Update to Appendix 3: Circular Arrangements of Ley Points

Michael Behrend, 2014

Computer simulations for Appendix 3A

Computer simulations for Major Tyler’s circles give results which agree well with the calculations in the original Appendix 3A. 

Side of square	=	25.000 miles
Alpha unit	=	0.180 mile
Width of circles	=	0.020 mile
Number of points	=	1000

The following are the results of 1,000,000 simulations (s.d.  = standard deviation):

14-pointers:	mean =	0	.000002	s.d.  =	0	.001414
13-pointers:	mean =	0	.000013	s.d.  =	0	.003606
12-pointers:	mean =	0	.000071	s.d.  =	0	.008426
11-pointers:	mean =	0	.000349	s.d.  =	0	.018678
10-pointers:	mean =	0	.001810	s.d.  =	0	.042529
9-pointers:	mean =	0	.007829	s.d.  =	0	.088542
8-pointers:	mean =	0	.032526	s.d.  =	0	.180133
7-pointers:	mean =	0	.120668	s.d.  =	0	.346250
6-pointers:	mean =	0	.402362	s.d.  =	0	.626791
5-pointers:	mean =	1	.186797	s.d.  =	1	.061859
4-pointers:	mean =	3	.062773	s.d.  =	1	.661144
3-pointers:	mean =	6	.796031	s.d.  =	2	.386468
2-pointers:	mean =	12	.720205	s.d.  =	3	.124147
1-pointers:	mean =	19	.778782	s.d.  =	3	.701221

From which the mean number of circles through at least 2 points =  24.331436, agreeing well with the 24.25 resulting from equations (6) and (7); and from which the mean number of circles through at least 8 points = 0.0426, agreeing well with the 0.04 resulting from equation (10). 

Theoretical treatment of Appendix 3B

The following is a more detailed examination of the Wooburn case in Appendix 3B.  Here the radii of the circles are allowed to vary continuously, rather than required to be integer multiples of a unit as in the Stonehenge case. 

Since Bob Forrest’s analysis ignores circles that fall partly outside the map, we suppose that n points have independent uniform distributions inside a disk with centre O.  The problem is to find, for integer r≥2, the expected number of rings of width w centred on O that contain exactly r points.  Here w is given in advance and is assumed to be small compared with the radius of the disk. 

Scale so that the disk has radius 1.  If a point P is uniformly distributed in the disk then the distance x=OP is distributed in [0,1] with p.d.f. 2x.  Suppose that x₁,…,x_n are independently distributed in [0,1] with p.d.f. ƒ(x).  An “r-pointer” is a subset of r points that are covered by an interval with a given small width w.  A maximal r-pointer is one that is not a subset of an (r+1)-pointer. 

For r≥2 let E_r(w) be the expected number of maximal r-pointers. To estimate E_r(w) on the assumption that w is small:

• we assume ƒ(x) is constant over the range x±w.
• we ignore the modifications which, in an exact analysis, are necessary for x<w and x>1−w.

Let p_r(w) be the probability that x₁,…,x_r form a maximal r-pointer with x₁>x₂ and x₃,…,x_r falling between x₁ and x₂.  Then E_r(w)=p_r(w)×n(n−1)×ⁿ⁻²C_r−2. 

To estimate p_r(w), write x₁=x and x₂=x−u, where 0<u<w.  The joint probability for x₁,x₂ is ƒ(x)²dxdu.  The probability that x₃,…,x_r fall in the interval (x₂,x₁) is ƒ(x)^r−2u^r−2.  To ensure a maximal r-pointer, the remaining n−r points must avoid the interval (x−w,x−u+w) of length 2w−u.  Integrating over u, we obtain as the contribution at x₁=x:

∫ w ƒ(x)rur−2{1−(2w−u)ƒ(x)}n−rdudx=ƒ(x)ndx ∫ w ur−2{ƒ(x)−1−2w+u}n−rdu. u=0 u=0

It is straightforward to show, by induction on s and integration by parts, that for non-negative integers s and t and constant λ:

∫ w us(λ+u)tdu= ∑ s (−1)i s! t! ws−i(λ+w)t+i+1 −(−1)s s!t! λs+t+1. 0 i=0 (s−i)! (t+i+1)! (s+t+1)!

Applying this with λ=ƒ(x)⁻¹−2w, and mutliplying by n(n−1)×ⁿ⁻²C_r−2 as noted above, we find the contribution at x to E_r(w) is:

[ ∑ r−2 (−1)i n!wr−2−i ƒ(x)r−i−1{1−wƒ(x)}n−r+i−1−(−1)rnƒ(x){1−2wƒ(x)}n−1 ] dx. i=0 (r−2−i)!(n−r+1−i)!

Restricting now to the particular case ƒ(x)=2x, we estimate E_r(w) as:

∫ 1 [ ∑ r−2 (−1)i n!wr−2−i (2x)r−i−1(1−2wx)n−r+i−1−(−1)r2nx(1−4wx)n−1 ] dx. 0 i=0 (r−2−i)!(n−r+1−i)!

By induction on s and integration by parts, we find that for non-negative integers s and t and constant μ:

∫ 1 xs(1−μx)tdx= s!t!{1−(1−μ)s+t+1} − ∑ s s!t! (1−μ)s+t+1−j 0 μs+1(s+t+1)! j=1 j! μs+1−j(s+t+1−j)!

Applying this with μ=2w and μ=4w and rearranging, we estimate E_r(w) as:

n! ∑ r−2 (−1)i r−1−i [ 1−(1−2w)n+1 − ∑ r−i−1 (2w)j(1−2w)n+1−j ] i=0 2w2 (n+1)! j=1 j!(n+1−j)!

−(−1)r [ 1−1(1−4w)n+1 − (1−4w)n ] . 8(n+1)w2 2w

We reduce the double summation to a single one by collecting terms with the same value of j.  Note that terms with a particular j occur for i=0,1,…,r−1−j.  For 1≤j<r define:

S(r,j)=(r−1)−(r−2)+(r−3)−(r−4)+…+(−1)r−j−1j.

It is easy to verify that

S(r,j)= { (r−j) / 2   if r,j have the same parity (r+j−1) / 2   otherwise.

The working is straightforward and gives E_r(w), assuming that w is small, as:

S(r,1) 1−(1−2w)n+1 − ∑ r−1 S(r,j) n!   2 j−1w j−2 (1−2w)n+1−j 2(n+1)w2 j=1 (n+1−j)! j!

−(−1)r [ 1−(1−4w)n+1 − (1−4w)n ] . 8(n+1)w2 2w

This can be simplified somewhat if we note that the contribution from the term with  j=1 is

−S(r,1) (1−2w)n . w

If we define a function g(n,ξ) for real ξ by

g(n,ξ)= 1−(1−2ξ)n+1 − (1−2ξ)n , 2(n+1)ξ2 ξ

and split off the term with  j=1, the expected number of maximal r-point circles for small w can be written as

Er(w)≈S(r,1)g(n,w)−(−1)rg(n,2w)− ∑ r−1 S(r,j) n!   2 j−1w j−2 (1−2w)n+1−j . j=2 (n+1−j)! j!

Note: Although this formula appears to contain terms in w⁻², in practice powers of w below w ^r−1 cancel out (not yet formally proved, 2014-01-07). 

The above formula was checked by running 1,000,000 computer simulations with the parameters assumed by Bob Forrest in his discussion of Major Tyler’s circles: 400 sites, diameter of disk 25 miles, permitted width 0.02 mile.  Normalizing to the unit disk we have n=400, w=0.02/12.5=0.0016.  The theory above agrees well with the results of simulations:

Points on circle (r)	Calculated expectation	Observed mean	Observed st. dev.
2	90.404935	90.373780	9.247583
3	44.778328	44.732816	6.491550
4	15.377190	15.344071	4.112452
5	4.089117	4.079850	2.266061
6	0.890479	0.887529	1.080800
7	0.164392	0.163295	0.462616
8	0.026348	0.026455	0.184285
9	0.003731	0.003639	0.067836
10	0.000473	0.000473	0.024099
11	0.000054	0.000058	0.008124
12	0.000006	0.000004	0.002000

If the parameters assumed by Bob Forrest are fair, there is nothing remarkable about Tyler’s system of circles centred on Wooburn Church (The Geometrical Arrangement of Ancient Sites, Fig. 3), which contains 20 2-pointers and 5 3-pointers.  This negative result remains if the number of valid ley points is halved.  Running another 1,000,000 simulations, with 200 sites, diameter of disk 25 miles, permitted width 0.02 mile, produced the following result:

Points on circle (r)	Calculated expectation	Observed mean	Observed st. dev.
2	42.714217	42.684971	5.593847
3	10.459765	10.447182	3.166102
4	1.782697	1.780030	1.430447
5	0.234897	0.234369	0.528051
6	0.025281	0.025112	0.171760
7	0.002300	0.002265	0.051146
8	0.000181	0.000168	0.013564
9	0.000013	0.000012	0.003464