Alignments of random points in a plane can be demonstrated by

statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

to be counter-intuitively easy to find when a large number of

random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...

points are marked on a bounded flat surface. This has been put forward as a demonstration that

ley line Ley lines () are straight alignments drawn between various historic structures and prominent landmarks. The idea was developed in early 20th-century Europe, with ley line believers arguing that these alignments were recognised by ancient soci ...

s 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 Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...

and

astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

. 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 a 1:50,000 scale

Ordnance Survey Ordnance Survey (OS) is the national mapping agency for Great Britain. The agency's name indicates its original military purpose (see ordnance and surveying), which was to map Scotland in the wake of the Jacobite rising of 1745. There was a ...

map, the corresponding width on the ground would be 50 m.

Estimate of probability of chance alignments

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: :''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'' 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 In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...

on a plane, or a

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...

on a sphere, or in general any

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...

on any other kind of

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

. 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 ''w''^''k''-2, where ''k'' is the number of points in an alignment. The following is a very approximate order-of-magnitude estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points. Consider a set of ''n'' points in a compact area with approximate diameter ''L'' and area approximately ''L''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'' ≪ ''L''). Consider all the unordered sets of ''k'' points from the ''n'' points, of which there are: :

\binom nk = \frac

(see

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...

and

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

for notation). To make a rough estimate of the probability that any given subset of ''k'' points is approximately

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...

in the way defined above, consider the line between the "leftmost" and "rightmost" two points in that set (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''/''L'', which can be seen by considering the ratio of the area of the line tolerance zone (roughly ''wL'') and the overall area (roughly ''L''²). So, the expected number of k-point alignments, by this definition, is very roughly: :

\frac   \left(\right)^

Among other things this can be used to show that, contrary to intuition, the number of k-point lines expected from random chance in a plane covered with points at a given density, for a given line width, increases much more than linearly with the size of the area considered, since the combinatorial explosion of growth in the number of possible combinations of points more than makes up for the increase in difficulty of any given combination lining up.

More precise estimate of expected number of alignments

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 Edmunds, M.G. & George, G.H., ''Random Alignment of Quasars'', Nature, vol. 290, pages 481-483, 1981 April 9 :

\mu = \frac   \frac  
 \left(  \right)^ n \left( n-1 \right) 
 \left( n-2 \right)

If edge effects (alignments lost over the boundaries of the square) are included, then the expression becomes :

\mu = \frac   \frac  
 \left(  \right)^ n \left( n-1 \right) 
 \left( n-2 \right) 
 \left( 1 - \frac  \left( \frac  \right) 
 + \frac  \left( \frac  - 1 \right) 
 \left( \frac  \right)^ \right)

A generalisation to ''k''-point alignments (ignoring edge effects) is :

\mu = \frac  
  \left( \frac  \right)^ 
 \left(  \right)^

which has roughly similar asymptotic scaling properties as the crude approximation in the previous section, with combinatorial explosion for large ''n'' overwhelming the effects of other variables.

Computer simulation of alignments

300px, 607 4-point alignments of 269 random points

Computer simulation Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be deter ...

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 Pizza (, ) is a dish of Italian origin consisting of a usually round, flat base of leavened wheat-based dough topped with tomatoes, cheese, and often various other ingredients (such as various types of sausage, anchovies, mushrooms, on ...

restaurant A restaurant is a business that prepares and serves food and drinks to customers. Meals are generally served and eaten on the premises, but many restaurants also offer take-out and food delivery services. Restaurants vary greatly in appearan ...

s or

telephone booth A telephone booth, telephone kiosk, telephone call box, telephone box or public call box is a tiny structure furnished with a payphone and designed for a telephone user's convenience; usually the user steps into the booth and closes the booth ...

s. It is easy to find alignments of 4 to 8 points in reasonably small data sets with ''w'' = 50 m. Choosing large areas or larger values of ''w'' makes it easy to find alignments of 20 or more points.

References

{{Portal bar, Mathematics, Physics, Astronomy, Stars, Spaceflight, Outer space, Science Euclidean geometry Statistical randomness Combinatorics Computer vision Astronomy Spatial analysis

Estimate of probability of chance alignments

More precise estimate of expected number of alignments

Computer simulation of alignments

See also

References