Journal of Geomancy vol. 4 no. 2, January 1980

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LANDSCAPE GEOMETRY NOTES

by MICHAEL BEHREND

Isosceles triangles among Llanfihangels

In a recent IGR paper (1) John M. Bullock draws attention to the large number of isosceles triangles, and other figures, formed by places with the same basic name: such as Llanfihangel, Hook, Hopton, etc.  He leaves open the question whether these figures are the result of deliberate design or mere chance.  Using a computer I have investigated the part of Mr Bullock’s study to which he has given the most attention, namely the occurrence of isosceles triangles among Llanfihangel place names.  The object was to discover whether there are significantly more isosceles triangles than we should expect from the workings of chance alone. 

From gazetteers, plus Crockford’s clerical directory, I extracted 48 places with the name Llanfihangel, or an English equivalent such as Michaelchurch.  (Incidentally, the English names can be tricky.  Mitchell Troy is a ‘Michael’ name – Llanfihangel Troddi in Welsh – whereas Mitcheldean is said to come from ‘mychel’ meaning ‘great’.) To this list was added the stone circle known as Mitchell’s Fold, used in the original study.  Following Bullock, I then measured the grid coordinates of the village church, dedicated to St Michael, on the 1:50 000 map.  There are disadvantages in using this small scale, since apart from the limited accuracy one is not always certain of choosing the old parish church rather than e.g. a Baptist chapel.  Since Mr Bullock suggests that evidence for deliberate design can be found it spite of these difficulties, I decided to press ahead.  In six parishes I couldn’t make a reasonable guess as to the correct church to measure, so I left these parishes out and ended with grid references for 43 sites.  {7}

The computer program first found all isosceles triangles among the 43 sites, allowing a discrepancy in the sides of up to 100 metres (2).  For purposes of comparison it was then necessary to count isosceles triangles among a collection of 43 random sites.  These random sites were produced by adding to each measured coordinate (N or E) a random displacement in the range ±10 km.  The procedure was carried out 100 times, and the number of isosceles triangles was counted each time.  This gives us a good idea of the number of isosceles triangles to be expected among random sites, having the same large-scale distribution over the map of Wales as the real Llanfihangels.  The results can be summarized as follows:

It appears that there are no more isosceles triangles among the real Llanfihangels than among an average collection of random sites.  However, there is still the possibility that a tolerance of 100 metres is too large, causing a few deliberately designed accurate triangles to be lost among the accidental ones.  Therefore, the program was run again with a tolerance of 20 metres - about the limit of accuracy for 1:50 000 maps.  The results this time were as follows:

The number of isosceles triangles among the real sites is now somewhat above the average for random sites, but not significantly so.  In fact 17 out of the 100 simulations produced 14 isosceles triangles or more. 

To those of us who remain sympathetic to theories of large-scale landscape surveying in antiquity, the results of the computer study are once again disappointing.  It is true that a fresh survey at the 6-inch scale would allow confident identification of churches and more accurate measurements; however, the results of this pilot study leave us little incentive to carry out the task. 

Other suggested geometrical patterns in Mr Bullock’s paper remain to be tested. 

Notes.

(1) J.M. Bullock, Studies in the Arrangement of Sites (IGR Occasional Paper No. 13, Bar Hill, 1979). 

(2) The curvature of the earth was allowed for.  The average correction required was about 1 part in 4000. 

Landscape Geometry of Southern Britain – an update.

In the IGR’s Occasional Paper No. 1 (The Landscape Geometry of Southern Britain, hereafter LGSB) published about 4 years ago, I described certain geometrical figures which I thought had been laid out in antiquity by highly skilled surveyors.  Since then, many other ‘significant’ relations between sites have turned up, but the suspicion has been growing in me that chance alone has played a large part in producing these geometrical patterns.  The same point was made of course by Robert Forrest (JOG 1/1) who pointed out a {8} regular hexagon and a vesica piscis among an arbitrary collection of modern sites in Wales.  In correspondence with Bob I queried his findings, because he was allowing much less accurate patterns than those in LGSB, but I have now applied the methods of LGSB to a few random sites, and the results still look bad for the ‘skilled surveyor’ hypothesis. 

Before going further I’d like to make it clear that I am discussing geometrical patterns, as in LGSB, and not simple alinements or ley-lines. 

First I took the grid references of 20 sites that were thought to be of importance in ‘real’ landscape geometry.  Then imaginary sites were produced from these by swapping the figures of the northing with the figures of the easting, except that the first figure of each coordinate remained fixed.  E.g. a site at ST 1234 5678 would be shifted to ST 1678 5234, the new site being in the same 10-km square as the old.  The reason for having a set procedure rather than a truly random process was to allow other people to verify or extend the results.  The sites used were as follows:

According to Dr Heinsch, the metre and its multiples were significant in landscape geometry.  Here we find that the distance between sites 4 and 8 (i.e. after the sites have been shifted as described) is 99·969 km, while 3 to 9 is 149·984 km; clear ‘evidence’ for a module of 50 km.  Multiples of 11 km were especially important.  And sure enough there is an isosceles triangle 2–16–18 with sides of 110·028 and 109·938 km.  In addition 14 to 19 is 220·002 km. 

For the alleged X unit of LGSB (295·32 metres) we need only look at the first three sites to find a triangle with sides of 809·95X and 450·05X, indicating a module of 90X units. 

A few more isosceles triangles are listed along with the errors in the sides; note that site 2 belongs to 4 isosceles triangles in all. 

Another, and the most disturbing to me, is the triangle 1–15–4.  Not only is this isosceles to 1 in 1000, but the angle at the apex is 36·04° or practically 1/10 of a circle.  So this triangle can be seen as a segment of a regular decagon like the one postulated, without very much more evidence, in LGSB.  There, it was thought significant that the short diagonal was 400X.  Well, using the mean radius, the second diagonal of the new and quite fortuitous decagon works out in terms of the Y unit (464·07 metres) as 600·02Y.  Considering that this polygon has been deduced from a simple test with only 20 random sites, it casts serious doubts on the patterns elaborated over several months’ work in LGSB.  I could add more details, e.g. on angular relationships, but I think this is enough to show that one must be very careful before assuming that geometrical relations between ancient sites, however accurate they may be, are the result of deliberate planning.