## 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.

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

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.

## Multiple Squares to form Flattened Circle Megaliths

above: a 28 square grid with double, triple (top), and four-square rectangles (red),
plus (gray again) the triple rectangles within class B

#### Contents

1.     Problems with Thom’s Stone Circle Geometries.

2.     Egyptian Grids of Multiple Squares.

3.     Generating Flattened Circles using a Grid of Squares.

#### ABSTRACT

This paper reviews the geometries proposed by Alexander Thom for a shape called a flattened circle, survivors of these being quite commonly found in the British Isles. Thom’s proposals appear to have been rejected through (a) disbelief that the Neolithic builders of megalithic monuments could have generated such sophistication using only ropes and stakes and (b) through assertions that real structures do not obey the geometry he overlaid upon his surveys.

## 1. Problems with Thom’s Stone Circle Geometries

Almost all of the different types of megalithic building[1] were evolved in the fifth millennium (5,000-4,000 BC), in the area around Carnac on southern Brittany’s Atlantic coast. This includes the many circles built later in the British Isles. When Alexander Thom surveyed these [2] he found them to be remarkably technical constructions, involving sophisticated geometrical ideas. It was only in the mid-seventies, when Thom came to Carnac, that the same geometries were found applied within Carnac’s stone circles which soon afterwards were found to precede those of Britain by at least a thousand years.

After an initial public and academic enthusiasm for Thom’s work[3] British archaeologists chose, with very few exceptions, to refute the entire notion that the Neolithic could have been constructing such technical geometries. As far as our History would have it, such geometries could only have been drawn after the development[4] of a functional mathematics which culminated in Euclid’s classical work on analytic geometry, Elements. Thom’s use of geometry was therefore anachronistic and Thom surely mistaken. For archaeologists to accept Thom’s geometries would have required a revolution in thinking about the megalithic; for which there was little appetite. It was easier to work instead to falsify Thom’s hypothesis with a new type work that argued against Thom’s arguments for geometry, a megalithic yard and astronomical alignments, concluding instead, for example, that “stone circles were distorted so that the audience could see all the rites; and the principals could occupy visually focal positions facing the spectators.”, clearly indicating the still current “comfort zone” within archaeology, in which unquestioned ideas about superstitious rites are used to supersede Thom’s accurate and well founded proposals, of a megalithic technical capability. The problem with inventing ancient rites as the primary purpose for stone circle building is that, whilst refuting Thom’s proposal, it cannot itself ever be proved in a scientific sense; Talk of rites as being the reason for stone circle construction is not delivering an evidence-based scientific proof and Thom’s proposals were not disproved by such ideas.

Figure 1 Thom’s site plans of two of Britain’s finest surviving Flattened Circles, above: Castle Rigg (Type A) and below: Long Meg (Type B). Castle Rigg’s axis of symmetry points (within a degree) to Long Meg, on a bearing which follows the diagonal of a two by one (east by north) rectangle, as if (despite some Lake District mountains in between) the two sites were related when built and hence contemporaneous. site plans by Alexander Thom.

Unlike many of his detractors, Thom surveyed stone circles and through this activity was to create the first (and only) extensive corpus of stone circle site plans. Through this he left a vitally important legacy by preserving their layout against further natural and man-made degradation. The geometrical overlays and typology found within Thom’s site plans have been dismissed as unlikely, on spurious technical grounds [*], usually by people with insufficient technical background in the technical issues within his work. Thom’s later work in Carnac has proven critical in providing further alternative explanations as to how the megalithic actually constructed these stone circle geometries without Euclidian geometrical methods, using instead the system of multiple squares found to be in use in the megalithic structures around Carnac[5]; this in the late 1970’s and after Thom’s surveying seasons earlier in that decade[6].

We will later show that such a system of multiple squares would have eliminated the use of a “compass” or arcing of ropes which Thom proposed to explain how different stone geometries were achieved. Instead, a grid of squares can locate the few key points on the perimeter of a flattened circle. A design method based upon a grid of multiple squares would eliminate the main objection to Thom’s proposal of: Euclid-like geometrical process was used to build stone circle geometries.

Figure 2 the geometries of Flattened Circles (left to right) called Types A, B and D

In the case of the Type A (flattened) stone circles proposed by Thom, I demonstrate below that accepting Thom’s interpretation of its shape is a necessary stepping stone to understanding how this could be achieved by a pre-arithmetic megalithic of the fifth millennium BC.

## 2. Egyptian Grids of Multiple Squares

The monuments of Carnac demonstrate the use of multiple squares and, because of their antiquity –one to two thousand years before the Pyramid Age – it appears likely that the later use of multiple squares in Dynastic Egypt demonstrates how such a technique could function as a pre-arithmetical geometrical framework. By the Middle Kingdom, the Egyptians had put stylus to papyrus to describe their mathematics in a document called the Rhind Manuscript. This recorded a system of geometry based around pre-Ptolemaic ideas, which included the use of a grid of multiple squares.

At Carnac, the angular extremes of sunrise and sunset, on the horizon during the year, followed the lesser angle of a 3-4-5 triangle whilst in the Rhind Manuscript one finds a “canevas” [*] or grid-based diagram, in which both of the acute angles of this 3-4-5 triangle, held primary to the Egyptians, are shown to be generated by the summed diagonal angles of either; two double squares or two triple squares. The resulting grid is then 14 squares by 14 squares, and this is exactly the grid upon which the Type A stone circles can most easily be constructed, if one excludes the use of ropes and stakes to achieve this design.

Figure 3 of a Rhind diagram showing evolution of a 3-4-5 triangle within a 14 by 14 grid of squares

Such a use of multiple squares, as a template on which to construct a stone circle geometry, raises the question of the side length used, since they all need to be identical and so the ability to create identical lengths would certainly suggest an accurate system of measures, or metrology, was in use. This leads into another bitter dispute, concerning the existence of Alexander Thom’s found measure, the megalithic yard, as being a primary unit of measure maintained accurately by the megalithic builders throughout the British Isles and Brittany. Thom did not know enough about historical metrology to see that the megalithic yard might well have been accompanied by systematic variations applied to its length or indeed, that other measures might also have been evolved. His proposal of an accurate megalithic yard, like that of exact stone circle geometries, also came to be rejected by archaeologists, who themselves knew very little about historical metrology[7] [*], pointed to cases where Alexander Thom’s hypothesis of a singular measure in use within megalithic Britain was muddied by the presence also of other standard units of measure.

## 3. Generating Flattened Circles using a Grid of Squares

One of the key objections for the megalithic concerning ropes to construct flattened circles is the necessity for measured radii and their centres. If Thom’s Type A or Type B flattened circles were instead constructed using a grid of squares, then some of the key points where a flattened circle’s radius of curvature changes (of which there are only four) must be points of intersection within the grid . This became clear through considering the Type A geometry and specifically its implicit double triangles, as possible right triangles.

Robin Heath has already noted[8] that these triangles are close to the invariant ratio, in their longest sides, of the (lengths of) time found between the eclipse year and the solar year, and this ratio is also to be found between the solar year and the thirteen lunar month year.

The baseline of such a right triangle is found to be 6/7 of the diameter MN of the Type A flattened circle and this implies, given the left-right symmetry of this form, that this key point at the end of the hypotenuse (where the radius of curvature changes) would sit on the corner of a grid point within a 14 by 14 square grid as a length equal to twelve grid units. The forming circle used by Thom, of diameter MN, would then inscribe the grid square.

Figure 4 Type A drawn on a 14 square grid

We also know, from Carnac, that the astronomers used a triple square to frame this right triangle which then relates the periods of eclipse and solar year. Since the vertical position of the key point is 12 units, then to left and right the key points either end of the central flattened arc are 4 units, either side of the central axis. To right and left of these triple squares can be found two four squares, that express with perfect accuracy the relationship of the lunar year to the solar year, as diagonal. These four squares have a baseline of twelve grid squares which exactly matches the number of lunar months within the lunar year.

One can then see within the 14 square grid that many multiple squares can be found, for example the triple squares either side of the vertical centreline have two four-square rectangles to the right and left (shown in red below, the ripple-squares being blue). These leave a row of 14 by 2 squares at the top which can be seen as a seven-square, the rectangle whose diagonal to side alignment is found between a double and a triple square.

## Conclusions

Clearly there are alternative ways of generating a flattened circle geometry that using stakes and ropes (geometry as we know it). We know that the Egyptians used grids within square grids of constant unit size and that multiple square rectangles were clearly used at Carnac in the megalithic (c. 4000 BC) before dynastic Egypt began, and by the time of the Rhind papyrus (Middle Kingdom) a system for containing irrationality of numbers had developed a school using grids, and what could be done with them. Ever since the Ancient world, this practice of “modularizing” buildings along rectangular or triangular “lines” became a key practical method outside of algebraic maths. It is therefore highly likely that grids gave the megalithic and later builders a canvas upon which to design and achieve accurate geometries not then rectalinear.

#### Some other resources.

More on the practical models of such early practices see my book Sacred Geometry: Language of the Angels. For more on flattened and multiple squares, please see chapter two of Sacred Number and the Lords of Time.

[1] ] Megalithic building types include standing stones, stone circles, stone rows, dolmen, chambered and other cairns.

[2] between 1934 and 1978

[3] during the late 1960s and early 1970s

[4] over two thousand miles away in the ancient near east

[5] [AAK and Howard Crowhurst]

[6] His survey can be found

[7] Historical metrology is a scattered remnant of the metrological system employed within the British stone circles and also within the Egyptian pyramids. It is this latter application of metrology in the ancient near east which spread metrology, though such an idea has also been opposed by archaeologists working in the near east.

[8] Sun, Moon and Stonehenge by Robin Heath 1998

## ﻿Astronomical Rock Art at Stoupe Brow, Fylingdales

first published 28 October 2016

I recently came across Rock Art and Ritual by Brian Smith and Alan Walker, (subtitled Interpreting the Prehistoric landscapes of the North York Moors. Stroud: History Press 2008. 38.). It tells the story: Following a wildfire of many square miles of the North Yorkshire Moors, thought ecologically devastating, those interested in its few decorated stones headed out to see how these antiquities had fared.

### Background

Fire had revealed many more stones carrying rock art or in organised groups. An urgent archaeological effort would be required before the inevitable regrowth of vegetation.

A photo of one stone in particular attracted my attention, at a site called Stoupe Brow (a.k.a. Brow Moor) near Fylingdales, North Yorkshire.

Continue reading “﻿Astronomical Rock Art at Stoupe Brow, Fylingdales”

## Chalk Drums to Symbolise Pi and Layout Monuments

December 2016 in numbersciences.org Hits: 3872

Perhaps as early as 4000 BC, there was a tradition of making chalk drums. Three highly decorated examples were found in a grave dated between 2600 and 2000 BC in Folkton, northern England and one undecorated chalk drum in southern England at Lavant in an upland downs known for a henge and many other neolithic features discovered in a recent community LIDAR project. The Lavant LIDAR project and the chalk drum found there are the first two articles in PAST, the Newsletter of The Prehistoric Society. (number 83. Summer 2016.) It gives the height and radius of both the Folkton drums 15, 16 and 17 and the Lavant drum, presenting these as a graph as below.

Continue reading “Chalk Drums to Symbolise Pi and Layout Monuments”

## The Approximation of π on Earth

π is a transcendental ratio existing between a diameter/ radius and circumference of a circle. A circle is an expression of eternity in that the circumference, if travelled upon, repeats eternally. The earths shape would be circular if the planet did not spin. Only the equator is now circular and enlarged, whilst the north and south poles have a shrunken radius and, in pre-history, the shape of the earth’s Meridian between the poles was quantified using approximations of π as was seen in the post before last. In some respects, the Earth is a designed type of planet which has to have a large moon, 3/11 of the earth’s size and a Meridian of such a size that the diverse biosphere can be created within the goldilocks region of the Sun’s radiance.

It would be impossible to quantify the earth as a physical object without the use of approximations to π, a technique seen as emerging in Crucuno between its dolmen and famous {3 4 5} Rectangle where the 32 lunar months in 945 days was used, through manipulation of proximate numbers to rationalize the lunar month to 27 feet (10 Drusian steps) within which days could be counted using one Iberian foot (of 32/35 feet) as described here and in my Sacred Geometry book.

John Michell (1983) saw that different types of foot had longer and shorter versions, different by one 175th part and corresponding to the north-south width of two parallels of latitude: 51-52 degrees, which is the mean earth degree, and 10-11 degrees. The ratio 176/175 is interesting as for its primes.

1. The harmonic primes {2 3 5} are 16/25 times 11/7.
2. The 11/7 is half of the pi of 22/7 and the harmonic ratio is the inverse of 25/8.

From this it is clear that these two latitudes are related by the approximation to 1 of a π (22/7) and a reciprocal 1/π (8/25).

But John Neal (2000) saw that some feet also expressed 441/440 which is the ratio between the mean radius of the earth and its polar radius, visually clear in the Great Pyramid. This ratio is also the cancellation of two different πs, namely 63/20 and 7/22 since 7 x 63 = 441 and 20 x 22 = 440. From this emerged an ancient model of the earth that was embodied within the ancient metrology itself. I call this the metrological model rather than the (earlier) geometrical model based upon equal perimeters and the singular π of 22/7.

The metrological model gave a set of regular reference latitudes that accurately defined the geoid of the planet’s meridian by 2,500 BC. One can ask how those developing the model came across the idea of using proximate ratios of π like 176/175 and 441/440, since the system works so well that one may say that the meridian appears to have been designed that way.

The geometric model already defined the mean radius as 3960 miles and so that gives a mean earth meridian of 22 x twelve to the power six. One 180th of this gives a degree length of 364953.6 feet and this is only found at the parallel 51-52 degrees. It is this that defines the megalithic in England, Wales, Scotland and Ireland, an obvious candidate for the metrological survey whose complementary latitude was probably 175/176 of this (362880 feet) in Ethiopia, south of the Great Pyramid. The parallel of the Great Pyramid is 441/440 longer (362704.72) than that of Ethiopia while Athens and Delphi are 440/441 of the mean earth and Stonehenge parallel, that is 364126 feet.

This system was first set by Neal in All Done With Mirrors 2000 as I was writing my first book Matrix of Creation. Are we to think Neal made it up or are we dealing with an exact science that had developed through the megalithic enterprise. And if the Egyptians had an exact science of the earth’s geiod, what are we to make of the fact that the earth appears to follow such a numerically inspired pattern of relationships still true today, in the age of global positioning satellites.

One clue lies in the mind, and how ancient number sciences focus holistically upon the balancing mean. A mean earth that did not spin never existed, since it was only the collision with another planet which created the Moon 3/11 smaller than the Earth. The mean earth radius is these days established as the cube root of the equatorial radius squared times the polar radius. This is less, by 3024/3025, than the geometric model’s mean earth radius of 3960 miles, again maintaining rationality.

It would appear that, in entering the physical and spatial, any planetary design might have been based upon precise rational approximations, about the mean size, of π. To this mystery must be added the musical harmony of the outer planets to the Moon, the Fibonacci harmony of Venus to the Earth itself and the extraordinary numerical relationships of planetary time created by the Sun, Moon and Earth documented by my heavily-diagrammed books and website. From this, more and more can be understood about our prehistory and about its monuments.

#### Books on Ancient Metrology

1. Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
2. Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.
3. Heath, Richard. Sacred Geometry: Language of the Angels. Vermont: Inner Traditions 2022.
4. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
5. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
6. —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.
7. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
8. —-. Ancient Metrology, Vol. 3, The Worldwide Diffusion – Ancient Egyptian, and American Metrology.  The Squeeze Press: 2024.
9. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

## π and the Megalithic Yard

The surveyor of megalithic monuments in Britain, Alexander Thom (1894 – 1985), thought the builders had a single measure called the Megalithic Yard which he found in the geometry of the monuments when these were based upon whole number geometries such as Pythagorean triangles. His first estimate was around 2.72 feet and his second and final was around 2.722 feet. I have found the two megalithic yards were sometimes 2.72 feet because the formula for 272/100 = 2.72 involved the prime number 17 as 8 x 17/ 100, and this enabled the lunar nodal period of 6800 days to be modelled and and the 33 year “solar hero” periods to be modelled, since these periods both involve the prime number 17 as a factor. In contrast, Thom’s final megalithic yard almost certainly conformed to ancient metrology within the Drusian module in which 2.7 feet times 126/125 equals 2.7216 feet, this within Thom’s error bars for his 2.722 feet as larger than 2.72 feet.

This suggests Thom was sampling more than one megalithic yard in different regions or employed for different uses. Neal [2000] found for Tom’s statistical data set having peaks corresponding to the steps of different modules and variations in ancient metrology, such as the Iberian with root 32/35 feet and the Sumerian with root 12/11 feet. It is only when you countenance the presence of prime numbers within metrological units that one breaks free of the inevitably weak state of proof as to what ancient units of measure actually were and, more importantly, why they were the exact values they were and further, how they came to be varied within their modules. However, the megalithic yard of 2.72 appears to outside the system in embodying the prime number 17 for the specific purpose of counting longer term periods which themselves embody that prime number.

The discipline of using only the first five primes {2 3 5 7 11} must have been accompanied by the perception that only if primes were dealt with could certain ends be served. This is crystal clear when we come to musical ratios in which the harmonic primes alone are used of {2 3 5} with an occasional “passenger” of the prime {7} as in 5040 which is 7 x 720, the harmonic constant.

## Using 2.72 feet to count the Nodal Period

The first remarkable characteristic of 2.72 feet is that 8 x 17 in the numerator means that the approximation to π of 25/8 = 3.125 can, in (conceptually) multiplying a diameter, provide an image of 25 units on the circumference of a stone circle. For example a diameter of 2 MY would suggest 17 MY on the circumference, which is quite remarkable. Further to this, we know that the 6800 days of nodal cycle is factored as 17 x 400 and that the MY was shown (acceptably) to have been made up of 40 digits (in conformance to the general tradition within metrology that there are 16 digits per foot and 40 for a step of 2.5 feet, which a MY traditionally is). The circumference of 17 MY is then 17 x 40 digits which means that a diameter of 20 MY would give a circumference of 17 x 400 digits equalling 6800 digits as a countable circumference in digits per day.

This highlights how prime number factors played a role, in the absence of arithmetical methods, in solving the astronomical problems faced by the late stone age when counting time periods in days.