Ancient Geometry & The Rule of Six
This treatise examines the way in which British Neolithic peoples might have used geometry to construct their Stone Circles and Henges. Diogenes NRest 10/05/2024
Ancient British architecture has its roots back in the Neolithic. The Neolithic farmers arrived in Britain around 5000 - 4500BC after a long migration in distance and time across Europe from the Fertile Crescent. Middle Eastern arable farming started around 10,000BC with the cultivation of emmer wheat and barley which grew natively in Anatolia. This spread along the Fertile Crescent which had the necessary geography to support these crops. Alongside farming, villages and towns emerged, along with more specialised trades.
Göbekli Tepe (excavations Klaus Schmidt 1996-2014) dates back to around 9500BCE. It is thought to contain the world's oldest megaliths in oval structures. The megaliths in the outer walls do seem to favour irregular 12 stone settings. It is probable that they used the Duodecimal system. 12 being handily divisible by 2, 3, 4 and 6. There is no reason to believe that they had progessed to the subtlety of Sexagesimal (based on 60) which later became prevalent in the Middle East. 60 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30 and also allowed for the handling of larger numbers and smaller fractions necessary for maths in the Bronze Age.
This Göbekli Tepe carving may contain the earliest representation of Handbags. The earlist Hunter / Gatherers would have needed woven baskets / handbags to be effective. Thus handbags may have their origins 3 million years ago amongst the tool-making Australopithecus. Handbags came to represent abundance of food. When humanity spread out from Africa, they took this concept with them. There are many depictions of handbags on carvings of Deities across the world. Deities being accredited as the source of all abundance. So there are ample evolutionary reasons why women may be attached to their handbags. It is mainly men who overstuff their built-in trouser pockets.
By the Bronze Age, geometry was documented on clay tablets. Tablet SB-13088 comes from the Elamite city of Susa and is thought to date to around 1894–1595 BC. Elam is situated to the SE end of the Fertile Crescent and to the East of Sumer. The tablet shows a Heptagon, part of which is very accurate. The cuneiform text relates to the construction of the Heptagon. I used NRest to model the Best Fit Circle from the vertices. This gives me the centre point of the Circle. The vertex E is clearly misplaced but still on the Circle. I then used NRest to connect the vertices to the centre. 4 of the 7 segments are accurate, including the inscribed lines. At least 2 of the Northern trio are less accurate. Images courtesy of the Louvre Museum. Angles courtesy of NRest.
A practical exercise in geometry, clay tablet Si 427 was discovered in the Sumerian city of Sippar and dates to between 1822-1762 BC. What we see is an estimation of the area of an irregular field by surveying rectangles and triangles. The numbers on it relate to lengths and areas. Sizes of fields were important matters for farmers. They would have known that a field measuring 6 by 5 units was the same size as a field of 5 by 6, i.e. 30 square units. Image courtesy of Istanbul Arkeoloji Muzeleri.
Babylonians approximated the area of a circle and consequently also knew the volume of a cylinder. Very handy for measuring quantities of liquid and grain. For this purpose they usually approximated Pi using the integer 3. They came to a better estimate of Pi by using a Hexagon to model a Circle - 25/8 or 3.125.
When it comes to Ancient British Architecture, we need to consider how they achieved the 5 / 10 / 20 Fold geometries we see in some Stone Circles. We can use a Hexagon to approximate a Pentagon by constructing a circular rope with 30 equal segments. We first construct a circle with a radius of 5 units. The rope can be divided into 6 linear sections which are 5 units in length. The vertices of this Hexagon will sit perfectly on the circle. Similarly we can set up the same rope to be divided into 5 linear sections which are 6 units in length. When we fit this onto the same circle, it doesn't quite work. We can either accept the small anomaly or increase the effective radius of the Pentagon by systematically moving every node slightly further out by the same amount until it does fit. This Pentagon works better for a circle of radius 5 6/60.
Approximating with a Hexagon can be used for other Regular Polygons. The Rule of Six can be applied to many different Regular Polygons with differing degrees of accuracy. The less accurate radii are 5, 7, 8, i.e. memorably the same as the no. of sides of the Polygon. The more accurate radii shown here are 5 +6/60, 6, 7 -5/60 and 8 -10/60. The more accurate radius for a 9-sided Nonagon would be 9 -14/60. There is a high degree of regularity in the way these more accurate numbers progress (as can be seen in the diagram), which could give rise to a general approximation formula for an n-sided Polygon (Enhanced Rule of Six):
- Radius = n - ( (n - 6) / 12 )
There is a far simpler method than Enhanced Rule of Six for generating better polygons using Rule of Six. Starting at 0, plot the points clockwise from 1 to 6 (red). Then starting at 0 again, plot the points anticlockwise from -1 to -6 (green). The true values for a Heptagon lie between the closely placed positive and negative nodes. For best accuracy, this should not be used for the first nodes 1 and -1. The initial error there we will denote as e. Using the midpoint between clockwise and anticlockwise subsequently, the error at 2 and -2 would be approx. 1.5 x e and the error at 3 and -3 would be approx. 0.5 x e. We don't need to plot the points 6 and -6, where the midpoint error would be approx. 2.5 x e. Note that if we plot the points up to 7 clockwise, we would have accumulated an error of approx. 7 x e from the starting point of 0. The old joiners' adage runs "Measure twice, cut once". This is more "Measure in two different ways, cut once".
On the reverse side of tablet SB-13088 we have a perfectly executed Hexagon. Most of us remember that the first thing we did when getting a compass, was to draw that Hexagonal flower within a circle. This is the easiest Regular Polygon to draw with a compass. The slot now becomes interesting as it appears from this side to be a 30° (1/12) template. In the associated exploration of symmetry, we start with the circumscribed Hexagon. The Radius and Sides are length 2.
To generate a Dodecahedron (12-gon) from a Hexagon, the rope with 2 sections already provides midpoints on the sides to double up the degree of Symmetry by projection (as shown in red). This also gives us the positons of East and West, in addition to the already established North and South, which gives us the Square (in green). To get an Octagon from a Square, we need to bisect the sides of the Square perpendicularly. This is the second exercise that we learn when we first get a compass. The rope of length 2 is also perfect for this task (in purple). The extra nodes for the Octagon lie where the purple line crosses the circle. An interesting implementation of these geometries can be found at Grange Embanked Stone Circle in the Archive.
Being part of the Fertile Crescent, Elam has a much older history of development than contemporaneous Sumer and hence probably a longer mathematical history. Susa dates back to around 4395BC. The history of mathematics in this corner of Asia at this time (3000-1500BC) is probably an unattributable mixture of Elamite, Sumerian and Babylonian. During these early days, there was contact between this area and Egypt / Indus Valley civilisations. After these early contacts, these civilisations went their own way. Egyptian mathematics may owe something to these early contacts.
The Plimpton 322 clay tablet from Babylon is thought to date back to 1900-1600 BC. On it we have a table of Pythagorean triples (Dr Daniel Mansfield, University of New South Wales). The Egyptians surveyed a right angle with a rope, using 12 identical sections in 3-4-5 configuration. This is the best known Pythagorean triple.
Another compelling way of thinking about this is that we are approximating the Polygons with a Circle. The Babylonians used 3 as an approximation of Pi. The circumference of the Circle is 2 Pi x Radius, which in Babylonian terms is 6 x Radius. For a Polygon of 11 sides we use a Radius of 11, which makes the circumference 6 x 11 or 11 x 6. We need the perimeter of the Polygon to be 11 x 6. Thus we get the Rule of Six directly from the bad Babylonian approximation of Pi. Using a bad approximation of Pi might seem a daft place to start. If we used the correct version of Pi, then the size of the circumference would be larger than 66. However when drawing the sides of the Polygons, we are taking shortcuts compared with going around the circumference properly. Thus using 3 as Pi counteracts the shortcut effect in a strangely coherent way.
Looking at this the other way around, Regular Polygons were used by the Greeks to estimate areas. This is called the Method of Exhaustion. Initally developed by Antiphon and Eudoxus of Cnidus, it was subsequently used by Euclid and Archimedes. Archimedes used the Method of Exhaustion as a way to compute the area inside a circle by filling the circle with a sequence of polygons with an increasing number of sides. The more sides, the more accurate the approximation. Thus the crude approximation of PI used in the Rule of Six deteriorates as the number of sides increases. Enhanced Rule of Six partially corrects for this, in a simplistic manner. Note by using mid points in the sides, you can double the number of points in a polygon without incurring any extra inaccuracy due to this effect.
I subsequently discovered this paper where the Rule of Six is applied to Regular Heptagons. If the segment length a is 6, then the radius is 7. The radius of the Heptagon is inscribed as 0;35 on tablet SB-13088. On the reverse, we have a Hexagon with an inscribed radius of 0;30 (0.5), giving a diameter of 1. Both polygons have a segment length of 0;30. The Hexagon is key to understanding the Heptagon and the 7a/6 formula. My main interest lies in the ancient British architecture which so far does not involve 7 fold symmetry.
In order to make Polygons with more sides, the Rule of Six already provides midpoints on the sides to double up the degree of Symmetry by projection (as shown with Hexagons). For odd numbered Polygons, we don't even need these as we can survey the diametrically opposed points on the circle to double up. So all the low numbered symmetries can easily be doubled up covering 10, 12, 14, 16 Fold symmetries. After that we could use compass methods to halve the angles and double the symmetries again, but I think it more likely that it would have been done by eye once the existing stones were close enough. Simply by 2 people, an observer and a mover, judging when the new stones were half way between the previous stones.
My aim in developing this treatise was to try to work out the simplest possible techniques that ancient peoples might have used in their architecture. In order to do this you need to divest yourself of modern notions on exactness. The Neolithic version of accuracy would be indistinguishable by eye on the ground, no matter what age you were in. Until the advent of Lidar and potentially drones, nobody had a decent view from directly above. There is evidence that the Rule of Six was used in Bronze Age Mesapotamia. Occam's Razor is our guide - the simplest explanation which covers all the fact is probably the correct one.
British Neolithic
Elm Decline (4710 - 4500BC) at Gors Fawr Bog is documented by T. Darvill, D.M.Evans, R.Fyfe & G.Wainwright, Strumble-Preseli Ancient Communities and Environment Study (SPACES): 4th report 2005, Archaeology in Wales 45: 17-23. We don't currently have any evidence of the direct causal agent. The least interesting, but plausible cause is Dutch Elm Disease. Otherwise it would seem to fall to man. Current notional dates might suggest that the Mesolithic people took to clearing the land for Farming. However it might be just as likely that this represents the arrival of the Neolithic farming peoples. The recent map (Detlef Gronenborn, Barbara Horejs, Börner, Ober) suggests that the Neolithic people were already in Brittany and Normandy by 5000BC, so the early date is feasible.
What Maths did the Neolithic farmers bring to Britain? Possibly Duodecimal and the calculation of area, but it is not clear what else. When the migration of Neolithic farmers started, their Maths probably wasn't very advanced. Initial British Neolithic Architecture consisted mainly of Long Cairns, Burial Mounds, Causewayed Enclosures and Cursus. Causewayed Enclosures, also found on the near Continent, usually take on irregular oval forms, but some take on very regular Circular (Salthouse) and Elliptical (Combe Hill) forms. These were the geometric precursors of later Henges and Stone Circles.
The earliest reference to Ellipses seems to be credited to the Greek Menaechmus around the years 360-350BC. Fortunately the most advanced feature of NRest is modelling Best Fit Ellipses. This might currently be the only way to identify them with any degree of certainty.
Neolithic people designed Circles, Ellipses and latterly more complex shapes as both Stone Circles and Henges. In the matter of Stone Circles, they usually favoured an even number of stones. Using diametrically opposed pairs of stones reduced the amount of work required and was pleasingly symmetric. The most common number of stones was 12, however we also get 8 and 10 stone settings which require a different geometry. There are circles with all sorts of different numbers of stones in them, but the most important aspect is their geometry. The stones have to belong to a defined pattern. We ignore associated stones that don't fit in and mentally include stones that might be missing in the pattern as long as the overall pattern is sound.
To design a Circle, we need 1 peg and a piece of rope. To design an Ellipse we need 2 pegs and a piece of rope. Ellipses are more complicated. They have an orientation, often NS / EW. They can have a wide variety of shapes from nearly circular to narrow forms. Any rope attached to 2 pegs can be used to create an Ellipse of some shape.
Further Mesapotamian Bronze Age Maths
Tablet YBC 7289 famously documents the Babylonian estimate for the Square Root of 2, marginally under 1.414213. The real value of Root 2 is approx. 1.41421356… This is documented on the diagonal of a square in sexagesimal, 1; 24, 51, 10. The badly drawn figure on the reverse side looks quite close to the 1, 2, Square Root 5 triangle. The only text on the tablet would appear to be a 1 and a 2. No sign of the text for Root 5 (2; 14, 9, 50). This would equate to an approx. value of 2.236065, whereas the true value of Root 5 is close to 2.236068. As we have seen previously, this would suggest a close relationship between each side of the tablet.
How did the Babylonians come to this highly accurate number for the Square Root of 2? It wasn't by using a ruler - this doesn't work to this degree of accuracy, no matter what century you work in. They probably used an Iterative Algorithm. In decimal, make a guess at Root 2, say 1.5. Square the number. It is way too high. Choose a lower number, 1.4 and square it. It is too low, but much closer. Choose a number between 1.4 and 1.5, but closer to the better 1.4, say 1.43 and square it etc. Carry on until you get to as many places as desired. The same procedure applies in sexagesimal, although the guesses would be sexagesimal guesses. It has also been suggested that Archimedes used an iterative approach to calculating Root 3. It is at this point that the Babylonians would have got a sense of the unending (or infinite as we call it). We know the Baylonians had tables of Squared and Cubed integers. By rights there should be a clay tablet somewhere tabulating all the Square Roots of Prime numbers. The Square Roots of all non-Prime numbers could be calculated from these.
We have already seen an example of an Iterative Algorithm when the area of an irregular shaped field was being estimated by using rectangles and right angled triangles. Start with the biggest internal rectangle and then fill in the remaining gaps with smaller rectangles ad infinitum (almost). There comes a point when the remaining space seems to be occupied by right angled triangles. Filling in right angled triangles with rectangles is another of those literally unending tasks. For every rectangle added to a triangle, you get 2 more triangles. We don't need to do this, as we know the area of right angled triangles (which are half rectangles). We simply sum up all the areas delineated to get the total area of the field.
There is an important lesson in this, even for modern computer algorithms. At some point, it may be more efficient to terminate an algorithm at a predetermined point and take a new approach from there on. The modern Quicksort algorithm is a good case in point. The original algorithm can be speeded up by stopping the Quicksort mechanism when the size of the subsets gets to around 24 - 20 items. The members inside these small subsets remain unsorted in the first phase. The members of each subset are all positioned correctly in relation to all the other subsets. Once the first Quicksort phase is complete on this basis, we simply run the whole data set through an Insertion Sort which is more efficient at sorting this nearly sorted data than Quicksort. In lots of ways, this enhanced version of Quicksort takes on the characteristics of the efficient Post Office Sort (before the advent of Post Codes and Sorting machinery).
Machine Learning algorithm Gradient Boosting Machine (GBM) might also benefit from this approach. There is no definite endpoint to the algorithm (unlike Quicksort), just increasing complexity producing diminishing returns. This increases the size of the model and makes it slower. GBMs also have a tendency to overfit. That is become less accurate when the modelling is taken too far. There is an argument for stopping the algorithm after a certain point is reached. There may be better ways of completing the modelling process.
There is much more maths originating from Bronze Age Mesapotamia which is beyond the scope of this treatise. People have written books about it. Here lies the significant foundations of the work which later flourished in Greece. Thanks to Charles Babbage and Ada Lovelace, there was a new age of mechanically executed algorithms. Now we are in the age of electronic algorithms pioneered by Alan Turing and others, but the basic Bronze Age principles still apply.