In February next year a new book (by me) will be released, as can be seen at the publisher website. The book will soon to be ready for laying it out for both printing and review (at Scribd.com, I’ll let you know).

figure: the punctuation of towers and western outlook. Possibly a funerial building for the king, it could be used as a living observatory and complex counting platform for studying the time periods of the sun, the moon, and even the planetary synods.

Some new material has recently been added, including chapters on the design of Angkor Wat (chapter 9) and St Peter’s basilica in Rome (chapter 10), and some early articles are visible on this site, through the links. As you can see, my books partly emerge from work presented on this website. This is important sol the subject can become something doable today since, at its heart. Though my results should not rely on modern techniques, I have to avail myself of modern techniques to work out what the ancient techniques must, instead, have been: for the ancients to have learned from the pattern of time in the sky, especially on the Horizon, but also of planetary motions amongst the stars.

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.

above: The Basilica plan at some stage gained a front extension using a golden rectangle. below: Later Plan for St. Peter’s 16thâ€“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/Ï† and the two together are 1 + 1/Ï† = Ï† (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 (Ï†)

Establishing the lengths from the unit square and point O, the center of the right hand side. OA’ is then âˆš5/2. When this is arced, the square is placed inside a half circle A’C, BC is âˆš5/2 + 1/2 = 1/Ï†.

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).

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 6^{2} = 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 Michelangelo, Carlo Maderno and Gian Lorenzo Bernini to become a transcept cathedral design (see below).

Later Plan for St. Peter’s 16thâ€“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]