## Diary Notes

#### 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.

## 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.

see also my youTube video of a talk at Megalithomania in 2015.

[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”