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:

# Category: Geometry and Metrology

Sacred numbers within metrologically-quantified geometries.

## 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 MuseumThe 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 r*elative 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).

## 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 (

Ernest McClain, Bibal Group: 18/03/2012Canevas) 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.

*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 e*b* 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.

## 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.

## Story of Three Similar Triangles

*first published on 24 May 2012*,

*Figure 1 Robin Heath’s original set of three right angled triangles that exploited the 3:2 points to make intermediate hypotenuses so as to achieve numerically accurate time lengths in units of lunar or solar months and lunar orbits.*

Interpreting *Lochmariaquer *in 2012, an early discovery was of a near-Pythagorean triangle with sides 18, 19 and 6. This year (2018) I found that triangle as between the start of the Erdevan Alignments near Carnac. But how did our work on cosmic N:N+1 triangles get started?

Robin Heath’s earliest work, *A Key to Stonehenge* (1993) placed his **Lunation Triangle** within a sequence of three right-angled triangles which could easily be constructed using one megalithic yard per lunar month. These would then have been useful in generating some key lengths proportional to the lunar year:

**the number of lunar months in the solar year,****the number of lunar orbits in the solar year**and**the length of the eclipse year in 30-day months.**

all in lunar months. These triangles are to be constructed using the number series 11, 12, 13, 14 so as to form N:N+1 triangles (see figure 1).

Continue reading “Story of Three Similar Triangles”n.b. In the 1990s the primary geometry used to explore megalithic astronomy was N:N+1 triangles, where N could be non-integer, since the lunation triangle was just such whilst easily set out using the 12:13:5 Pythagorean triangle and forming the intermediate hypotenuse to the 3 point of the 5 side. In the 11:12 and 13:14 triangles, the short side is not equal to 5.