Category: Geometry and Metrology

Sacred numbers within metrologically-quantified geometries.

Diary Notes

Gurdjieff’s Diagram of Everything Living

This popular post has been brought forward in the midst of other related posts, including Gurdjieff, Octave Worlds & Tuning Theory , an unpublished paper from 2018-19.

Numbers of a Living Planet

I have started to serialize a book idea online, since it throws light on the central theme of living on a planet with Life, including me, but in a culture that has lost its understanding (from the deep past) that the world was forged into a set of special number relations. These numbers gave the earth and its large moon resonant relationships with the other planets that are largely dismissed by science because causation is by forces and not through the properties of numbers. It is also problematic that astronomy today sees the sun as gravitational center (which it is) and that the traditional viewpoint of all pre-scientific civilizations and cultures was based upon a planetary universe that was earth-centered (geocentric), rather than sun centered (heliocentric).

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets

New Beginning in providing videos

Since ClipChamp is available on Microsoft 365, I have been able to replace my broken video editor with this simple devise for making intermediate educational videos. To do this I wrote a few lines if text about how the Moon’s orbit could be tracked in the sky and how this naturally lead to 28 lunar mansions in some earlier astronomies, before the 12 signs of the Zodiac (thought to come from Mesopotamian astronomy to join our present constellations which collided with the Greece Myths to make our present organization of the stars in the night sky. This earlier 28-fold system of the lunar orbit appears to have been recognized as similar, upon the ecliptic, to the 28 synodic loops of Saturn in 29 practical years of 365 days.

Saturn’s “Measuring” of the Lunar Month

The Coronation Pavement of Westminster Abbey

Mosaic Pavements for crowning of kings and queens probably derive from a northern European and Mediterranean traditions, (a) of sacred king-making stones, in which a king would (for example) place his foot into a foot-shaped depression, and (b) the mosaic pavements of Roman villas and Orthodox churches. But an even older tradition seems to inform the Westminster pavement: a geometrical model of a circle and square of equal perimeter. This geometry conforms to the relative sizes of the earth and moon as an 11-unit side length and the equal-perimeter circle’s diameter (14 units ) minus 11 units so that the moon is diameter 3. for more, see this article on the design of King Charles III coronation pavement.

Grids of Squares & Flattened Circles

There is a common approach in ancient building based upon the establishment of a grid of squares, as a framework for the geometrical construction of buildings, from stone circles to Egyptian and Greek temples, to Roman and Orthodox Basilicas, and to Gothic and Enlightenment buildings, plus in Indian temples. Just as one builds foundations, all that is inside a building is controlled by numerical ideas. I have therefore published some work I did to show how flattened circle, in megalithic times, could have used what came to be Egyptian methods for laying out building works and to not always depend on the ropes and stakes of the free style geometrical construction which led to analytical geometry, compass and straight edge.

Peat Fires revealing Rock Art

There have been a number of large peat moor fires in England and one of these in North Yorkshire revealed a few megalithic sites. I have republished my own interpretation of a significant pattern made on a major flat stone as part of an egg-shaped stone circle. The egg can be seen in the work of Alexander Thom as based on the near-Pythagorean triangle with sides {17 17 24.0416}. When Thom’s plan is laid over that of the excavation (Rock Art and Ritual by Brian Smith and Alan Walker), one can see there is a close fit to the excavated site. When the egg is expanded to fit the line drawn by the excavators, the units of the geometry are 1/2 foot (6 inches) so that 17 = 102 inches (8.5 feet), 24 = 144 inches (12 feet) and 12 = 72 inches (6 feet), possible by overlaying different plans, one with the scale shown!

Geometry of the stone egg where the rock art was found on one of its stones. Note the alignment of the egg’s axes to the cardinality of the sun’s solstice extremes at that latitude.

Chalk Drums to generate pi

When I joined the Prehistoric Society for a year, an article about megalithic chalk drums being found with strange decoration which may depict PI, since their diameter allows rolling them to count out a given type of foot measure. This may be why some are not carved because they were heavily used while others could have been metrological standards, not as rods but as cylinders that do not required end-to-end counting but continuous counting, providing one can count!

Angkor Wat as west-facing observatory

I have been doing work on Angkor Wat, something I never got around to after a first introductory post about nested squares there. Both Lords of Time and Language of the Angels were to have included it. Eleanor Mannikka, spent 20 years on a numerical analysis of its architecture and there is an amazing set of French plans by G. Nafilyan. I looked at the temple as an observatory, since it looks west as aligned towards the sun and moon setting on the horizon, which appears to have been part of its intended use. Settings are easier to work with that risings, since there is plenty of warning of settings as sun or moon slowly travel every day towards the western horizon.

Dun Torcuill: The Broch that Modelled the World

image above courtesy Marc Calhoun


Script

This video introduces an article on a Scottish iron-age stone tower or brock which encoded the size of the Earth. 

In the picture above [1] the inner profile of the thick-walled Iron-Age broch of Dun Torceill is the only elliptical example, almost every other broch having a circular inner court.

Torceill’s essential data was reported by Euan MacKie in 1977 [2]: The inner chamber of the broch is an ellipse with axes nearly 23:25 (and not 14:15 as proposed by Mackie).

The actual ratio directly generates a metrological difference, between the major and minor axis lengths, of 63/20 feet. When multiplied by the broch’s 40-foot major axis, this π-like yard creates a length of 126 feet which, multiplied again by π as 22/7, the simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods., generates 396 feet. If each of these feet represented ten miles, this number is an accurate approximation to the mean radius of the Earth, were it a sphere.

If we take the size of the moon in that model, as being 3/11 of 396 feet this would give a circle radius 108 feet and one can see that, using the moon, the outer perimeter of the brock was probably elliptical too.

Thank you for watching.

You can view the full article on sacred dot number sciences dot org, searching for BROCK, spelt B R O C H.

Earth and Moon within Westminster’s Coronation Pavement

Our modern globes are based upon political boundaries and geographical topography yet they had geometrical predecessors which described the world as an image, a diagram or schemata. By some act of intuition, an original Idea for the form of the Earth had become established as a simple two-dimensional geometry, very like eastern mandalas.


Figure 1 Photo of the Cosmati Pavement at Westminster Abbey
[Copyright: Dean and Chapter of Westminster]

Such a diagram came to be built into the Cosmati pavement of Westminster Abbey, this installed during the reign of Henry III as a gift from the Pope and one or more Cosmati master craftsmen. It was dedicated to the Saxon King (and Saint) Edward, the Confessor. This exotic pavement became the focus for the Coronations of subsequent English then British monarchs. Its presence at the heart of English then British king-making is part of what is called the Matter of Britain, one of many Mysteries as to the meaning of its design.

Continue reading “Earth and Moon within Westminster’s Coronation Pavement”

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).