In 1972 John Michell first inferred an enormous ten-sided form nearly sixty three miles across, in which important historical and neolithic sites had been intended as ten vertices around an ancient centre, signified by a Whiteleafed Oak (see previous article extract).

Figure 1 The Decagon of Perpetual Choirs, anchored upon Stonehenge, the Solstice sunrise in summer and set in winter

Michell had previously [1991] developed the idea of the enchantment of the land as an actual practice; land areas were enchanted by using a geometrical pattern integrated with myths and ritual calendars, enacted within that framework. This was long before, around 930, such a pattern was being established of thing-places in Iceland. The idea of thing places is still findable in English names such as Gor-ing, the centre northeast of Stonehenge, where the summer solstice sun arose.

"Perpetual choirs were a Celtic institution, from pagan into early Christian times. In Iola Morganwg's Triads of Britain, translated from Welsh, it is stated that 'in each of these three choirs there were 24,000 saints; that is, there were a hundred for every hour of the day and the night in rotation, perpetuating the praise and service of God without rest or intermission.' " from *The Measure of Albion*

"Three of the choirs were located at Stonehenge, at Glastonbury, and near Llantwit Major in Wales. Others appear to have been at Goring-on-Thames and at Croft Hill in Leicestershire, a traditional site of ritual, legal, and popular assemblies." from *The Dimensions of Paradise*

Michell inferred a decagon because, from Stonehenge, the angle between the westerly bearing to Glastonbury and that to Goring-on-Thames was 144 degrees, the internal angle of the vertices of a regular decagon. Being regular, this decagon of perpetual choirs is inscribed within a circle that can be viewed as a scale model of the mean earth in which 80 feet on the circle's circumference is equivalent to a geographical mile of 5068.8 feet on the surface of the mean earth, then having been reduced by 63.36, the scaling involved (5068.8 equalling 80 times 63.36). Michell also found that the sides of the decagon, between the choir sites, were also a metrological model of the mean earth radius and I noticed that, in this case, it is the inter-site distance (the ten sides) which are 63.36 of the mean earth radius. There is therefore some common factor, of 63.36, operating to make the sides and outer circle of the decagon relate to the mean earth. We must remember Michell's assertion here; that the mean earth was ancient man's conception of the spiritual world, and also note that 63.36 is one fiftieth of 3168, the number he found associated with the boundaries of ancient sacred spaces.

Figure 2 Ten mean earths surround the perpetualchoirs and the Whiteleafed Oak lies at the Centre of another mean earth.

The decagon symbolically replaces the circular norm of the mean earth whilst presenting 10 smaller models of the mean earth between adjacent sides, each a thing-place. This is therefore an advanced geometric modelling of the mean earth containing unfamiliar features. It models the mean earth in two ways, each of different scale. Whilst using the metrological model of the mean earth, it seems to have broken with the ancient metrology in that Michell found a unit of 2.0412 feet (100,000 between choirs sites) and this unit does not belong to ancient metrology, as it is not based solely upon the primes 2, 3, 5, 7 and 11. This metrological anomoly is due to the usage of geometrical properties of the regular decagon, and whilst Michell was aware of those properties it is necessary to understand this decagon as a new type of numeracy arising out of geometrical methods, visible in designs as early as Archaic Greece and possibly even earlier, in Egypt.

*Figure 3 The two forms of isosceles triangle generated by the golden rectangle (sides 1: 1.618), which are formative within geometrical structures based upon 5: the left hand structure can be tiled with ten others to form the decagon, and the right hand triangle can, with it, form a pentagon. (see sri yantra research)*

The decagon has ten identical segments and the radii are the Golden Mean relative to outer side lengths as unity. This enables the mean earth radius to "be" the golden mean (Greek PHI) whilst introducing PHI as a different invariant to the Greek PI found in circular structures, an invariant for which many integer approximations exist including those found in the Fibonacci series (0-1-1-2-3-5-8-13-21-34- etc). Understanding the decagon, Michell used the approximation 160/99 = 1.6162 to model phi and we note that 160 is twice the factor of 80 found modelling the Greek mile above and that 99 contains 11 as a factor, a factor also of 3168 and hence 63.36, the scaling factors found between radius and side length.

Whilst more could be said, a general truth has perhaps emerged:

Any regular decagon, of any scale relative to the mean earth will relate to the mean earth in both radius and side length.

Is this true?

- The side length of 204120 feet divides into the circle's radius as 160 / 99 = 1.6162 .
- The radius divides the mean earth's as 63.36 whilst the sides divide it as 102.4 and 102.4/63.36 = 1.6162 .

This means that the ratio of the different sides are a model of the scale chosen for the decagon relative to the mean earth. It follows that the invariant nature of the decagon is only ideally suited to be a double model of the mean earth at this exact size: since 63.36 gives the 316.8 factor required for the decagon to have a 3168 perimeter.

*Figure 4 The invariant context of the perpetual choir Decagon to the mean earth, seen through the invariant nature of its triangular segments.*

The decagon of perpetual choirs demonstrates the mean earth can be usefully seen as a decagon, whose nature can be transmitted to earth using the decagon's unique invariance, at a scale of 63.36. In this ability, it is the triangular segment which has the visually direct (but analytically confusing) ability to reciprocally reflect the sides of the decagon back to the circle's radius, then modelling the mean earth radius. So what are the characteristics of the triangular segment?

*Figure 5 the idealised side lengths of the decagon's internal segments*

Firstly, 512 / 316.8 equals 160 / 99 and is merely inflated numerically by 16 top and bottom. Secondly, by this means the Decagon is transferring the attention away from the circle and onto the chord of the circle and the sum of the decagon's sides sum to 3168, Michell's idealised spiritual circumference. This was usually modelled in the metrological circumference of sacred circles but was transferred to the decagon's side lengths as ten chord lengths. Importantly, this allowed segments to be alignments between sites.

When Michell identified the side length (204120 ft) as 1/102.4 of the mean earth radius (20,901,888 ft), he was seeing a numerical transmission of the 512 length, where 512/5 = 102.4. The same transformation of the side length is 316.8/5 = 63.36, the mystery factor alluded to earlier.

The true unit for the decagon then emerges, as that required to make the perimeter of the whole decagon 3168 units. There must be 316.8 in each side so 204120/316.8 = 644.3182 feet or 675 units of 21/22 ft - a unit then belonging to metrology as 175/176 of the Roman foot (a root reciprocal Roman foot in John Neal's classification). This new unit makes the radius of the perpetual choirs a canonical 345600 feet of 21/22 feet which, divided by 675, equals 512 such lengths so as to give the side length of 316.8 of those lengths (of 644.318 feet.) The units were after all ideal and metrological but, because the key geometrical transformation was re-entrant and reversible, the familiar relationship of radius, circumference and scaling found in circular models of the mean earth was broken. Instead we find a dynamic symmetry (meaning "analogy") similar to that found in archaic Greek designs. *note: This interpretation has been superceded in the next posting by adopting the proper furlong of the 5000 foot mile, 625 feet long. *

* *

Therefore, when asking when such a landform might have been organised, one is attracted towards the late Bronze Age, when Stonehenge existed and the doctrine of perpetual choirs and en-chantment may well have been developed by proto-Celtic cultures, then an advancing religious concept connected to megalithic geomancy. This form allowed Stonehenge and the solstice sun to have been used as an anchor for other sites, the alignment to the summer solstice sunrise giving the bearing to Gor-ing, then -144 degrees of azimuth relative to Glastonbury, from where a further +36 degrees of azimuth would find Llantwit Major.

John Michell wrote about the perpetual choirs in Twelve-Tribe Nations but then found them to be ten-fold. He packaged up his own work in progress in The Measure of Albion and later wrote a less numerical view of it in the second edition of Dimensions of Paradise (see above figure from that).

### Discussion

The analysis above adds validity to John Michell's proposal, not least because one can see how the decagon can achieve a very elegant modelling of the mean earth upon the earth. We have learnt something from it, which may be "the proof of the pudding" as to whether numerical interpretations are likely to have been intended. We can also now question whether the pattern was actually completed and how it could have been achieved in practice, whilst noting that the sun's solstice angle is increasing away from east-west, over the pattern's two degrees of latitude north of Stonehenge.

The circle around the outside of the decagon is 207360 feet long and to make this 216,000, like the radius of Einar Palsson's Icelandic image, only requires that length to be modelled in root Roman feet of 24/25 = 0.96 feet! This means the arc over each decagon side (or chord) is 216,000 Roman feet whilst the chord is 63/64 less than that which translates (as above) into 316.8 units of 644.318 feet each equalling 675 units of 24/25 x 175/176 - the reciprocal Roman foot.

A further point to reflect upon is that the three choirs referred to in Iola Morganwg's*** *Triads of Britain*, each of 24,000, add up to 72,000 which is half of the 144,000 singers in the Choir referred to in *The Revelation of St John, *a number with special properties for ancient tuning theory - an apparent anachronism unless such harmonic knowledge prefigured the Christian era yet was transmitted into it.

APPENDICES

#### Wikipedia: Golden Mean and Decagon

#### Data

- Mean Earth Radius: 20901888 feet (ancient model)
- Radius of decagon: 329,890.9091 feet (345600 RRR feet, 512 units)
- Side length of decagon: 204,120 feet (316.8 units)
- Proposed foot: 21/22 feet (root reciprocal Roman foot = RRR)
- Proposed unit: 675 such feet (644.318 feet)
- Circumference of out circle 2,073,600 feet
- Perimeter of decagon 2,041,200 feet (3168 units)

#### John Michell on Perpetual Choirs.

- 1972.
*City of Revelation: On the Proportions and Symbolic Numbers of the Cosmic Temple.*Garnstone Press:London 1972. ISBN 978-0-85511-040-6 - 1991.
*Twelve-Tribe Nations and the Science of Enchanting the Landscape*. (reprinted 2008.) 68-70. - 2004.
*The Measure of Albion: The Lost Science of Prehistoric Britain*. 112-115. - 2008.
*The Dimensions of Paradise: Sacred Geometry, Ancient Science, and the Heavenly Order on Earth*. (reprinted 2006 as*Lost Science of Measuring the Earth*) 101-104.