, for to do so would be to drink from a polluted pool.
Oh, and lest i be misunderstood, this is not to say that i personaIly have no interest in
metaphysics or eccentric religions -- for i do find them
intriguing. But that is another subject...for another web page. One i call Keyword: Nut-Case.
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