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Here's a handwaving order-of-magnitude argument for the probability of ley lines.
Consider a set on n points in an area with approximate diameter d. Consider a valid line to be one where every point is within distance w/2 of the line (that is, lies on a track of width w.
Consider all the unordered sets of k points from the n points, of which there are
What is the probability that any given set of points is co-linear in this way? Let's very roughly consider the line between the "leftmost" and "rightmost" two points of the k selected points (for some arbitary left/right axis: we can choose top and bottom for the exceptional vertical case). For each of the remaining points, the probability that the point is "near enough" to the line is roughly w/d. Write ε = w/d
So, the expected number of k-point ley lines is very roughly
So, set w = 50m, d = 30 km, (pencil line on an OS map) thus ε = 0.001666...
Then, we can tabulate the expected number of k-point lines found in a single map as follows:
w = 50m, d = 30 km
k n expected points
3 from 10: 0.20 4 from 10: 0.00 5 from 10: 0.00 6 from 10: 0.00 7 from 10: 0.00
3 from 20: 1.90
4 from 20: 0.01 5 from 20: 0.00 6 from 20: 0.00 7 from 20: 0.00
3 from 50: 32.67 4 from 50: 0.64 5 from 50: 0.01 6 from 50: 0.00 7 from 50: 0.00
3 from 60: 57.03 4 from 60: 1.35 5 from 60: 0.03 6 from 60: 0.00 7 from 60: 0.00
3 from 70: 91.23 4 from 70: 2.55 5 from 70: 0.06 6 from 70: 0.00 7 from 70: 0.00
3 from 100: 269.50 4 from 100: 10.89 5 from 100: 0.35 6 from 100: 0.01 7 from 100: 0.00
3 from 200: 2189.00 4 from 200: 179.68 5 from 200: 11.74 6 from 200: 0.64 7 from 200: 0.03
Notice the rapid increase of the number of expected lines with n.
Ley line proponents generally view five point ley lines as definitive proof that a line is real. Note that as there are roughly 400 OS maps covering the UK, if there were 50 points in each, there would be an expected 256 4-point leys and 4 five-point leys found nationally.
Increasing the value of w dramatically increases the number of expected lines on a single map, with effects that increase with the value of k.
Increasing the size of a map will increase d, thus reducing ε, but will increase n: for example, if a 30 km map has 50 points, a 60 km map with the equivalent density of significant points will have 200, although it will have a value of ε that is half of that for the 30 km map.
w = 50m, d = 60 km
3 from 100: 134.75 4 from 100: 2.72 5 from 100: 0.04 6 from 100: 0.00 7 from 100: 0.00
3 from 150: 459.42 4 from 150: 14.07 5 from 150: 0.34 6 from 150: 0.01 7 from 150: 0.00
3 from 200: 1094.50 4 from 200: 44.92 5 from 200: 1.47 6 from 200: 0.04 7 from 200: 0.00
3 from 250: 2144.17 4 from 250: 110.34 5 from 250: 4.52 6 from 250: 0.15 7 from 250: 0.00
3 from 300: 3712.58 4 from 300: 229.72 5 from 300: 11.33 6 from 300: 0.46 7 from 300: 0.02
Note how the probability of finding a 5-point ley on the 60km map with n=200 is much more than four times that of finding a 5-point ley on the 30km map with n=50. With 100 such maps nationally, the expected national number of 4-point leys would be 4492, with 147 5-point leys.
Combining all these 100 maps into a single national map 600 km in "diameter" containing 20,000 points gives the following expected number of leys:
3 from 20000: 111094445.00 4 from 20000: 46282408.68 5 from 20000: 15424384.07 6 from 20000: 4283479.99 7 from 20000: 1019570.23 8 from 20000: 212336.12 9 from 20000: 39305.78 10 from 20000: 6548.01 11 from 20000: 991.63 12 from 20000: 137.65