Proportions: A study of its origins and how civilizations manifest them

Whenever the Ancient Egyptian artists sculptured, inscribed or painted figures, their proportions would be determined by a canon of proportions. Up until the end of the New Kingdom's 26th Dynasty, the Ancient Egyptians used a grid that measured 18 units to the hairline, or 19 units to the top of the head.  The height of the figure was usually measured to the hairline rather than the top of the head, this part of the head often being concealed by a crown or head piece making it difficult to base a canon of proportions on. 1

This separation of the crown of the skull from the rest of the body reduces the height of the figure to 18 units and provides a consistent point upon which a figure's proportions could be based. 

In the Old Kingdom a more simple canon was used, from which the later grid of 18 squares evolved. Also based on the height of the forehead or hairline, this canon had generally six lines, five of which form the basis of, and therefore corresponded to the later 18/19 canon. Occasionally a line level with the top of the head corresponding with the later canon's 19th line was added, though in many Old Kingdom examples this line is omitted.

"In other words, these horizontals in the (18/19) grid system correspond to (the Old Kingdom) guide lines. Clearly, therefore, the squared grid system in which a standing figure consisted of 18 squares from the soles to the hairline must have developed out of the guide line system.  It was able to incorporate all of the earlier lines except those marking the armpits and the crown of the head....The old vertical axial guide line became incorporated as a vertical guide line." 2

This vertical axial line usually passed in front of the ear.  In the grid that evolved out of this earlier guideline system, the vertical line immediately in front of this axial line runs through the eye. 

This more simple system of horizontal guide lines may have developed into the grid of 18 squares during the Old Kingdom.  Grids have been found dating to the third dynasty or possibly earlier.  Gay Robins writes:

"There is no doubt that grids had already been employed for other purposes in the Old Kingdom....Certainly with the majority of surviving tombs decorated in relief, evidence for the artist's original layout on the wall must have been lost in most cases.  By contrast, painted tombs, which were more likely to show evidence of the initial stages of working, have on the whole not been well preserved.  So the number of tombs known at the moment to have guidelines is a very small portion of all surviving Old Kingdom tombs.  It is possible therefore, that evidence for figures drawn on grids has simply not survived..." 3

In his paper, Rudolf Gantenbrink established that the King's chamber 'air shafts' theoretically meet at a point that is 11/18 of the horizontal distance between the outer openings of the two shafts on the face of the pyramid.  He illustrates this with a diagram of the pyramid's cross section in which the shafts are contained in a grid that is 18 squares in width. 4

By laying a hypothetical grid over figures from early dynasties it can be demonstrated that their proportions are identical to those of later dynasties.  This would of course be expected if the grid was based upon this earlier system of horizontal lines.

The three figures above have a hypothetical grid of 19 squares overlayed 
to show the 18:11 relationship between the height of the hairline and navel

It must be said, however, that the canon of proportions did vary over the thousands of years of Egyptian civilisation.  By applying the hypothetical grid of 19 squares to figures from different eras, Gay Robins demonstrates that though different systems were used in different eras, it is possible to speak of what she terms "classic proportions".    

These classic proportions began to appear in royal figures of the Third Dynasty and were found almost universally in the Fifth and Sixth dynasties.  She adds that draftsman deliberately returned to these proportions from time to time throughout history after periods of political upheaval and artistic change. 

Japanese:

Though expensive, I believe this book is pretty much the gold-standard reference for this kind of thing (you can do a "search inside" on it to get some nuggets).

Traditional Japanese architecture was highly modular, and the modules weren't so much based on the tatami size as the tatami count. Traditionally there are 2 tatami sizes:one for eastern Japan, one for Western (slightly smaller in the east), plus a still-smaller size used in modern apartment-blocks (danchi), but they're all about 3' x 6'. There are only a few customary tatami counts, though: 4.5, 6, 8, and (I think) 15, and everything else about the room is pretty much derived from that.

Greek Proportions

Greek Proportions.pdf

Geometrical figures overlaid on an architectural drawing

Caption: 
Fig. 11. Root-3 geometric proportions in the Jefferson Rotunda. Thomas Jefferson, University of Virginia: Rotunda, South Elevation, 1819. Image courtesy of the Thomas Jefferson Papers, Special Collections, University of Virginia Library (N-328, K No. 8). Geometric overlay: Rachel Fletcher.

Context: 
The Jefferson Rotunda, South Elevation: Root-3 Proportions ( Fig. 11 ). As in the Pantheon, a circle traces the exterior surface of the Rotunda dome. In fact, Jefferson draws such a circle, dotted in ink ( Fig. 2 ). A new circle of equal radius is drawn from the top of the dome. The result is a vesica piscis , with vertical and horizontal axes in 1:sqrt(3) ratio. The horizontal axis locates the base of the dome, which spans 120°“. A notation by Jefferson specifies that half the dome's surface spans "60°“. [ 13 ]

A square is drawn about the circle. Its base locates the baseline of the Rotunda. In addition, the top of the dome locates one apex of an equilateral triangle. The remaining two apexes touch the right and left sides of the square, while locating the floor level of the portico. The half-side of the equilateral triangle and its altitude are equal in length to the axes of the vesica piscis , in 1:sqrt(3) ratio.

Source: 
Fletcher, R. (2003). An American Vision of Harmony: Geometric Proportions in Thomas Jefferson's Rotunda at the University of Virginia. Nexus Network Journal. Architecture and Mathematics Online, 5(2).

The Mayan Venus Calendar and the 
Golden Proportion

The Mayan Venus Calendar - this is what my latest study is all about. It is called Tzolkin: Visionary Perspectives and Calendar Studies . I'll try to give an overview of this complicated subject, and then answer the inevitable question: "So what?"

The ancient Maya tracked the morningstar risings of Venus, which occur every 584 days. Venus was the centerpiece of Mayan cosmology and mythology. They created a framework of cycles for predicting the future morningstar risings of Venus - for centuries to come. With this system, they could also predict solstice and eclipse dates. The two cycles involved in this "calendar" system were the vague year of 365 days (the haab) and the sacred cycle of 260 days (the tzolkin). These two cycles synchronize with each other every 52 years, and this period of time was known as the Calendar Round. The Venus cycle fits in nicely with multiples of the Calendar Round, such that all three cycles synchronize once every 104 years (2 Calendar Rounds). This was the Venus Round, and the sacred day which began the Venus Round was known as the Sacred Day of Venus. From evidence in the Dresden Codex, one of the few surviving Mayan books, we know that 1 Ahau was the traditional Sacred Day of Venus for the Maya. Linguistically, the word Ahau means Lord, Sun, or Marksman, implying that the three cycles are "shot forth" from the sun on this day.

The Maya used this sacred framework of days to track the astronomical movements of Venus. Mythologies developed around Venus and the day-signs associated with its risings. The Popol Vuhof the Quiche Maya is one example of how the cycle of Venus was mythologized. In early Western civilization - in Mesopotamia - we know that Venus worship gave rise to astronomy, science, math and probably civilization itself. As did the Babylonian astrologers, gazing out from their ziggurats, the early skywatchers of Mexico noticed that 5 Venus cycles equals 8 years. This is just to say that if Venus emerged in the east as morningstar on, say, Christmas, then Venus would again emerge on Christmas eight years later. 13 of these Sun/Venus cycles equal one Venus Round. Another interesting fact related to this is that, because of this 8:5 ratio, Venus traces a five-pointed star around the ecliptic every 8 years. The occult symbol of the pentagram probably has its roots in the discovery of this fact in ancient Mesopotamia.

The moon cycle was also incorporated into the Mayan Venus Calendar, by way of the 9-moon sacred cycle of 260 days. So this amazing calendrical system of the Maya structured the movements of Sun, Venus and Moon, the three brightest celestial lights.

Unfortunately, the Maya were unable to perfect their Venus Calendar. A slight difference between the true cycle of Venus and the 584-day approximation employed by the Maya made long term predictions unreliable. The Maya had been collecting observational data and were working on this problem for many centuries by the time the Spaniards arrived in the 1500's. Around this time, the Venus tradition became fragmented and no Mayan groups surviving today follow a Venus Calendar. However, the basic framework Šof the 260-day and 365-day cycle (the tzolkin and haab) is still used by the Maya living in the Highlands of Guatemala.

So what? Well, the Venus tradition is dead. But with the deciphering of the Dresden Codex and the acceptance of the standard count of days still used by the Guatemala Maya, we should be able to reconstruct and resurrect the Venus Calendar of old. And with modern data on the Venus cycle available, we could even perfect it! We may imagine this to be the central dream of the ancient Mayan skywatchers, to create a self-adjusting perpetual Venus Calendar. I have endeavored to do this in my book, and propose that the next Sacred Day of Venus to begin a 104-year Venus Round occurs on the Venus rising of April 3rd, 2001 A.D., which is the Sacred Day of Venus 1 Ahau. My argument is well documented and discusses various opposing viewpoints on the question. The value in doing this lies with the resurgence of interest in Gnostic beliefs and Goddess Religions. As we know, civilization itself was spurred forth by the goddess - the study of the Venus cycle in ancient Babylonia. And now, with patriarchy running its course, perhaps it is time again to get in touch with our roots. The practical Venus Calendar I propose can be followed for some 2500 years with great accuracy, and is based upon the system of the Dresden Codex.

That covers the academic facet of my book. In the second section, "Visionary Perspectives," the more magical and mysterious properties of the Sacred Calendar are explored. This involves elucidating its essential structure and meaning. And what an amazing accomplishment it was! Some say that it arose simply to structure the planting and harvesting of crops - and indeed, this it does. But so many planetary cycles fit neatly into the system, and at the same time the 260-day cycle is said to correspond with the 260-day human gestation period! It is really an amazing cosmology of numbers, having uses on many different levels of reality. How can it have such universal appeal? The second part of my book answers this two-part question: What is the essential nature of the Sacred Calendar and to what does it owe its amazing properties? In the end, I relate the findings by which I conclude that the Golden Proportion, which gives rise to the spirals of seashells, pinecones, and DNA, is the core of the Sacred Calendar. This is true for numerological as well as philosophical and cosmological reasons. I will relate this in brief:

On one level the Calendar is an oracle used in divination. The 13-day periods, 20-day periods, along with other cycles of 260 days, 52 years and 104 years all make up the Venus Calendar system. And yet, the mystery is compounded because the ritual days are also used in casting auguries. In this way it is similar in function to the I Ching, the 64-hexagrams of the ancient Chinese Book of Changes.

The I Ching is a philosophy which explains the emanation of the world from the Tao and the primary duality of yin and yang. It does not focus so much on objects, but rather the process of change which "objects" undergo. As the world emerges, phenomena differentiate into 2, 4, 8, 16, 32, and 64 stages. This is called an exponential expansion, and is one model of terrestrial evolution. Yet a different factor needs to be considered - the involution process. This consideration allows for the apparent fact that there is a direction to the phenomenal evolution around us, that an inward dimension and its unfolding must be accounted for. This is just to say that spirit is the flower of the terrestrial process. Earth gives birth to spirit. While some readers may have objections to this, I don't have the space here to elaborate, so let's just assume that Darwin was either partly wrong, or that consciousness is the ultimate adaptive strategy . Now the I Ching only describes change - an endless, natural flow of forces, one giving way to the next - an ever repeating phenomenal flux in 64 stages. The I Ching, incredible though it is, is somehow lacking in its role as a cosmological model.

The Golden Proportion is the key to a fuller understanding of phenomenal unfolding - for it manifests in organic forms such as seashells and pinecones, as well as our own DNA. The involuting spiral of consciousness - the unfolding of the spiritual flower - this is what the Golden Proportion describes. Yet, the I Ching is fortunate in that it has a philosophical number scheme that goes with it; it has the 8 trigrams and the 64 hexagrams to encode and make known its secret workings. The Golden Proportion, being essentially a ratio usually approximated as 1.618, does not have a conventional system of numbers to encode its workings. Or does it? What if we combine the spiral unfolding of the Golden Proportion (PHI=1.618) with the exponential expansion of the I Ching? This is what we get:

These are all the essential numbers of the Mayan Venus Round system! So it seems that the Maya were able to intuit the profound workings of the Golden Proportion and transform it into an incredible cosmological system, having biological, mythological, spiritual, occult, calendrical, astrological and astronomical uses. The simple table given above contains profound implications, and with it the essential nature of the Mayan Venus Calendar becomes a bit more clear. The Sacred Calendar is such a rich vein of hermetic and esoteric truths that I find it hard to imagine that it is not the focus of greater interest. The reason for this, I believe, is that there has been some amount of disinformation and confusion about its nature. I hope that my book is a step towards clarifying the problem, so that we can celebrate and embrace the profound implications of this work of Native American genius.