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Megalithic Mathematics 7

The Compound Rings. 3.7.1  A further step in stone ring design was recognised by Thom in only ten sites and he has termed these the compound rings. Although each of these rings is constructed with unique procedures they still conform to the abstract numerical imperatives illustrated in the simple circles, D rings, ellipses and eggs. With the compound rings each design is an original solution to the various mathematical constrictions imposed on the less complex sites,

A.Thom,'Megalithic Sites in Britain' p.84 The Compound Rings We shall discuss Avebury in this chapter, but before doing so it is advisable to look at three rings whose designs lead up to the Avebury construction. These three sites seem to the author to be amongst the most important in Britain. Their geometrical construction shows a mastery of the technique of finding designs which, while possessing an elegance of symmetry and proportion, yet incorporate a hidden significance in that integral lengths were obtained for the basic dimensions and the perimeters were multiples of 2.5 MY. It is true that today we can be petty and apply our short-cutting knowledge of trigonometry to show that their lengths were only approximations. Their 13.5 is our l3.503, their 15 our l4.99, but this does not show that they failed. Within their limitations they succeeded. To our modern thinking they were attempting the impossible, but in more advanced spheres so are we.

3.7.2  Kerry Pole stone ring, Montgomeryshire.

In the Kerry Pole design superb symmetry of number has been achieved with all but one vital length being near a perfect integer. Only the perimeter arc GCBDH shows a clear variation from exact whole units.

	The total perimeter length of (97.38 MY =) 38.952 Megalithic Rods is short of the perfect integer 39 by only 3.92 inches, (98 mm).		A.Thom, Megalithic Sites in Britain p.88. Note that radius KT lies close to the north/south meridian.

Thom 'Megalithic Sites in Britain' p88. Ring near Kerry Pole On the ground this is a very unimpressive site, but when it is surveyed (Fig. 7.5) and the geometry studied it turns out to be another member of the group we are examining. The construction is again based on two circles. Here their diameters are definite, 32 and 16 MY. Two points are then established on the outer circle, E at 5 MY from the axis and G at 14 (!). Bisect the angle GOE by the line LOK. Draw KP/2 T and KP/1S. The corner arcs ES and TG are centred on P/1 and P/2 and the closing arc on K. A little trigonometry gives the radius KS of the closing arc and the perimeter. The remarkable thing is that these are 29.98 and 97.38 MY. Thus all the radii are integral, 16, 8, and 30 MY, and the perimeter only 0.l2 different from a multiple of 2.5.

(!). Error. Here, in the text, Thom has given the distance between G and B as 14 MY although his surveyed plan makes this length closer to 17 MY. Perhaps this slight textual error obscured the more obvious method of locating G and H which is to first describe the trisection of the main circle. The length of arc GB will then be 16.75 MY whilst the length of chord GB will of course be 16 MY,- a radius.

	3.7.3   Construction set for nominal Kerry Pole design. This is a construction procedure which can be followed using rope, pegs, and rods measuring Megalithic Yards and Megalithic Rods, (2.5 MY).

Careful rope-and-peg scribing would be sufficiently accurate at this scale of drawing.

	3.7.4  Tri-section points on the perimeter can be found by laying together at, OB, two equilateral triangles with sides equal to a radius. Finding the deci-section points of the perimeter of a circle needs careful measurement along the perimeter, however, by selecting 16 MY as the radius for the large circle the builders would know that the resultant perimeter length would be very close to 100 MY, (100.531). Then, drawing a chord of 10 MY, EF, the necessary deci-section of the perimeter could be achieved with very good precision without the need to measure along the curve. The chord is located by measuring two perpendiculars, each 2 MR, from AO which is OB produced.

3.7.5  Here, as elsewhere, we may be seeing the practicality of employing a unit of measurement which has 100 sub-divisions. 1 Megalithic Rod = 100 Megalithic Inches. Perhaps the original devising of this design was arrived at by working to a scale of 1:10 or 1:100, on sand or close cropped turf or rock slabs. For a similarly devised ring design see the Avebury costruction set at Maths 10.paragraph 3.10.1

A simple method of finding the angle bisector is to measure two arcs of the same radius centred P and Q.

Draw small arcs ES and TG centred P and Q with KP produced to S and KQ to T.

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With closing arc ST centred K, half the design perimeter is completed.

Follow the mirror of the procedure to complete the other half of the perimeter.

These are the stone positions as surveyed by Alexander Thom.

This is the design as derived from Thom's algorithm, (with correction), overlaid on the survey of the stones.

This computer re-generation is accurate to + & - 1.91 inches, (48 mm), on the ground.

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