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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 article will explore the statistics of ley lines. Leys are a
controversial topic, and objections to their being intentional
have 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.
What's In a Ley?
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).
Ley Statistics
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).
Personal Investigation
"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.
A First Attempt
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 n2 / k3
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.
The Hough Transform
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:
the points map into Hough curves that
look like the following:
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 instead 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 resolution.
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!
The Results
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".
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.
What Does It Mean?
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, "complexifying"
will not affect rules 1 or 2.
Possible Problems
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
reasonable extent they agree with other computer simulations and
actual mapwork.
Conclusion
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 article 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.
Appendix: More on the Hough Transform
(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 û2 to +.5 û2. 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 é). 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.
References
Clarkson, K.L, H. Edelsbrunner, L.J. Guibas, M. Sharir and 1988 E.
Welyl. "Combinatorial Complexity Bounds for Arrangements of Curves
and Surfaces." 29th Annual IEEE Symposium on the Foundations of
Computer Science: 568-579.
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
Oruc, A. Yavuz. 1995 Interview (combinatorics). Electrical
Engineering Dept, U. of MD. College Park.
Szemeredi, E. and W. Trotter. 1983 "Some Extremal Problems in
Discrete Geometry." Combinatorica 3(3-4):381-392
Watkins, A 1948 The Old Straight Track, 4th Ed. Methuen & Co.,
London.
Williamson, T. and L. Bellamy 1983 Ley Lines in Question. World's
Work, Surrey.
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