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Megalithic Mathematics 6

The Egg rings 3.6.1   Thom has identified ten megalithic rings constructed in egg shapes. At the heart of these designs, as with the ellipses, he has detected the employment of mirrored Pythagorean triangles governing the construction. The corners of these triads subtend the necessary arcs which interlock to make the final perimeter. If the length of sides of these triangles are truly a number of whole units and the principal radius scribing the large end is also integral then the radii of the other three arcs must also be integral.

A.Thom, Megalithic Sites in Britain p.29.

3.6.2   Allen Water, Borders. This solstice orientated ring has been constructed with a triangle which is very near Pythagorean. The units used are half Megalithic yards. The lengths of the sides are;- a = 5.5 MY = 11 units. b = 6.5 MY = 13 units. c = 8.5 MY = 17 units. For a perfect Pythagorean triangle- a sq. + b sq. must equal c sq., but in this case;- a sq. + b sq. = 290     and c sq. = 289 .......a near perfect result. This triangle is so close to perfection it would seem unlikely that the constructors could tell the difference and probably assumed they had discovered another example in the special triangle series.

A.Thom, Megalithic Sites in Britain p.70.

3.6.3  Re- survey of nominal design for Allen Water stone ring Following is a re- generation of the alleged design for the Allen Water egg ring. Accepting the 11, 13, 17 triangle the nominal design has been drawn with the major radius = 16 units, (1 unit = 1/2 Megalithic Yard).

If the adjacent sides of this triangle are laid out accurately to 5.5 and 6.5 MY then the hypoteneuse must be 8.5898 MY, an 0.0898 MY discrepancy from ideal. 0.0898 MY is 2.93 inches, 73 millimetres or 1 pixel in this computer image.

If the radius of the main arc is set to a precise number of units, here 16, and the other arcs tacked to the ends consecutively all arcs will have integral radii.

	Comparison of nominal design with survey of remains of stones setting.

3.6.4  This digital reconstruction is accurate to + or - 1.17 inches, (29 mm) on the ground. Despite the ruinous condition of this site it can be seen how closely the design sits on the stones. Originally the inner faces of the stones were probably set to the design. Here we may be seeing another of the fine 'tweaks' applied to the perimeter in order to idealise this length. The length of the perimeter as shown is 55.68 MY or 22.27 Megalithic Rods. If the desired perimeter was intended to be 23 MR then the radii would have to be expanded slightly to achieve precise integrality and this outer perimeter would fall within the bodies of the stones closer to their centres. Thus the final stone settings were a compromise between marking the ring with the ideal radii and the ring with the ideal perimeter.

3.6.5  Woodhenge, Wiltshire. In the geometry of the construction set for the Woodhenge rings we can see strong correlations both for the Pythagorean special triangle basis of egg rings and the demand for integrality of perimeters. The perimeter lengths of the six rings rise in even multiples of 20 MY with one omission- the 120 MY.

A.Thom, Megalithic Sites in Britain p.74.

Ring perimeters;- I = 40 MY II = 60 MY III= 80 MY IV = 100 MY V = 140 MY VI = 160 MY

The base construction Pythagorean triangle, as at Allen Water, is expressed in units of half yards as ;- a = 6 MY = 12 units b = 17.5 MY = 35 units c = 18.5 MY= 37 units As 12 sq. + 35 sq. = 1369             and 37 sq. = 1369 then this is the sixth and largest of the series of true Pythagorean special triangles.

3.6.6  Thom made a re-survey of the positions of the concrete markers erected by archaeologists in the holes believed to have contained timber posts. When using an axis of symmetry which aligned with the first flash of the upper limb of the rising Sun at the summer solstice in 1800 B.C. he deduced that all of the arcs in this design were struck from only four centres and that there is a strict relationship between the small and large arcs.

A.Thom, Megalithic Sites in Britain p.73 A very careful survey, using a steel tape and theodolite, was made of the concrete posts which the excavators placed in the post-holes in the chalk. A reproduction to a very much reduced scale is shown in Fig. 6.16. The axis drawn is chosen to be along the azimuth of the point on the horizon where the midsummer Sun first appeared about 1800 B.C. Using centres on this axis we then find;- (1) the arcs at the large end have a common centre at A, (2) the arcs at the small end have a common centre at B, (3) the distance AB between these centres is 6 MY, (4) the arcs are equally spaced with one gap, (5) the radius at the small end is in each ring 1 MY smaller than the radius at the large end. These facts are indisputable but in themselves they do not explain the construction, because the radii are not integral multiples of the yard.

3.6.7  The construction triangles. On further examination Thom noted that a mirrored pair of special Pythagorean triangles with sides measured in half Megalithic Yards could support the construction centres for all of the 24 arcs of the Woodhenge design. The small and large end arcs would be struck from the ends of the common bases of the triangles whilst the larger side arcs would each be struck from the apex of the triangle on the opposite side of the design.

3.6.8  A.Thom, Megalithic Sites in Britain p.75 The lengths of the sides of the construction triangle are a= 6, b= 17.5, c= 18.5 The fact that these are rational numbers shows that we are dealing with a Pythagorean triangle. In units of half-yards the triangle is 12sq +35sq = 37sq. The discovery of this triangle must be considered as one of the greatest achievements of the circle builders. That they themselves considered it important is shown by the use they made of it at Woodhenge. Its use at another site will be discussed later. But we have yet to show how the radii of the rings were chosen. The scheme used only becomes apparent when we realize that the rings were intended to have perimeters which were multiples of 20 MY. The values selected were 40, 60, 80, 100, 140, and 160 MY.

3.6.9  The perimeters/ radii relationship. Thom thinks that the perimeters of the six rings were designed to increase in close multiples of 20 MY. This demand would necessitate abandoning the integrality of the radii, however a degree of integrality was maintained in the progression of these measurements by having the large end radius of each ring close to 1 MY greater than the corresponding small end radius. 3.6.10  How close the ring designers came to achieving the ideal nominal perimeters and radii in this ambitious plan can be seen from the following table where actual measurements are given along with the nominal value. Pi for each ring is also calculated.

	Rings	Perimeters nominal (MY)	Perimeters actual (MY)	Radii differential r1 - r2. Nominal=1.(MY)	pi
	I	160	161.0	24.35 - 23.46 = 0.89	3.02
	II	140	138.2	21.18 - 20.29 = 0.89	3.00
	III	100	104.2	14.7 - 13.82 = 0.88	2.95
	IV	80	79.9	11.54 - 10.53 = 1.01	2.90
	V	60	61.3	8.24 - 7.23 = 1.01	2.83
	VI	40	39.4	5.07 - 4.06 = 1.01	2.70

3.6.11  Ring III, the 100 Megalithic Yard ring. It can be seen that the greatest discrepancy from a nominal perimeter is with ring III. Instead of 100 MY we have 104.2 with a radius differential, 0.88, farthest from the ideal 1 MY. Here the constructors were dealing with the most difficult ring to regularise in the desired series. If the radius were to be taken closer to 1 then the perimeter would be further yet from ideal, and vice-versa, a closer solution to the 100 MY perimeter would have dragged the radius down further from nominal.

Thom speculates that here we might be seeing another ingenious compromise, similar to the tweaking of the perimeter of the Allen Water ring. The actual position of the posts of ring III are centred some 17.25 inches, (0.43m), outside the true ideal 100 MY perimeter. Archaeologists believe these posts were very large- nearly 3 feet thick, (0.85m).

Woodhenge detail.

Here the surveyors had the greatest difficulty in avoiding transgressing the 1 MY radius differential. To have centred these posts on the 100 MY perimeter would have decreased the differential unacceptably. Thom surmises that posts were here employed which, although centred closer to the nominal radius, were of such exceptional thickness that the inner  faces of the posts would define the ideal 100 MY perimeter, (Or advantage taken of the traditional method of supporting a large roof if this is the remains of a timber building. If this were the case the initial planning of the building would have been based on this solution to establishing an egg ring of 100 MY perimeter).

Thom, 'Megalithic sites in Britain' p75. It will be seen that ring III is some 4 per cent large. This ring is very nearly represented by taking r1 = 15 and r2 = 14, which gives a ring about 0.53 MY or l.44 ft outside the hypothetical 100-MY ring everywhere. It thus appears that if the posts were 2.88 ft (or about 1 MY) diameter the inside of the structure would be a perfect fit. The excavators found that there were deep ramps to all the holes in this ring, indicating that very large posts had been used carrying perhaps a platform or roof.

3.6.12  Integrating pi at Woodhenge. Ring II. Examining the final column in the above table we see that pi for ring II is exactly 3. This ring carries the second largest diameter posts. Thom suggests that a fundamental motive behind the construction of Woodhenge was to display a progression of study into the rationalisation of pi. At Woodhenge we have a series of rings progressing in 20 MY multiples with each having a value of pi close to 3, the more heavily marked ring II being a perfect, (or near perfect), solution to the rationalising or integrating of pi.

	Rings	Perimeter nominal. (MY)	Perimeter actual. (MY)	Radii differential r1 - r2 nominal=1MY	Pi
	II	140	138.2	21.18 - 20.29 = 0.89	3.00

Further weight to Thom's suggestion is given when we see that the constructors may have refrained from 'tweaking' the radius of ring II to achieve a perfect solution for the perimeter. It can be seen that the radius differential is short of nominal, 0.89 MY, whilst the perimeter is also short at 138.2 MY. Having a perfect 140 MY perimeter with the radii differential closer to the nominal 1 is entirely possible but this would have moved the value of pi away from the ideal 3.

Thom 'Megalithic sites in Britain' p75/76. In the above table pi is the theoretical ratio of P, the nominal perimeter, to the greatest diameter (2r1+5). It will be seen that pi gradually increases as the rings get larger until at ring II it is 3.00. A more exact calculation gives 2.9994. No matter how carefully the builders made their measurements they could never have detected the difference between this and 3. One is tempted to surmise that the whole set of rings may be a permanent record of an elaborate empirical determination of a geometrically constructed ring which would have as it were pi= 3 and at the same time have a circumference a multiple of 20 yds. Certainly none of our modern circle squarers have obtained a closer approximation. It may be remarked that ring-Il post-holes are better marked than ring I which overshot the mark with pi = 3.02. Presumably the inner ring was laid out first. One wonders how many rings were set out before the builders discovered that every 20 yds they added to the circumference gave them the same increment to the radius (actually 1O/pi. Did they notice this after four rings and then attempt an extrapolation? It is much more likely that they already possessed this kind of knowledge, because this cannot have been their first attempt.

3.6.13  Verification of the value for the Megalithic Yard. The precise dimension of the Megalithic Yard had been derived by Thom before his study of Woodhenge. He used this as the scale unit when laying out his nominal construction lines. Subsequently he made an analysis of the actual size of the Woodhenge rings and derived a value for the unit of measurement used at Woodhenge. This matched, very closely, the figure deduced from many other sites around Britain.

Thom 'Megalithic Sites in Britain' p75 We can, by the statistical method described and used earlier, find from P(actual), neglecting ring III, the value of the Megalithic yard which best fits Woodhenge. This turns out to be about 2.718, a value so close to 2.72 (used in drawing the rings) as to show that we can be quite certain we are using the identical geometric construction to that used by the builders. 3.6.14  Special Pythagorean triangle no. 6.

Thom 'Megalithic Sites in Britain' p77. They had probably experimented with many other triangles before arriving at the 12, 35, 37. One is entitled to reject the above reason for making the structure, but everyone must be impressed by the laborious, painstaking work which preceded the discovery of the sixth member of the list of perfect Pythagorean triangles and the construction of a set of rings based on this triangle with perimeters exact multiples of 20 yds.

3.6.15  Woodhenge. If Thom's analysis seems reasonable it would be fair to say that Woodhenge, of it's own, demonstrates much of the mathematical thought of the British stone ring builders. Here we see the assiduous efforts to maintain integrality in as many measurements as possible. We also see their attempts to rationalise pi and eliminate an incommensurable. We clearly see the use of the final triangle in the six-series special Pythagorean triangles list, and we also find a good proximity to the previously derived value of the Megalithic Yard.

3.6.16  Originality of design seemed to be a high priority throughout the principal period of stone ring building. There are few exact duplicates. Variety was achieved in the simple circles and D rings by seeking a range of appropriate diameters. Variance in egg rings and ellipses was managed by choosing as their construction base from the range of Pythagorean special triangles. Hundreds of unique designs based on only four classes of rings were produced throughout Britain.

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