Origins of the Olmec/Maya Number Sciences

ABOVE: Stela C from Tres Zapotes roughly rebuilt by Ludovic Celle and based on a drawing by Miguel Covarrubias.

Introduction

The policy of archaeology regarding the Maya and their root progenitor the Olmec (1500 BCE onwards) is that its cultural innovations were made within Mexico alongside an agrarian revolution of the three sisters, namely squash, maize (“corn”), and climbing beans. This relationship of agriculture to civilizing skills then reads like the Neolithic revolution in Mesopotamia after 4000 BCE, where irrigation made the fertile loam able to absorb agricultural innovations from the northern golden triangle leading to writing, trade, city states, religion, arithmetic and so on. However, the idea that the ancient near east or India could have been an influence through ocean conveyors, of currents and trade winds, has never been accepted when proposed. Yet there are good reasons to think this since the astronomy and monumentalism of the pre-Columbian Mexican civilizations has precedents in the ancient near east and other locations.

The timing of the Olmec and the strangeness of immediately building sacred cities with an almost captive population of around 10,000 people, such as La Venta and San Lorenzo, with strong Jaguar imagery and practices, implies a cultic basis was present from the beginning. And it is now looking likely that the ancient near east was similarly prefigured, not just by agriculture but also by know how involving numbers for the building of sacred buildings with astronomical aspects – a tradition that goes back at least to the megalithic of the Atlantic seaboard of Europe.

Since Columbus, the native populations of North and South America have been largely displaced or marginalized. It may be for this reason that the notion that people from an advanced population had initiated the Olmec civilization requires a high, possibly impossible, level of proof. This Isolationism***, perhaps to avoid “adding insult to injury”, is against the Olmec having derived from the Old World, where the historical records are not that much better. The Olmec origin date is around the time of the quite sudden collapse of the Bronze Age in the Mediterranean around 1200 BCE. And the Olmec, Maya and Aztec appear to have had a definite myth concerning someone called Quetzelcoatl bringing civilizing skills to found their culture, though their culture was also seen as arising from a group of seven underground caves.

***The opposite of Diffusionism: Diffusionism is an anthropological school of thought, was an attempt to understand the distribution of culture in terms of the origin of culture traits and their spread from one society to another. Versions of diffusionist thought included the conviction that all cultures originated from one culture center (heliocentric diffusion); the more reasonable view that cultures originated from a limited number of culture centers (culture circles); and finally the notion that each society is influenced by others but that the process of diffusion is both [subject to chance] and arbitrary . read more

Long Counts and The LUNAR Calendar

Having sketched this background, this article will explore a strange coincidence between the calendrical origins of the Megalithic in Brittany, of a 36 lunar month, 3 lunar year calendar, and the 18 month calendar found in the some of the later Olmec Great Counts, called after the Supplementary Glyphs appended to record the local time in an 18 lunar month calendar. The correlation between long counts and the supplementary data has been invaluable since the long counts can be ambiguous between one or more possible dates but we can predict the sun and moon that far back can compare the glyphs with the alternative dates. Counts have also been found that were eclipses of the sun or moon, resolving a given long count date. It is therefoe interesting to compare the two calendars using the geometrical fact that 36 lunar months is both 2 x 18, 4 x 9 and 3 x 12 since 36 is 4 x 3 x3.

The implication is that the megalithic calendar over three years, which was based upon noticing that three solar years was the diagonal of a four square triangle whose side length is three lunar years, appears to have resulted in an Olmec/Maya calendar in which each square is 9 lunar months. As was noted in previous books (2004, 2016, 2018), the range 9 to 18 years contains a single lunar month {12}, the Jupiter synod {13.5}, the Saturn synod {12.8} and the Uranus synod {12.5}. This octave range between 9 and 2 x 9 = 18 was therefore possible to manifest as a Mexican city design (Teotihuacan) and as the Parthenon of Athens. A number of other examples can be found as one of the proposed major models used from the megalithic onwards, as discussed in Sacred Number: Language of the Angels (2021).

Developmental Roots below 6

Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This can be shown graphically as:

The first two roots “open up” the possibilities of
three-dimensional space.
Continue reading “Developmental Roots below 6”

Umayyad Mosque: Golden Rectangles from Squares

photo above of Umayyad Mosque, Damascus by Bernard Gagnon for Wikipedia CC BY-SA 3.0.

In previous articles on double squares and then St Peter’s Basilica, it became clear that squares and double squares have been embodied, within sacred buildings and art, because circles can then spawn golden rectangles from them. A golden rectangle has one dimension related to its other dimension as the golden mean {1.618034…}. Firstly, the original square plus golden rectangle is a larger golden rectangle but, secondly, the new golden rectangle (beside the square) shares its side length as one unit {1} but its other side is then the reciprocal of the golden mean (0.618034).

The golden mean is the only irrational number whose reciprocal, and square share its fractional part {0.618034 1.618034 2.618034}: there can be only one real number for which this is true. But it is in its geometrical expression, living structure and aesthetics (as in classical architecture) that lead its uniqueness to be seen as a divine ratio. Therefore, it seems, ancient human civilizations sought this golden form of harmony within the form of the Temple, especially in Dynastic Egypt and Classical Greece. The planet Venus must have reinforced this significance since its synod {584 days} is 8/5 of the solar year {365 days} and its manifestation such as evening and morning stars, move around the zodiac tracing out a pentacle or five-pointed star, the natural geometry of the golden mean.

The natural geometry of the Golden Mean is the Pentacle, traced out by planet Venus upon the Zodiac as evening and morning star. (from Sacred Number and the Origins of Civilization)

In the renaissance, the Classical tradition of Ancient Greece and Rome was reborn as neoclassicism, a famous proponent being Palladio, and further neo-classicism arose in the 19th Century and continues in the United States. From this, the previous article on St Peter’s saw its original square become rectangular in a golden way. The whole basis for this is due to the nature of squares and circles, that is: golden rectangles are easily formed geometrically through squares and circles.

The extension of St Peter’s from a square, by adding a golden rectangle, can be seen to also apply within the original square. Furthermore, there is a medium-sized square within the golden rectangle plus a small golden rectangle (see below).

The overall golden rectangle of St Paul’s of a square and golden rectangle below. Using the square within the golden rectangle, the original square above can have four such overlapping squares, to create a cruciform pattern, the upper part of which was used to lay out the Umayyad Mosque.

The medium square can be tiled four times within the large square to overlap the other medium squares, as shown above. This creates a small central square while the four regions that overlap are smaller golden rectangles. The lower golden rectangle is also repeated four times with overlapping, twice horizontally and twice vertically. It is seen that squares and golden rectangles can recede within a square, into smaller sizes, or expand around a square. It is as if all levels of scale hold a kind of fractal, based upon the golden mean.

The top six elements of the square can be seen to match the site plan of the Great (Umayyad) Mosque of Damascus, built 900 years before St Peter’s Basilica, on the site of an Orthodox Cathedral and, before that, a Roman temple to Jupiter. In other words, any golden rectangle design can contain resonances of somewhat different golden mean designs, that may express a different meaning or context; in this case the Mosque gives the notion of two squares overlapping to generate an intervening region of blending and the rectangle of overlap will then be phi squared in height (shown yellow below) relative to the width being unity – the central square’s side length.

The geometry of the Umayyad Mosque

My thanks to Dan Palmateer, for his emails and diagramming whilst on this theme of golden rectangles. One of his own pictures (below) shows the central square of the main square, by tiling the main square with the small golden rectangle.

The central square within the greater square is revealed in St Peter’s as a square within a circular area, noting that this plan (held by The Met Museum) was made after the building had been completed.

There was obviously a vernacular of golden rectangular building in Islam which was carried forth in Renaissance Europe. The potential for golden rectangular building can be all-embracing, as it is a property of space itself, due to numbers.

St Peter’s Basilica: A Golden Rectangle Extension to a Square

HAPPY NEW YEAR

above: The Basilica plan at some stage gained a front extension using a golden rectangle. below: Later Plan for St. Peter’s 16th–17th century. Anonymous. Metropolitan Museum.

The question is whether the extension from a square was related the previous square design. The original square seems quite reworked but similar still to the original square. The four gates were transformed into three ambulatories defining four circles left, above, right and centre, see below.

Equal Perimeter models at the center of St Peter’s Basilica

Equal Perimeter Models

The central circle can be considered as 11 units in diameter so that its out-square is then 44 units. The circle of equal perimeter to the square will then be 14 units in diameter and the difference of 3 defines a circle diameter 3 units. The 11-circle represents the Earth while the 3-circle represents the Moon, to very high precision – hence making this model a representative of the Mysteries inherited from deep antiquity; at least the megalithic age and/or early dynastic Egypt, when the earth’s size can be seen in Stonehenge and Great Pyramid. This inner EP model, is diagonal so that the pillars represent four moons.

An outer Equal Perimeter model is in the cardinal directions (this alternation also found in the Cosmati pavement at Westminster Abbey, and inner models are related to the microcosm of the human being relative to the slightly larger model of Moons). The two sizes of Moon define the circles at the center, around St Peter’s monument. The mandala-like character of the Equal Perimeter model give here the impressions of a flower’s petals and leaves.

Golden Rectangles

You may remember a recent post about double squares and golden rectangles, where a half-circle that fits a Square has root 5 diagonal radius which, arced down, generates a golden triangle. It is therefore possible to fit the square part of the original design and draw the circle that fits the half-diagonal of the square as shown below.

The golden extension of the Basilica’s Square Plan

By eye, the square’s side is one {1} and the new side length below is 1/φ and the two together are 1 + 1/φ = φ (D’B’ below) which is the magic of the Golden Mean. This insight can be quantified to grasp this design as a useful generality:

Quantifying how the golden mean rectangles are generating phi (φ)

Establishing the lengths from the unit square and point O, the center of the right hand side. OA’ is then √5/2. When this is arced, the square is placed inside a half circle A’C, BC is √5/2 + 1/2 = 1/φ.

The rectangle sides ACD’B’ are the golden mean relative to the width A’B = 1, the unit square’s side, but that unit side length A’B is the golden mean relative to the side of the golden rectangle BC. In addition the length B’D’ is the golden mean squared relative to BC, the side of the golden rectangle.

Commentary

It seems that the equal perimeter models within the square design of Bramante were adjusted. The golden mean was used to extend the Basilica (originally an Orthodox square building named after St Basil) into a golden rectangle. This could be done by adding the equivalent lesser golden rectangle, relative to the unit square through the properties of the out half-circle from O.

The series of golden rectangles can travel out in four directions, each coming naturally from a single unitary square. The likely threefold symbolic message, added by the extension seems to be the primacy of the unitary square, of St Peter (on whom the Church was to be founded) and of the Pope (as a living symbol of St Peter).

St Peter’s Basilica: Starcut & Equal Perimeter

In Malcolm Stewart’s book on Sacred Geometry, his starcut diagram was applied to Raphael’s painting The School of Athens to create radiants to the people standing around the Athenium Lyceum. “If the starcut was the central geometrical determinant for Raphael’s formal depiction of classical philosophy” it was a “known authoritative device” or framework for geometrical understanding. Stewart found a potential antecedent for such a technique Donato Brahmante’s plan for St Peter’s (see above) which was square like a starcut diagram.

left: Stewarts book cover right: The simplest version of the starcut square where the sides are divided by two and the outer square is four squares of nine, which is 62 = 36 squares and there an octagon within the crossing lines. If there were 72 squares, then the octagon’s vertices would all be on crossings.

A starcut diagram works as a linear interpolator of lines drawn between its sides which are then divided by a number of points that radiate out to other points. The inner lines in this one are eight in number, three per side. Malcolm Stewart shows (see below) the number of coincidences between the plan and a starcut, as if the design was partly arrived at by establishing this pattern. The cardinal cross between its four entrances could have be arrived at, as could the corner octagons with their entrance and side circles lying on starcut radiants. And the central square has corners defining the central space and pillars for supporting the dome.

There seems to be other signs of starcutting such as Honnecourt’s Man, that masons were using such frameworks to build all manner of buildings, sculptures and designs. To investigate further, I made a diagram of my own, over Bramante’s plan and used the method of modular analysis, based on the fact that the central cross of walk ways is one fifth of the square’s side length so that 5 by 5 squares (in red) will define that feature. But there also seems to be a 3 by 3 grid of squares at work (shown in blue) to define the central space in the standard style of the Basilica from the Orthodox (Eastern Church) tradition, this then accounting for most of Stewart’s dotted lines.

Reconstructing most of Malcolm Stewart’s fig. 8.18 using grids of five and three, and applying modular analysis to the Basilica, to quantify it in relative units 1/120th of its side length.

The plan has no scale from which metrology can be deduced, but the smallest number able to hold these two grids together is 60. But to resolve the width of the corner octagons (as 15) I have used a side length of 120. The squares of 24 divided by the octagon width is 24/15 = 8/5 = 1.6. On can see that the starcut diagram was probably part of modular analysis, a technique popular in modern studies of cathedrals which, of necessity, can’t have been designed except as a meaningful whole. But this design would go through many hands including  MichelangeloCarlo Maderno and Gian Lorenzo Bernini to become a transcept cathedral design (see below).

Later Plan for St. Peter’s 16th–17th century. Anonymous. Metropolitan Museum.

My own book on sacred geometry found a different framework was often present in such capital buildings, a model called Equal Perimeter which is a model of pi as 22/7 but is also the basis for a cosmological model of the Earth and the Moon, as 3/11ths of the Earth in size. This model is principally a circle the same perimeter size as a given circle’s circumference, the square being symbolic of the earth in its side length, as a scaled down mean diameter for the Earth. The basilica square limits could then the Earth and the circle of equal perimeter and size of the Moon, as shown overlaid below. Just as the presence of starcut or modular frameworks were linked to a medieval tradition, perhaps parts of that tradition were conscious of this long lost knowledge of the size of the Earth and Moon.

The Equal Perimeter model seems quite clear within the Basilica as originally conceived by Bramante.

It would seem that the equal perimeter design was in use in medieval times because the Cosmati pavement of Westminster Abbey holds it very clearly, and it was the Pope who sent Cosmati guildsmen for its construction. If the basilica was completed on 18 November 1626, the Westminster pavement was completed by 1268 for king Henry III. Its mosaic is depicted in Hans Holbein’s The Ambassadors. The interpretation I gave to it is in my Sacred Geometry book was first published here.

In summary, sacred geometry became a repository for esoteric information and techniques useful for laying out the capital buildings and other religious artifacts in which the exoteric aspects of religion are performed. Rituals often have a deeper meaning, only accessible when one seeks to understand rather than merely know them. It may be that this was a necessary compromise between the outer and inner meaning of life in those times.

Cosmati Great Pavement at Westminster Abbey as a model of the Earth and Moon.
[Copyright: Dean and Chapter of Westminster]

Starcut Diagram: geometry to define tuning

This is a re-posting of an article thought lost, deriving in part from Malcolm Stewart’s Starcut Diagram. The long awaited 2nd edition Sacred Geometry of the Starcut Diagram has now been published by Inner Traditions. Before this, Ernest McClain had been working on tuning via Gothic master Honnecourt’s Diagram of a Man (fig. 2), which is effectively a double square version of the starcut diagram.

The square is the simplest of two dimensional structures to draw, giving access to many fundamental values; for example the unit square has the diagonal length equal to the square root of two which, compared to the unit side length, forms the perfect tritone of 1.414 in our decimal fractional notation (figure 1 left). If the diagonal is brought down to overlay a side then one has the beginning of an ancient series of root derivations usually viewed within the context of a double square, a context often found in Egyptian sacred art where “the stretching of the rope” was used to layout temples and square grids were used to express complex relationships, a technique Schwaller de Lubitz termed Canevas (1998). Harmonically the double square expresses octave doubling (figure 1 right).

Figure 1 left: The doubling of the square side equal 360 units and right: The double square as naturally expressing the ordinal square roots of early integers.

Musical strings have whole number lengths, in ratio to one another, to form intervals between strings and this gives geometry a closer affinity to tuning theory than the use of arithmetic to calculate the ratios within a given octave range. The musicology inferred for the ancient world by Ernest G. McClain in his Myth of Invariance (1976) was calculational rather than geometrical, but in later work McClain (Bibal 2012-13) was very interested in whatever could work (such as folding paper) but was especially interested in the rare surviving notebook of 13th century artist Villard de Honnecourt, whose sketches employed rectilinear frameworks within which cathedrals, their detailing, human and other figures could be drawn.

“I believe we have overlooked Honnecourt as a prime example of what Neugebauer meant in claiming Mesopotamian geometry to approach Renaissance levels illustrated in Descartes. If Honnecourt is 13th c. then he seems more likely to be preserving the ancient picture, not anticipating the new one.”

This draws one into significant earlier traditions of sacred art in Egypt (Canevas) and in Indian temple and statue design, and to Renaissance paintings (see end quote) in which composition was based upon geometrical ideas such as symmetry, divisions into squares and alignments to diagonals. Figure 2 shows one of Honnecourt’s highly stylised sketches of a man, using a technique still in use by a 20th century heraldic artist.

Ernest McClain, Bibal Group: 18/03/2012

Figure 2 The Honnecourt Man employing a geometrical canon.

The six units, to the shoulders of the man, can be divided to form a double square, the lower square for the legs and the upper one for the torso. The upper square is then a region of octave doubling. McClain had apparently seen a rare and more explicit version of this arrangement and, from memory, attempted a reconstruction from first principles (figure 3), which he shared with his Bibal colleagues, including myself.

Figure 3 McClain’s final picture of the Honnecourt Man, its implied Monochord of intervals and their reciprocals.

To achieve a tuning framework, the central crossing point had been moved downwards by half a unit, in a double square of side length three. On the right this is ½ of a string length when the rectangle is taken to define the body of a monochord. McClain was a master of the monochord since his days studying Pythagorean tuning. Perhaps his greatest insight was the fact that the diagonal lines, in crossing, were inadvertently performing calculations and providing the ratios between string lengths forming musical intervals.

Since the active region for octave studies is the region of doubling, the top square is of primary interest. At the time I was also interested in multiple squares and the Egyptian Canevas (de Lubitz. 1998. Chapter 8) since these have special properties and were evidently known as early as the fifth millennium BC (see Heath 2014, chapter two) by the megalith builders of Carnac. In my own redrawing of McClain’s diagram (figure 4) multiple squares are to be seen within the top square. This revealed that projective geometry was to be found as these radiant lines, of the sort seen in the perspective of three dimensions when drawn in two dimensions.

Figure 4 Redrawing McClain to show multiple squares, and how a numerical octave limit of 360 is seen creating lengths similar to those found in his harmonic mountains.

Returning to this matter, a recently developed technique of populating a single square provides a mechanism for studying what happens within such a square when “starcut”.

Figure 5 left: Malcolm Stewart’s 2nd edition book cover introducing right: the Starcut Diagram, applicable to the top square of Honnecourt’s octave model .

Malcolm Stewart’s diagram is a powerful way of using a single square to achieve many geometrical results and, in our case, it is a minimalist version that could have more lines emanating from the corners and more intermediate points dividing the squares sides, to which the radiant lines can then travel. Adding more divisions along the sides of the starcut is like multiplying the limiting number of a musical matrix, for example twice as many raises by an octave.

A computer program was developed within the Processing framework to increase the divisions of the sides and draw the resulting radiants. A limit of 720 was used since this defines Just intonation of scales and 720 has been identified in many ancient texts as having been a significant limiting number in antiquity. Since McClain was finding elements of octave tuning within a two-square geometry, my aim was to see if the crossing points between radiants of a single square (starcut) defined tones in the just scales possible to 360:720. This appears to be the case (figure 6) though most of the required tone numbers appear along the central vertical division and it is only at the locations nearest to D that eb to f and C to c# that only appear “off axis”. The pattern of the tones then forms an interesting invariant pattern.

Figure 6 Computer generated radiants for a starcut diagram with sides divided into six.

Figure 7 http://HarmonicExplorer.org showing the tone circle and harmonic mountain (matrix) for limit 720, the “calendar constant” of 360 days and nights.

Each of the radiant crossing points represents the diagonal of an M by N rectangle and so the rational “calculation” of a given tone, through the crossing of radiants, is a result of the differences from D (equal to either 360 or 720) to the tone number concerned (figure 8).

Figure 8 How the tone numbers are calculated via geometrical coincidence of cartesian radiants which are rational in their shorter side length at the value of a Just tone number

It is therefore no miracle that the tone numbers for Just intonation can be found at some crossing points and, once these are located on this diagram, those locations could have been remembered as a system for working out Just tone numbers.

Bibliography

Heath, Richard.

  • 2014. Sacred Number and the Lords of Time. Rochester, VT: Inner Traditions.
  • 2018. Harmonic Origins of the World: Sacred Number at the Source of Creation. Inner Traditions.
  • 2021. Sacred Geometry: Language of the Angels. Inner Traditions.

Lubitz, R.A. Schwaller de.

  • 1998. The Temple of Man: Apet of the South at Luxor. Vermont: Inner Traditions.

McClain, Ernest G. 

  • 1976. The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas Hays.

Stewart, Malcolm.

  • 2022. Sacred Geometry of the Starcut Diagram: The Genesis of Number, Proportion, and Cosmology. Inner Traditions.