Jason Papadopoulos
"So, recently folks have been connecting up all these ancient sites with straight lines. They can get four, five, even six of them, all in a row." "Sure, but it's all by chance, isn't it?"
Is it?
Every time I brought up the ley phenomenon to the uninitiated, I was immediately cut off by this question. The English countryside is full of very ancient sites and earthworks, and many of these fall on alignments over ten miles long, with eerie accuracy. Examples can be found of these "ley lines" that seem to defy the law of averages, and one would be hard-pressed to believe that such cases could possibly arise by chance.
This paper will explore the statistics of ley lines. Leys are a controversial topic, and objections to their being intentional have been arisen from many directions. I will examine what physically constitutes a ley line and then review previous work on how likely a ley line is to arise by chance. A sizeable portion of this paper will then be devoted to my own computer experiments. Note in passing that to focus on ley statistics requires being sketchy in other areas.
The simplest answer is "prehistoric standing stones and earthworks", but already there is controversy. Prehistory spans a very long period, and when two random "ancient" structures are chosen it is possible that these features were constructed in periods separated by almost 4000 years (WB 1983:31). Thus those two structures could have been constructed by very different societies, and have had nothing to do with each other.
Many ley hunters when including (non-prehistoric) churches in leys, cite a letter from Pope Gregory in 601 AD stating that pagan "temples" ought not to be destroyed, but purified and converted to churches (Watkins 194:117). Indeed, there are standing stones in country churchyards, as Watkins shows. Including all churches because of this, however, is unrealistic. Devereux and Thomson list a ley through London that consists of five medieval churches, despite the fact that
"...the city site, while not completely deserted...was of no special importance until the Romans founded their settlement" (WB 1983:138)
Likewise with castles: a mound is more defensible than flat ground, and many prehistoric mounds exist; castle keeps, then, belong on ley lines. It is again unrealistic, however, that all medieval nobility would have their choice of castle site dictated by existing terrain, when they could simply have the terrain modified to suit their wishes exactly.
Watkins hypothesized that ley lines were the sighting points for a vast network of "straight tracks" that covered prehistoric England, and his book includes several crossroads used as ley points and instances of dirt pathways uncovered in the course of sewer excavation (Watkins 1948:38-39).
The impression received is that deciding whether a given site is a viable candidate for being a "ley point" is a difficult matter and would often require archaeological evidence. Ley hunting is typically an easy matter, however. Most ley hunters would only connect the ley points on an Ordnance Service (OS) map and then confirm the ley points in the field. Most do not perform more orthodox research which would tell, for example, that the straight paths through England are mostly "Planned Countryside" enacted by Parliament in the 18th and 19th Centuries, while older tracks than these are "notoriously devoid of straight lines" (WB 1983:88). Notice also that many of the citations in this paper are from Williamson and Bellamy, both archaeologists; this is because they include historical evidence where others do not.
In fairness to the existing material, "questionable" ley sites (small mark stones, trees, stretches of modern road) are usually ignored in a published ley.
How wide must an "old straight track" be? Watkins insisted that ancient tracks be just wide enough to travel on foot, perhaps two to four yards (DT 1979:72). Using a very sharp pencil on an OS map produces an effective line about 30 feet wide; this would be about the best one could expect without doing fieldwork. Statistical studies often could not work with widths less than a hundred yards (see Appendix).
Watkins was the first (1925) to attempt answering the question of whether ley lines of significant size could arise by chance (Watkins 1948:203-204). The OS sheet of Andover contains 51 churches that can be organized into 1 five-point, 8 four-point and 29 three-point leys. To see how many leys could be expected by chance he marked out 51 crosses "haphazardly" on a similar size sheet, and found no five-point, 1 four-point and 33 three-point leys. He concluded from this that with 50 sites, finding a four- point ley by chance was unlikely, and a five-point ley was ironclad evidence that the placement was deliberate.
From this he developed a rating system (DT 1979:31) that assigned points to possible ley features: "ancient sites" got a full point, and incidental features like stretches of road, "mark stones", or "ancient trees" fractions of a point. If the total summed to 5 or more the ley was deemed to be deliberate.
Peter Furness in 1965 derived a closed-form expression (details unavailable) for the probability of a given size ley existing (DT 1979:38), and from this declared that a seven-point ley would only arise in 1 out of 1000 OS maps. Further (WB 1983:94), assuming a given map had 200 ley points, he calculated that one could expect 1570 three-point, 72 four-point and 2 five-point alignments to occur by chance. Confirmation of a sort came from Robert Forrest (WB 1983:95), whose computer study is the only one of its kind available. His 200 random point run found that 752 three-point, 33 four-point and 2 five-point ley lines existed by chance alone, and suggested that Watkins' criterion of a five-point ley being almost impossible was unrealistic for large collections of points.
Both these studies required many assumptions (WB 1983:96-98): that there were only 200 points in the average map (the average is 300 to 400), that they were all small (some earthworks can reach 10 acres in area), all evenly distributed, etc. Accounting for these factors theoretically would have been next-to-impossible, so Forrest instead ran a simulation. This involved looking at a sample map, randomizing the points in it but keeping their distribution the same, and plotting all the ley lines by hand. This time many more lines were found: 39 five-point, 10 six-point and 1 seven-point alignments.
There is also a famous study by John Michell, but I omit it due to doubts about its assumptions. The interested reader should consult (WB 1983:102-106).
"The past evidence for leys is statistically poor. It is to be hoped that future evidence will be of a much more rigorous nature."-Robert Forrest (DT 1979:39) Computer work on ley line statistics seems to have stopped, and I wondered if more could be learned with modern computers and recent mathematical results. I therefore have tried to analyze the available evidence based on my own numerical experiments. Though I have attempted to make assumptions as realistic as possible, getting answers requires ignoring a lot of information, like the length of a given ley or the topography of the sample region.
Combinatorics is the mathematical branch of counting and estimation, and there are several combinatorial results that appeared useful. One is that (ST 1983:389, Clarkson et. al. 1988:570)):
Given n points in the plane, the number of lines formed that connect k or more points cannot exceed 8*( n^2 / k^3 )
Unfortunately, bounds like this aren't close enough to the actual numbers of ley lines to tell anything. The above, for example, predicts that a group of 200 points can have at most about 240 lines that connect 5 of those points; this is far in excess of anything found in the field. These results are worst-case; similar results for a random distribution do not exist.
The alternative, then, is to perform experiments with random groups of points. Care is necessary, because counting lines from a large random distribution can involve A LOT of work. Until now the method used was the naive one: connect every plotted point to every other plotted point with a line, and look for equations of lines that were almost the same. The computation and storage required grows quadratically with the number of points used; thus repeating experiments for large problems may take too long even for a very fast computer.
A better method is needed, one that can pick out lines from very large distributions of random points quickly, and that can also be used for non-random distributions and "blobs" instead of points.
Choose two random topics that have as little as possible to do with each other. One can hardly do better than "prehistoric Britain" and "image processing"! Nevertheless, lines often need to be found in pictures, and the Hough transform is exactly what ley computation requires.
Ordinarily the Hough method is used to pick out lines and shapes in the presence of noise; ley simulation requires looking at the noise itself, without any underlying pattern. This makes the transform very simple, especially since analyzing ley lines doesn't require processing colors like analyzing images does.
The details of the Hough transform are in the Appendix. For now, it is sufficient (Duda and Hart 1972:12-13) to know that given a group of points in the plane,
1. The Hough transform maps a single point in the plane onto a sinusoidal curve in a "Hough array".
2. When several points in the plane are on a line, their curves pass through a single point in the Hough array.
3. Thus, lines in the plane become "spikes" in the Hough array. The more points a line connects, the taller the spike for that line. Likewise, the larger the Hough array, the more lines it can detect (higher resolution).
Here's a (simple) example of the Hough transform in action. Given a bunch of spikes in the plane that look like this:
- - - - - 1 - x=3 y=3
- - - - 1 - -
- - - 1 - - -
- - - - - - - x=3 y=0
- 1 - - - - -
1 - - - - - -
- - - - - - -
the points map into Hough curves that look like the following:
0ø 90ø 180ø - - - - - - - - - - - - - - - - 1 1 1 - - - - - - - - - 1 - - - 1 - - - - - - - 1 - - - - - - - - - - 1 - - - - 1 - - 1 - - - 1 - r=3 1 - 1 1 - 1 1 - - - - - 1 - 1 - - - - - 1 1 - 1 1 - r=2 - - - - - - - - 1 - 1 - - 1 - - 1 1 1 1 1 1 5 - - - r=1 - 1 1 - - - - - - - 1 1 - 1 - - - - - - - 1 - 1 - 1 r=0 - - - - - - - 1 1 - 1 1 - - - - - - - - - - - - - 1 r=-1 - - - - - - 1 1 - - - 1 - - - - - - 1 - - - - - - - r=-2 1 1 - 1 1 - 1 - - - - - 1 - - 1 - - - - - - - - - - r=-3 - - - - - 1 - - - - - - - 1 - - - 1 - - - - - - - - - 1 1 1 - - - - - - - - - - - - - - - - - - - - - -
Note the prominent "spike" at (135ø,1). The Hough array actually conveys quite a bit of useful information: there's a line connecting five points in the picture, and the closest that line gets to the origin is 1 unit, angled 135ø from the horizontal (actually it's .707 units and but the resolution is too coarse).
The Hough transform has many advantages over brute-force methods. First, it's fast; the computation only grows linearly with the number of points. Secondly, each point in the plane can be treated separately; only one point at a time is needed, reducing the storage requirements drastically. Even better, different distributions of points can be "layered" over each other (i.e. start with a scattering of "small" points, then add a scattering of "large" points, then add more small points to one corner of the plane). Finally, since the Hough array "digitizes" the plane, each point in the plane has an implicit size...just like its counterpart on an OS map (see appendix).
The only real disadvantage of the Hough transform for the ley case is the size of the Hough array. Transforming a photograph is easy because if lines exist in a digitized picture they contain many points; the Hough array can be small because the "spikes" will be large enough to show up even at low resolution. In the case of ley lines, however, it is required to see lines containing only 4 or 5 points intead of 40 or 50. The Hough array must be large enough to plot hundreds of curves without many curves overlapping, to keep true results from being hidden by poor resoluton.
Lastly, any numerical results should correspond to actual work performed on real maps. Watkins found 33 three-point and 1 four-point alignment out of 51 random points on paper, so under similar conditions the Hough method should as well. This is to avoid my confidently looking up from the computer and declaring the English countryside to be wrong!
For ease of comparison, the computed results are all grouped together. Each set of numbers is an average over three to four trials (an advantage of using the computer), and rounded to the nearest integer, or to one decimal place if the number of leys found was small. All except the 400-point case used the same size Hough array (the array used had to be enormous, and even on a workstation each run took three or four minutes; larger collections of points would have needed an even bigger array).
"Vanilla" results used only small points, randomly and evenly distributed over the plane.
"Concentrated" results used only small points, but half of the points were scattered within one-fourth of the plane.
"Oversize" results were distributed evenly over the plane, but for half of the points, two parallel Hough curves were plotted instead of one. This simulates, to some degree, these points being "larger" in area.
Finally, "Realistic" results combine the previous two categories: half the points are distributed in one-fourth of the plane, and within each region of the plane (the three-fourths or one-fourth region) half the points are "large".
50 Random Points
Alignments:
3-point 4-point
Vanilla 35 .5
Concentrated 38 .75
Oversize 119 1.7
Realistic 143 3
100 Random Points
Alignments:
3-point 4-point 5-point
Vanilla 246 8.5 .5
Concentrated 315 12 0
Oversize 854 47 .7
Realistic 1007 58 4
200 Random Points
Alignments:
3-point 4-point 5-point 6-point 7-point
Vanilla 1915 130 7 .3 0
Concentrated 2154 196 15 2.3 1
Oversize 5641 625 56 6 .5
Realistic 6125 788 88 8.3 .7
300 Random Points
Alignments:
3-point 4-point 5-point 6-point 7-point
Vanilla 5663 628 55 4.5 .5
400 Random Points
Alignments:
3-point 4-point 5-point 6-point 7-point 8-point
Vanilla 12949 1899 225 20 2.3 .3
Note that the vanilla 50-point case mirrors the observations of Watkins. Until larger groups of points are tested, this would seem evidence enough that a four-point ley is statistically unusual and a five-point ley impossible to achieve by chance. In larger groups of points, however, it is seen that both become commonplace. "Statistically unusual" leys become harder to find as the number of points grows; if these results are accurate, in a field of 300 points one would need to find an eight-point ley to be sure something is going on, and the burden may be worse in a non-vanilla case.
One interesting observation about the data is that although adding complexity to the group of random points almost always causes many more lines to be found, it does not add leys with more points. Thus, even in a pseudo-realistic case with 50 random points, the Hough transform predicts no 5-point leys will be found, just as in the vanilla case. This is important, because it shows that including Williamson's and Bellamy's objections to computer simulation will not change the essential size of lines to expect by chance, although the number of lines may change greatly.
Another pattern involves the relative numbers of lines. In many cases, the number of ley lines with "n" points is approximately 1/10 (between 1/5 and 1/15) the number of ley lines with "n-1" points, at least to order of magnitude. In addition, adding complexity to the collection of points does not change the number of leys found, at least to order of magnitude. Thus, if only one or two leys of a certain size were found by chance in a uniform distribution, making the point distribution more realistic will not increase the number of leys found beyond ten.
It has been shown that the computer results agree with those of Watkins from a map. What about the "high end"? Williamson and Bellamy also computed a full-scale simulation from a map, one with many more points than Watkins used (WB 1983:102). Their test accounted for nonuniform point distributions and different point sizes; the agreement between their results and mine depends on how many ley points were on the map of Wiltshire used.
Williamson and Bellamy found 127 six-point, 48 seven-point, 12 eight-point and 5 nine-point leys when they moved the regular map features around randomly. If the Wiltshire map contained about 500 points the computer results would agree rather well; they would predict a small number (perhaps 0-3) of nine-point leys and a larger number (perhaps 5 to 15) of eight-point leys, in keeping with previous trends. The agreement, of course, degrades if the Wiltshire map had fewer points. More realistic cases could not be tested for large point distributions; the computation involved is prohibitive, even for the Hough transform.
More disturbing is the fact that the number of Wiltshire leys found only increases by a factor of two to four as the leys get smaller, instead of a factor of about ten as the computer results predict. The importance of this is unclear, since Forrest's work on the South Wales map (1 seven-point, 10 six-point, 39 five-point leys) does conform somewhat to the "10 to 1" rule.
Compared with Forrest's computer study using 200 points in a vanilla distribution, the Hough transform in "vanilla mode" found almost three times as many 3-, 4- and 5-point ley lines. Whether this is realistic (since the Hough array gives each point an implicit size and so increases the likelihood of alignment), or an oversight on my part is unclear. Checking the "oversize" results for 200 points shows that making some ley points bigger can increase the number of leys found dramatically (but by less than a factor of 10), so it's quite possible that the Hough results are more realistic than Forrest's. The comparison with Furness' theoretical results is closer: the Hough method found about 1 1/2 times more leys.
Based on the computer's results, it seems that
1. A six-point ley is statistically unlikely if there are 100 points in a given region. For every 100 points more that a region contains, a ley one point bigger is needed to be statistically unlikely.
2. However many large leys are found in a uniform distribution, the number of smaller leys is greater by a factor of about ten for every point shorter. Thus there are approximately ten times as many leys that are one point shorter, 100 times as many that are two points shorter, etc.
3. Including effects such as uneven distribution and variable size of ley points will increase the expected number of ley lines of a given size, but by less than a factor of 10; further, "complex- ifying" will not affect rules 1 or 2.
Like any statistical study, there are doubts about how true the results really are. For instance, I have no way of being sure exactly how large a ley point used in the Hough transform really is, or if "large" ley points were really much larger (guesses for both are in the Appendix). Likewise, the implicit width of a connecting ley line is almost completely unknown. The Hough array used is large enough to avoid overlapping Hough curves for small collections of points, but for large collections of points with fancy things happening there may have been noticeable overlap, and hence "false lines" detected.
Still, there are patterns to be found in the results, and to a resonable extent they agree with other computer simulations and actual mapwork.
I would like to believe that ley lines are by design, that they are not 20th century man playing "connect the dots" with a random landscape. However, based on the results found, the case for ley lines seems hopeless to me, and the statistical aspect is only one facet of a whole range of problems with ley theory.
Notice that with enough points in a given area, large ley lines can appear completely at random. Half of the Ley Hunter's Companion is devoted to listing out 41 exceptional ley lines throughout England, and most of these have seven or eight points. If dozens of OS maps are needed to cover England and many maps average 300 or 400 viable ley points, then on these grounds alone, in light of the mapwork and computer results, those 41 leys are as likely products of coincidence as of neolithic man. Worse, that statement is only the numbers talking; nothing has yet been said of the ley sites themselves being unsuitable (for instance, a castle that excavation shows was never built on a prehistoric mound, and hence has no reason to be on a prehistoric line).
This paper has necessarily been overly mathematical and lacking in archaeology, but I feel the topic- and understanding the results- demands it. I know very little statistics, and much of the Hough transform information in the appendix is based on crude approximations and trial-and-error. Despite this, there is information contained herein that I believe to be new, and that makes some progress towards answering the question posed at the very beginning with some confidence.
(Specifics on the Hough method paraphrase Duda and Hart.)
Given any line in the plane, a perpendicular can be drawn which intersects the origin. The length "r" of that perpendicular and the angle "A" it makes with the horizontal then uniquely specify that line, if A is between 0ø and 180ø.
The coordinates (x,y) for a point on a line with fixed (r,A) then satisfy (r can be negative):
x cos(A) + y sin(A) = r
Paul Hough realized that if a group of points (x(i),y(i)) are on a line, then plotting
x(i) cos(A) + y(i) sin(A)(1)
for A between 0ø and 180ø would yield, eventually, a value of r that is the same for all the points. The Hough transform divides the (r,A) plane into discrete values of r and A; for a given point (x(i),y(i)) equation (1) is evaluated for the whole range of values, and 1 is added to every (r,A) cell that the resulting curve passes through. The "count" in some cells will increase as the curves for more points are added; when all the points have been dealt with in this way, the "count" in a given (r,A) cell is then the number of points that lie on a line described by (r,A).
The Hough array I used was 1000 by 1000, and the x and y values of random points were uniformly distributed between -.5 and +.5 units; thus r varied from -.5 * SQRT2 to +.5 * SQRT2. Since the Hough array is "digitized", those random points have finite size (a small range of x and y values will fall in the same (r,A) cell). A very rough estimate of how large a random point will be can be found from the area of an (r,A) cell:
max. area = (max. r)*(size of cell) = .00000222 square units.
On a 25 mile by 25 mile OS map this translates to about .84 acres maximum; "large" points used in computer runs had an area of twice this much, very roughly (since two values of r were plotted for every A). This area is enough for standing stones, mounds, and small earthworks but not enough for hillforts and such. Also, this is a maximum; there were points with zero size in all the computer runs.
A related question is how wide a line the computer "drew" to connect sites. Frankly, I'm at a loss to find out; my first idea was simply the width of a single (r,A) cell, but on a real map this comes out to a whopping 120 feet. However, a 1 mm wide line is over 50 meters wide in OS scale (DT 1979:39), so this may not be as disastrous as it seems.
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Devereux, P. and I. Thomson. 1979 "The Ley Hunter's Companion". Thames and Hudson, London.
Duda, R. and P. Hart. 1972 "Use of the Hough Transformation to Detect Lines and Curves in Pictures." Communications of the ACM 15(1):12-16
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