## 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.

## Recent Posts

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===>LOOK AHEAD: SEARCH “4-square” OR “triple-square”

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===> SEARCH on “pi” (and so on)

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## Double squares: Venus and the Golden Mean

The humble square, with side length equal to one unit, is like the number one. It’s area is one square unit and, when we add another identical square to one side, the double square appears. Above right the Egyptian Djed column is shown within a double square. The Djed is the rotating earth which the gods and demons have a tug of war over. This is also a key story in the Indian tradition, called The Churning of the Oceans, where the churning creates both the food of the gods (soma) and every wonderful thing that emerges upon the Earth. In this, the double square symbolized the northern and southern hemispheres of the Earth. The anthropomorphic form Djed shown above has elbows indicative of the Double square.

Figure 1 The churning of the ocean (Samudra Manthan in Sanskrit)

The Djed appears to be the general principle of rotation of, and apparent motion around, the earth.

The god Isis is (as a planet) Venus and is shown (fig.2) offering up the sun disk: another Djed is below, with her Ankh symbol of Life atop the Djed, now having female arms . This sun most probably points to the practical year as 365 days which is 5/8 of the Venus synod of 584 days. (This ratio of 1.6 is the sixth note of the octave 1 to 2.)

In figure 2, two female attendants provide the duality which one might take to be her two famous manifestations of (firstly) the brightest Evening Star, as the sun goes down, and then (after that) the brightest Morning Star before the sun rises. Above there is duality again with three baboons either side of the sun, perhaps representing the six visible planets: Moon, Venus and Mercury: Jupiter, Saturn, Mars and their “tug of war”.

Figure 2 The creation of Horus-Ra from out of an ankh with female arms atop a djed. from Budge 1899, also fig. 7.8 of Richard Heath, The Harmonic Origins of the World.

The numbers 5 and 8 are Fibonacci approximations {1 2 3 5 8 13 21 34 …} to the golden mean, a transcendent number {1.618034…} which rational numbers can only approximate. Venus and the Earth have clearly settled into orbits around the sun resonant with Fibonacci ratios since the Venus orbital period (224.701 days) is 8/13 of the solar year. And it is this fact that eventuates in what we see on Earth, namely the manifestations of Venus every 8/5 of a practical year. of 365 days.

Figure 3 The double square, its in-circle and out-circle manifesting golden rectangles around itself.

In this post, I developed a result sent to me, that a square drawn within the upper hemisphere of a circle must define a golden mean rectangle either side from its height of 1 and the remaining radius of 0.618034… and so it can be seen that the divine principle of the Golden Mean emanates from the double square, either side of each square, when the double square is embraced by a circle drawn from its center. Obviously, on Earth and between orbits (of Venus and Earth), the Golden Mean (also called Phi) has to be approximated by whole number ratios but the principle is present within the geometry and its out-circle. Schwaller de Lubicz thought the dynastic Egyptians held the Golden Mean to be “the fundamental scisson” (literally “scissor cut”) in the range one to two and, its reciprocal can be seen to share the portion over 1 (figure 3).

One can see that geometry and the early numbers would have been seen as two aspects of what we call space and time, in which “things” are separate from each other in Existence but somehow conjoined within Eternity. What we call order is in fact an achievement of harmony made possible by the arranging and fitting of parts to form a coherent whole. It is this insight which gave meaning to their study of geometry and numbers from the prehistoric onwards, into the recorded history of early civilizations. The meaning for Life on Earth became encoded within ancient and prehistoric symbols, whose geometrical and numerical language of expression went to the heart of phenomena.

## Double Square and the Golden Rectangle

above: Dan Palmateer wrote of this, “it just hit me that the conjunction of the circle to the golden rectangle existed.”

Here we will continue in the mode of a lesson in Geometry where what is grasped intuitively has to have reason for it to be true. It occurred to me that the square in the top hemisphere is the twin of a square in the lower hemisphere, hence this has a relationship to the double square rectangle. So one can (1) Make a Double Square and then (2) Find the center and (3) a radius can then draw the out-circle of a double square (see diagram below).

The diagonal from the centre would be the square root of 5 if the top square is seen as two double squares of unit size, that is (4) Identify the units as nested double squares. One can then see (5) a cross within the circle holding 12 squares, but when (6) the root 5 comes down to the right horizontal then the familiar formula (root(5) – 1)/2 = 0.618 so there are many transcendent (not Fibonacci) versions of the Golden mean within in the diagram as shown below.

The in-circle of the cross, radius 2, shows how one can divide that circle into twelve equal portions as with the Zodiac, matching the twelve squares. The out-circle shows Dan’s insight as eight golden rectangles which, overlap over the four “missing” squares of the 16 square grid, which is a simpler framework for generating this geometry as a Whole.