Alignments of random points
Alignments of random points in the plane can be demonstrated by statistics to be remarkably and counter-intuitively easy to find when a large number of random points are marked on a bounded flat surface. This has been put forward as a demonstration that ley lines and other similar mysterious alignments believed by some to be phenomena of deep significance might exist solely due to chance alone, as opposed to the supernatural or anthropological explanations put forward by their proponents. The topic has also been studied in the fields of computer vision and astronomy.
A number of studies have examined the mathematics of alignment of random points on the plane. In all of these, the width of the line - the allowed displacement of the positions of the points from a perfect straight line - 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 the 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, the corresponding width on the ground would be 50 m.
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 combinations of sets of points in that area overwhelms the decrease in the probability that any given set of points in that area line up.
One definition which expresses the generally accepted meaning of "alignment" is:
More precisely, a 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. The number of alignments found is very sensitive to the allowed width w, increasing approximately proportionately to wk-2, where k is the number of points in an alignment.
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