Alignments of random points
Statistics shows that if you put lots of random points on a bounded flat surface you can find many alignments of random points. Some people think that this shows that such things as ley lines exist naturally as coincidences, and are therefore not interesting phenomena. Other people see this as a failure to understand scientific method which is about setting particular criteria for particular comparisons which can then reveal levels of probability.
One precise definition which expresses the generally accepted meaning of "alignment" as:
One simple definition of "straight path of width w" is 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 an uncountable number of infinitesimally different straight paths will contain any given set of points that are aligned in this way, so only the existence of at least one straight path is important to consider 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 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.
For example, using a 1mm pencil line to draw alignments on an 1:50,000 Ordnance Survey map, a suitable value of w would be 50m.
An estimate of the probability of alignments existing by chance
Statistically, finding alignments on a landscape gets progressively easier as the area to be considered increases. One way of understanding this phenomenon is to see that the increase in the number of possible combinations of points in that area overwhelms the decrease in the probability that any given set of points in that area line up.
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.
For those interested in the mathematics, the following is a very approximate estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points.
Consider a set of 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 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.
So, the expected number of k-point ley lines is very roughly
For n >> k this is approximately
Now assume that area is equal to d2, and say there is a density α of points such that n = αd2.
Then we have the expected number of lines equal to:
and an area density of k-point lines of:
Gathering the terms in k we have an areal density of k-point lines of:
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.
Computer simulation of alignments
Computer simulations 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 non-magical features such as pizza restaurants.
It is easy to find alignments of 4 to 8 points in reasonably small data sets with w = 50m. Choosing large areas or larger values of w makes it easy to find alignments of 20 or more points.