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| [[Image:Ley lines.svg|thumb|300px|right|80 4-point near-alignments of 137 random points]]
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| '''Alignments of random points''', as shown by [[statistics]], can be found when a large number of [[randomness|random]] points are marked on a bounded flat surface. This might be used to show that [[ley line]]s exist due to chance alone (as opposed to supernatural or anthropological explanations).
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| One precise definition which expresses the generally accepted meaning of "alignment" as:
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| :''A set of points, chosen from a given set of landmark points, all of which lie within at least one straight path of a given width w''
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| "Straight path of width w" may be defined as the set of all points within a distance of ''w''/2 of a [[straight line]] on a plane, or a [[great circle]] on a sphere, or in general any [[geodesic]] on any other kind of [[manifold]]. Note that, in general, any given set of points that are aligned in this way will contain a large number of infinitesimally different straight paths. Therefore, only the existence of at least one straight path is necessary to determine whether a set of points is an alignment. For this reason, it is easier to count the sets of points, rather than the paths themselves.
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| The width ''w'' is important: it allows the fact that real-world features are not mathematical points, and that their positions need not line up exactly for them to be considered in alignment. [[Alfred Watkins]], in his classic work on ley lines ''[[The Old Straight Track]]'', used width of a pencil line on a map as the threshold for the tolerance of what might be regarded as an alignment. For example, using a 1 mm pencil line to draw alignments on an 1:50,000 [[Ordnance Survey]] map, a suitable value of ''w'' would be 50 m.<ref>{{cite isbn|0349137072}}</ref>
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| == An estimate of the probability of alignments existing by chance ==
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| [[Contrary to intuition]], finding alignments between randomly placed points on a landscape gets progressively easier as the geographic area to be considered increases. One way of understanding this phenomenon is to see that the increase in the number of possible [[combination]]s of sets of points in that area overwhelms the decrease in the probability that any given set of points in that area line up.
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| The number of alignments found is very sensitive to the allowed width ''w'', increasing approximately proportionately to ''w''<sup>''k''-2</sup>, where ''k'' is the number of points in an alignment.
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| The following is a very approximate order-of-magnitude estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points.
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| Consider a set of ''n'' points in a compact area with approximate diameter ''d'' and area approximately ''d''². 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'', where ''w'' << ''d'').
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| Consider all the unordered sets of ''k'' points from the ''n'' points, of which there are:
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| :<math> \frac {n!} {(n-k)!k!}. </math>
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| What is the probability that any given set of points is [[collinearity|collinear]] in this way? Let us very roughly consider the line between the "leftmost" and "rightmost" two points of the ''k'' selected points (for some arbitrary left/right axis: we can choose top and bottom for the exceptional vertical case). These two points are by definition on this line. For each of the remaining ''k''-2 points, the probability that the point is "near enough" to the line is roughly ''w''/''d'', which can be seen by considering the ratio of the area of the line tolerance zone (roughly ''wd'') and the overall area (roughly ''d''²).
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| So, the expected number of k-point alignments, by this definition, is very roughly:
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| :<math> \frac {n!} {(n-k)!k!} \left({\frac{w}{d}}\right)^{k-2}.</math>
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| For ''n'' >> ''k'' this is approximately:
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| :<math> \frac {n^k} {k!} \left({\frac{w}{d}}\right)^{k-2}.</math>
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| Now assume that area is equal to <math>d^2</math>, and say there is a density α of points such that <math>n = \alpha d^2</math>.
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| Then we have the expected number of lines equal to:
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| :<math> \frac {\alpha^k d^{2k}} {k!} \left( {\frac{w}{d}} \right)^{k-2}</math>
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| and an area density of k-point lines of:
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| :<math> \frac 1 {d^2} \frac {\alpha^k d^{2k}} {k!} \left( {\frac{w}{d}} \right)^{k-2}.</math> | |
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| Gathering the terms in ''k'' we have an areal density of k-point lines of:
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| :<math> d^k \frac {\alpha^k} {k!} w^{k-2}.</math>
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| Thus, contrary to intuition, the number of k-point lines expected from random chance increases much more than linearly with the size of the area considered.
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| == A more precise estimate of the expected number of alignments ==
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| A more precise expression for the number of 3-point alignments of maximum width ''w'' and maximum length ''d'' expected by chance among ''n'' points placed randomly on a square of side ''L'' is <ref>Edmunds, M.G. & George, G.H., ''Random Alignment of Quasars'', Nature, vol. 290, pages 481-483, 1981 April 9</ref>
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| :<math> \mu = \frac {\pi } {3} \frac {w}{L}
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| \left( {\frac {d}{L}} \right)^{3} n \left( n-1 \right)
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| \left( n-2 \right) </math>
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| If edge effects (alignments lost over the boundaries of the square) are included, then the expression becomes
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| :<math> \mu = \frac {\pi } {3} \frac {w}{L}
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| \left( {\frac {d}{L}} \right)^{3} n \left( n-1 \right)
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| \left( n-2 \right)
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| \left( 1 - \frac {3}{\pi } \left( \frac {d}{L} \right)
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| + \frac {3}{5} \left( \frac {4}{\pi } - 1 \right)
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| \left( \frac {d}{L} \right)^{2} \right) </math>
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| A generalisation to ''k''-point alignments (ignoring edge effects) is<ref>G.H. George, [http://www.engr.mun.ca/~ggeorge/astron/thesis.html] "Alignment of Quasars", ''Summary of Ph.D. Thesis'', 1983</ref>
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| :<math> \mu = \frac {\pi n \left( n-1 \right) \left( n-2 \right)
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| \cdots \left( n - \left( k-1 \right) \right) }
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| {k \left( k-2 \right) !} \left( \frac {w}{L} \right)^{k-2}
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| \left( {\frac {d}{L}} \right)^{k} </math>
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| == Computer simulation of alignments ==
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| [[image:leylines.png|Image of ley line simulation|thumb|300px|right|''607 4-point alignments of 269 random points'']]
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| [[Computer simulation]]s show that points on a plane tend to form alignments similar to those found by ley hunters in numbers consistent with the order-of-magnitude estimates above, suggesting that ley lines may also be generated by chance. This phenomenon occurs regardless of whether the points are generated pseudo-randomly by computer, or from data sets of mundane features such as pizza restaurants.
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| It is easy to find alignments of 4 to 8 points in reasonably small data sets with ''w'' = 50 m.
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| Choosing large areas or larger values of ''w'' makes it easy to find alignments of 20 or more points.
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| == References ==
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| {{Reflist}}
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| == See also ==
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| * [[Ley line]]s
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| * ''[[The Old Straight Track]]''
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| * [[Coincidence]]
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| * [[Procrustes analysis]]
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| [[Category:Euclidean geometry]]
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| [[Category:Statistical randomness]]
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