Talk:Ley line/archive 4

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Removed mathematical section:

Clarity

Latest comment: 15 years ago1 comment1 person in discussion

This is obviously quite a technical article, but somewhere along the line we seem to have lost any clarity for a lay reader. The introduction gives us the description of ley lines being "hypothetical alignments of a number of places of geographical interest" before launching into the technical stuff and I can't be the only one who feels the need for some brief overview of what a "hypothetical alignment" actually is in this context. Blankfrackis (talk) 23:34, 11 February 2009 (UTC)

Expected numbers of ley lines through random points

For those interested in the mathematics, the following is a very approximate estimate of the probability of "ley line"-like alignments, assuming a plane covered with uniformly distributed "significant" points.

Image of leyline simulation
80 4-point leylines pass near 137 random points

Consider a set of n points in an area with area $d^{2}$ and approximate diameter d. Consider a valid ley line to be one where every point is within distance w/2 of the line (that is, lies on a straight track of width w).

Consider all the unordered sets (known as combinations) of k points from the n points, of which there are (see factorial for the notation used):

{\frac {n!}{(n-k)!k!}}

What is the probability that any given set of points fits a ley line? Let's very roughly consider the line drawn through 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

{\frac {n!}{(n-k)!k!}}\left({\frac {w}{d}}\right)^{k-2}

With the assumption of constant density, and for values of d where n >> k this value can be shown to be roughly proportional in $d^{k+2}$ . 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.

In addition, it can be noted that even small increases in w greatly increase the number of expected lines. For example, allowing the use of "large" features such as stone circles has the effect of increasing the effective value of w.

Numerical results

Latest comment: 18 years ago1 comment1 person in discussion

Image of ley line simulation
607 4-point ley lines pass near 269 random points

Plugging in some typical values for the variables, and using the exact formula for combinations, rather than the approximation:

w = 50m, the width corresponding to a 1mm pencil line on a 50000:1 map

d = 100km

α = 5.5 × 10^-8m^-2 (50 points in a 30km square area)

then n is 550 and the number of leys predicted is of the order of 1000, with the longest leys expected to be 6-point leys. The number of predicted ley lines increases sharply as the size of the area considered increases.

According to this argument, the existence of long-distance ley line alignments in the English landscape should not surprise us.

(End of removed section.)

As I say above, if the mathematical derivation is published somewhere else, a description (not a regurgitation) of that research should be included, with a reference; if it isn't, it shouldn't be mentioned at all. -- Oliver P. 06:33 13 Jul 2003 (UTC)

Hmm, I see this thing is getting a bit out of hand - the article appears to have been totally rewritten since I last looked! Seems to me that it is proving hard to strike a balance between those who are determined to believe in leys and those (like me) who are completely skeptical. I found the maths argument pretty interesting, and thought it helped to balance the (original) article pretty well, so my vote is that it should stay - if it needs making more rigorous then perhaps it could be tightened up. However I would like to make one point about maps - the statement:

Others have pointed out the disparity between the two-dimensional representation of the map with the fact that the most accurate representation of the surface of the world is the geoid, which has very different qualities, particularly in relation to the question: what is a straight line running along the ground?

Is pure rubbish, as far as I can tell. Which "others"?: Sounds like somebody putting forward their own misconceptions dressed up as NPOV. Map projections answer this question perfectly adequately - the article could do well to link to other pages about them if anything like the above remains in. The key thing is that map projections used for OS (conic with two std. parallels, e.g) and other maps give TRUE BEARINGS - that is, an angular measure between features on the map gives the same angle as actually sighting the features on the ground. Just as well otherwise navigating using maps would be rather difficult, especially in an aircraft! (I'm a pilot, so trust me on this). This applies over pretty long distances - certainly on the scale of the UK it holds true. On continental scales you might (just about) have a point, though AFAIK there are no leys claimed for this sort of scale. Thus trying to call this into doubt in order to support the idea of leys is specious and has no place here. GRAHAMUK 09:51 13 Jul 2003 (UTC)

Yes, straight lines on the OS map are very good approximations to great circles over tens of kilometer distances, certainly good enough for ley lines. Note that the Earth is not a geoid, any more than it is a sphere. On the local scales ley lines lie in, local topography dominates (or we'd all have the sea lapping at our feet).

To Oliver: you seem keen to take out any mathematical statements from this article, on the grounds that it is "original research", but allow unattributed statements like the above to stay, no matter how silly. Yet you allow the waffle such as the above to say.

For example, take the imprecise argument of "extending a line", which is so imprecise as to be meaningless. Think: how many lines are there on a plane: an infinite number. What's the probability of a subset of points of interest lying near enough to that line? Greater than zero. What's the expected number of ley lines, by this definition? Infinite.

Are you proposing a ban on using mathematical statements in articles? -- The Anome 10:13 13 Jul 2003 (UTC)

Of course not. I'm as much of a fan of mathematical statements as anyone. I think I was fairly clear: we are supposed to be summarising human knowledge, not adding to it. If the mathematical statements have been published elsewhere, then including some of them might be appropriate, but the article should say whose research they come from. For example, "Fred Bloggs showed in his 1976 book A Mathematical Analysis of Alleged Ley Lines that the following formula should hold," and then include the formula. You see? And as for allowing silliness to stay, well, just just because I don't touch something, it doesn't mean that I support its inclusion. I can't deal with everything. -- Oliver P. 10:50 13 Jul 2003 (UTC)

To Oliver Pereira: Your point of "think about it!" being unencyclopaedic in tone is well taken. I fear I lapse into an audience-oriented style far too often. However the fact of meandering roads having been based on cattle-tracks though is so far from being original research as to be not funny. This is such an uncontroversial statement that I didn't even think, nor do I now think it needs to be qualified. In most countries there is no shortage of this fact being enshrined in the names of the roads themself. An American or British name escapes me at this point, but closer to home we do have frex. "Hämeen Härkätie" (literally Ostrobothnian Ox-track) which is the name of a vital (if meandering) strand in the Finnish Highway-network. Anyway, I'll mull this over for a bit and see if I can base a more substantive edit on this observation in a while. I am confident that the article on ley lines will wait for it -- Cimon Avaro on a pogo-stick 12:02 13 Jul 2003 (UTC)

As for drovers tracks, there is the English word Donga which was appropriated by the Dongas whenthey occupied St Catherine's Hill, just by Winchester.

As for the mathematical discussion this has not just involved Alfred Watkins but also Major F. C. Tyler, who played a significant role in the Old Straight Track Club particularly after the death of Watkins. Then of course there is also Forrest and Furness.

As for country crossing leylines the claim for the St Michael's ley line includes points: Cairn Les Boel, Penzance, Hurlers, Cheesering, Brentnor, the Wellington Monument, Trull, Creech St Michael, Burrow Mump, Glastonbury Tor, Trowbridge, Bowerhill, Oliver's Castle, Avebury, Ogbourne St George, Dorchester, Whipsnade, Royston, Bury St Edmuns, St Margaret's Church Hopton, i.e 20 locations, making the previous statistical analysis futile (only dealing with 4 locations) whereas the key issue here are a) how wide is a Ley line, b) how wide is an "interesting location", and c) what is meant by a straight line.

On the social history of Ley lines, John Michellwas much involved in publicising them. In his first book The Flying Saucer Vision, 1967, there is but one rather bald reference to ley lines. It is only with The View over Atlantis that he really goes to town - or should that be the countryside - to deal with Ley lines. His original angle was that they were connected with UFOs, something often missed out in later books on the topic.Harry Potter 13:49 13 Jul 2003 (UTC)

Perhaps I misunderstand the formula, but it seems that all three of these questions can be answered by simply adjusting the width of the ley line (w in the formula). A less-than-straight line is simply a straight line that covers a wider swath. The variance in width of significant points again can be accounted for by adjusting the width of the line, which is accounted for in the formula. (Can a mathematician or statistician confirm this?) Consequently, I don't see how a 20-point ley line makes statistical analysis "futile"; if the analysis can be improved to demonstrate that an adjustment in allowable width makes such 20-point lines possible (or even likely), then the analysis is quite useful. Again, HP, if you find the statistical analysis to be flawed, please help us improve it. -- Wapcaplet 15:21 13 Jul 2003 (UTC)

By, the way, HP, to get 20-point leys, just try:

w = 500m
d = 200,000m
density as before, 5.5e-8 m^-2
expected numbers of 20-point leys: 38

Note how most of the points on your 20-pointer are towns or villages, justifying the larger value of w. Push w up to 1km, and you can even expect 20-pointers for d = 100,000m.

Since the expectation is proportional to w^k-2 (see previous equations at the top of the page), increasing w by a factor of 10 from 50m to 500m increases the expected numbers of 20-point leys by a factor of 10¹⁸! Wiggle room indeed.

-- Anon.

For a good reference for National Grid vs. GPS etc. in the UK see: http://www.gps.gov.uk/guidecontents.asp -- Anon.

- - Ha. Several points here. What I've been thinking about is a statistical model which looks at the frequency of interesting points per unit of distnace travelled along a line. I haven't quite worked out how to realise that, but it certainly simplified the question about longer lines. - - - - Secondly, perhaps the results should be expressed in terms of liklihoods rather than expectancy, i.e. the probability of having p points in a line of distance d is x%. - - - - And yes it precisely the issue of wriggle room which needs to be clarified, and perhaps turned the other way around too - i.e. in order to get a significant result w must be less than whatever, and the location size must be less than whatever. But then if the w size (wriggle size?) is kept small, then the sooner the difference between the geometric nature of the projection on a 2D surface comes out of kilter with what is happening in the field. - - - - <blockquote>Others have pointed out the disparity between the two-dimensional representation of the map with the fact that the most accurate representation of the surface of the world is the geoid, which has very different qualities, particularly in relation to the question: what is a straight line running along the ground?</blockquote> - - - - Wapcaplet suggests that these problems have been solved by the "various projection models we have for generating 2D maps of the Earth." No so. There is an irredemiable loss of information, so some projections for instance preserve land area, others preserve different features. But the fundamental geometry is different i.e. the internal angles of a triangle only add up to 180 degrees on a flat surface. And if you have not got a flat surface, the question arises what do you mean by a straight line. This can be answered as a Great circle when dealing with a sphere, but this is not what we are dealing with. What HAS happened is that through computers it is now possible to generate a £D representation through GPS systems. So the statement "The key thing is that map projections used for OS (conic with two std. parallels, e.g) and other maps give TRUE BEARINGS - that is, an angular measure between features on the map gives the same angle as actually sighting the features on the ground." is incorrect, even if the disaparaties only become apparent when considered over longer distances than usually found on a large scale map. - - - - Also I note that some people seem to be concerned over whether the article should instuct people as to whetehr ley lines are "real" or not. This is not the role of NPOV writing. The problem is what to effectively describe what a ley line might be. As there are a range of different conceptions layered on top of each other, from trade toutes to lines of mystic energy to flight guidelines for UFOs, then I feel that the historic secretion of these conceptions, from Watkins through Michell to the likes of Devereux and even Pennick needs to be described. I do not have time to write anymore just now, but will return later. Harry Potter 21:13 13 Jul 2003 (UTC) - - - - Right. "On continental scales you might (just about) have a point, though AFAIK there are no leys claimed for this sort of scale. Thus trying to call this into doubt in order to support the idea of leys is specious and has no place here. GRAHAMUK 09:51 13 Jul 2003 (UTC)" Perhaps GrahamUk would fare better if he did not draw conclusions which, as he admits fall outside his sphere of knowledge. In fact there are a range of different systems: some relate to magnetic lines of declination, planetry lines . . . indeed there has been a cottage industry in constructing system, some applying astrological formulations. There are such things as the Unified Vector geometry Polyhedron Sphere developed by William Becker and Bethe Hagens . . . hey reading Steve Cozzi's Planets in Locality he extends the 20 point ley mentioned above across the North Sea and Jutland to somewhere on the coast of Sweden. On the other hand, if we go back to the twenties we come across a faction of British Israelites criticising the British Israelite Federation for refusing to see that British measurements are cosmic as a matter of fact (this is from a leaflet entitled "Hurrah, for Great britain, the New Jerusalem" probably published by Hamish McHuisden, author of The Great Law. He developed a theory of enormous circles centred on London (identified with Christ, Glasgow (identified with God and the place where the Lusitania sank off Cork identified with Lucifer. Now many people might just want to wave their hands in the air and say why must this sort of material appear in an encyclopedia!!! However there is no reason to think that any individual should be interested in every article in an encyclopedia and indeed such conjecture are reminiscent of those made by David Deutsch at the beginning of The Fabric of Reality. In fact behind a veneer of naive realism to often there is a desire to defend a "comon sense" which merely uncritically reproduces elements of the dominant ideology as being of universal validity. But despite all of this, whatever the difficulties the topic of ley lines will have to dealt with properly. Not only do we have to deal with the British Israelites, but also with Freemasonry? Why, you might ask? Because we know that operative masons were the first to be involved with gunnery in Britain (this can be found in any good account of the bombadment of Thrieve Castle). And we know that Gentleman masons Robert Moray and Alexander Hamilton are first recorded as being admitted to the brotherhood by Covenanter Army for their ballistic skills. Indeed, this connection with the artillery (Hamilton was General of Artillery) is born out by the fact that it was Ordnance Survey which produced the maps, primarily to aid in firing at an enemy by having accurate maps.(this is also born out by the role played by military organisations sometimes called (in true Orwellian fashion Defense. - - - - All roads lead to Rome. as the saying goes, and any mapping system (reduction of the world to two dimensions) will privilege certain areas over others, in that there is generally an origin out from which the rest is defined. This structure then creates a gradient in accuracy with more inaccuracy as you move away from the centre. Therefore none of them can be NPOV, as they depict the world froma centralised point whose view becomes less exact the further one moves away from the omphalos (this is the Greek expression meaning navel but also refering to the centre of the world - hence the omphalos at Delphi for the Greeks. - - - - In this context we can see the Heilige Linien of Wilhelm Teudt in his 1929 book Germanische Heiligtumer, being taken up by the Deutsche Ahnerbe, a Nazi outfit which then employed Teudt to run Externsteine as a nazi cult centre, laid out according to Teudt's ideas concerning alignment. These were also taken up by Kurt Gerlach who introduced the Heilige Lines into his studies of German settlement in Bohemia in the tenth and eleventh centuries. As these researches were being published by the nazi Ahenerbe at a time of twentieth century invasion of Czechoslovakia, they clearly had a highly political role to play. Nigel Pennick has frequently refered to these, and the Institute of Geomantic Research (with which he is deeply involved) has published some of the Nazi tracts. His attempts to shrug of the political context of his researches runs counter to the glowing review of Kurt Saxon's Wheels of Rage which Pennick had published in the anarchist Cienfuegos Press Review. (I don't know whetehr the person who introduced teh question of "Holy Lines" was deliberately refering to Tuedt's lines or not?) - - - - Wheer does this leave us? Well we have really only just started to scratch the surface.Harry Potter 23:14 13 Jul 2003 (UTC)

Please Harry Potter, don't scratch any deeper!! It's hard enough to read through that screed of stuff and make much sense of it as it is. This is not a personal attack - but your argument would be easier to follow if you formatted it a bit more clearly. Now, you requote me, so I feel it is only right to respond.

1. I stated that as far as I knew there are no leys suggested on a continental scale. If there are, please refute this with some data! Watkins' leys are definitely on the "local" scale, that is, within the area bounded by typical OS maps of an area - that is by his own admission. On this scale, my second point is true:

2. That maps give true bearings. Yes they do. Fact. As I stated, a large scale map that shows significant areas of the earth will not give true bearings, but OS maps do. Flight navigation maps for local areas do (e.g. the southern UK CAA map, which I use regularly). Any typical map that you may encounter (e.g. road atlas) does. The projection distorts the representation of the sphere subtly to make it so. This means that straight lines on maps are straight lines on the ground, up to several hundred kilometres in length. Simply stating that this isn't true seems to me to be wishful thinking on your part - even if the earth was such a small sphere that discrepancies existed over very short distances would not magically (pun intended) make the argument any stronger. This whole bit about map projections is a red herring - it neither supports nor refutes the ley argument. It is simply specious and should be removed.

3. I have trouble with the attribution to "others", i.e. "Others have stated that... blah blah". What others? Please tell me who they are. I suggest that this is just a way of stating your opinion without any backup, and dressing it up as NPOV. That won't do.

I am going to remove the offending paragraph, it adds nothing one way or the other. I feel that with the latest reworking by The Anome the article is very fair and balanced to both sides of the argument. May I modestly propose that the edit war cools down a bit before any further gross hacking is done? GRAHAMUK 00:37 14 Jul 2003 (UTC)

There's some mention of John Michell comparing Feng Shui to Ley lines. This might be a bit of a red herring, if I misremember what I've heard of F.S., but isn't at least part of the basic principles of F.S. being the avoidance of straight lines? If so, then what have ley lines to do with Feng Shui? Malcolm Farmer 12:01 14 Jul 2003 (UTC)

It's not so much that the 2 are the same, but the see also suggests only that there's some similarity - in Feng Shui, if I understand correctly, there's also the idea of some force (chi) floating around between objects...

Likewise, the link to the The Bible Code is there because that's another case of things seeming to have meaning but actually being generated by chance. Evercat 13:12 14 Jul 2003 (UTC)

My feeling is that we need an Earth mysteries page where [[Feng Shui}} can nestle with Geomancy. I think it may be useful to out Landscape alignements in order to deal with a range of phenomena of which Ley lines constitute one element. I think it is important to recognise that the relationship between Ley lines and probability is something recognised by Alfred Watkins and which has been worked upon by subsequent generations. My own position is that the problem with statistics is more with how it is used to represent reality. i.e. the idea that a ley line should be expanded ten fold just because it goes through a town is redolent of the sort of ideology which promotes the current proposed expansion of the M25. What i am more concerned with is the use of statistics to show what level of accuracy would create what significance rather than have someone pluck a figure out of hat to produce the answer 38 (which Douglas Adams no doubt would accept as a close enough approximation to 43 if he were still alive. As for the inaccuracies of the O.S. maps, what I say is patently true: they are inaccurate. The fact that the level of inaccuracy is irrelevant for a pilot is also patently true. The question remains, as I have said before, the criteria used. So whacking out the width of a ley line to 1km might work if the aim is to amuse, however if it is to inform then I must admit to feeling short changed. Far better, says I, to conduct some fieldwork. So the fact that a place mentioned is a town, lets examine one town - Royston. Take it back through a thousand years and we have the cross roads of Ermine Street and the Icknield Way, something which falls within the 17.5 metres width that Williamson and Bellamy discuss. I don't know whetehr this alledged Ley line goes directly through the cross roads or not, and I suspect no-one does, because I suspect no-one has done the research. However with the new availability of GPS sensors it is only a matter of time before Cyber Ley Hunters get to work. Until we have some decent figures to calculate with the rest is speculation and any statictical activity not geared to specifying the probability of events in a carefully defined manner can only be regarded as pseudoscience.Harry Potter 18:58 14 Jul 2003 (UTC)

I'd support an Earth mysteries page - no problem, it means I can dutifully ignore it. ;-) A couple of things - Douglas Adams' number is 42, not 43. Now, OS maps - I don't wish to harp on about this but I would like to know what you mean by inaccurate. Yes, they are inaccurate, but to what degree and type? They are inaccurate in some gross aspects - they sometimes don't show huge complexes such as nuclear research establishments and so on, but that might be considered understandable. They occasionally contain out of date errors, or lack the resolution to show very small features. While these are all inaccuracies they are all different types of inaccuracy - which did you mean? In general terms, they are accurate in that the locations of towns, villages, hamlets, homesteads, earthworks and so on are placed correctly given the projection in use. Straight lines drawn through them will be seen as straight on the ground up to a few hundred ks in length. How wide is a drawn line? With a sharp pencil even on a 1:50000 map, that represents a width of 10-20 metres, on the 1:25000 map resolutions of 5-10 metres are possible. Notwithstanding my use of maps as a pilot (where the most useful maps show half the country), another area I have had a lot of experience in is car rallying. On many events I have used the 1;50000 maps to navigate a car and driver on a rally. On such an event, I am usually able to give the driver almost pacenote like information simply by looking closely at the map. If you're not familiar with rallies I could direct you to my article about Rally Navigation techniques, tips and tricks - the point here is that the detail extractable from the map goes far beyond "turn left at the next junction" - you can give the driver every little kink, bend, gradient change, and so on - in this respect the maps are astonishingly accurate. In fact so much so that the occasional error can cause major problems because you come to rely on the map almost totally. Anyway, this is a digression - the pilots' maps are possibly more germaine to the ley line argument because they are used to plot tracks of many hundreds of kilometres in length, which is why I mentioned them. I can draw a line on the map from A to B and then fly that track (allowing for wind, magnetic variation, etc. naturally) spot on by measuring the angles off the map and lining them up on my flight instruments. Strangely enough the straight line I fly across the ground matches what I see on the map! Never deviates! Yet at a height of 2000 ft or so, even the tiniest ground feature is clearly visible. Curiously enough I find that typical flight tracks very often join up a number of random towns and villages along the way - handily I can make use of this for visual navigation. It's common for there to be five or six notable places I fly directly over on a typical flight leg. These are chance alignments which are pure coincidence, but if somebody of a mystical tendency drew such a line and noticed that it passed through all these places, would they assume that the alignments were somehow evidence of some ineffable grand plan? Sadly I suspect so. Surely the chance alignment theory is clearly the simplest, and by Occams Razor, therefore the most likely to be correctGRAHAMUK 09:55 15 Jul 2003 (UTC)

Sadly I don't share the happy-go-lucky world of the pilot or rally driver. I lost my innocence early: I was exposed to Ley lines as a child by my father who was a physicist and electical engineer who worked on travelling waves tubes, and microwaves after a stint in the admiralty during the Second World War. Even back in the forties there narrow beam infra-red beacons which were used by submarines to contact people on the shoreline of hostile countries, and by the sixties the telecommunication techniques my father was working on were concerned with narrow beams linking up such prominent features as the British Telecom Tower. My father always treated people of the mystical tendency with tolerance, although he did laugh about them in privacy of his own home. He also engaged with Alexander Thom's theories about Stonehenge and other Stone circles. From these I feel the case for astronomical alignments in stone age monuments has been stoutly made. (I visited St Cristina's Well in Paulilatino, Sardinia last year and it was clearly designed an observatory.) The chance alignment theory is not really a theory, but a null hypothesis. As for being "correct", I am not sure what you mean by that. The discussion about probability does not show this or that to be correct but produces values, probabilities according to which people can decide whether to pay such phenomena any more attention. Fine. My problem is with something like ths St Michael ley line is whetehr it is something which is just slapped together, or is their a real alignment which exists within a precise framework. I am more interested in the use of statistics which aims to keep lines narrow and then check if there is a real alignment than when people start talking about 1km wide Ley lines. Another point which has not yet come into this discussion is post-fieldwork additions to ley lines. This is prominent in Williamson and Bellamy's critique of John Michell's The Old Stones of Land's End, 1974 and which raises more queries. Also it is important to note that not only was Alfred Watkins aware of the statistical critique of Ley lines, he was anxious to forestall it, as previously discussed, and also in his point system which gave different features different weights from 1/4 to 1 point, and which required a sum of at least 5 points. The probablistic discussion here was just of poor quality, mixing up circles and squares and when applied to the St Michael line descended into ridiculousness when much of the area of the poorly conceived squircle is sea.Harry Potter 19:14 15 Jul 2003 (UTC)

To HP:

Agree, it's a null hypothesis. That why it's called that in the article.
Clue: Passing a 50m wide path over 1km wide objects is (to first order) equivalent to passing a 1km wide path over point objects.
The square/circle error is a factor of 1.4 or so: small compared with other terms in the equation, which is an order-of-magnitide calculation. More accuracy is easy, but the same general results would be obtained.
Numerical experiments are the gold standard for checking these sorts of calculations: please feel free to generate your own ley line checking programs.
An argument you can't understand is not necessarily false.

To anyone:

does anyone have a list of megalith coordinates?

-- The Anome 00:14 16 Jul 2003 (UTC)

This online resource may be of interest: how accurate it is, and what the copyright situation is, I don't know. Perhaps an approach to the site authors would be a good idea? -- The Anome 23:20 16 Jul 2003 (UTC)

What a load of old tosh. The "clue", aha suddenly all the objecyts have sprouted in size to 1km across. what childishness. The suircle error is far more serious than the meaningless "a factor of 1.4 or so". In fact it is (4-PI), I.E. about 0.8584. However as this has to raised to the power 18 (=k-2), this then becomes 2.433690014 lines expected. (OK where are the other one and half ley lines then might be the obvious response!!) However why was the choice made to have the line 1km thick (or os some new megalythic kilometre) and what happens if we reduce it even a little. Take the line down to 900m and we get and expectation of 0.365283815, 750m and we get 0.013720439. I have known i am free to generate ley line checking programs, but I don't see where such a stupid numbers game gets us short of having more information on the real world to work with. However, another approach might be to invisiage an observer at a particular site noticing alignments. Perhaps we could work on the line being a ray from this vista, so that greater accuracy occurs nearer the site, and it decreases with distance, thus objects further afield must be larger which also affects their visibility. This would also deal with the problems of 2D vs 3D inaccuracies which increase over distance.

I don't know how to repond to the ignorance about what I said about judging places more accordinbg to how they were, rather than taking into account the growth of urban sprawl (or shall we have some wag paraphrasing Watkins by saying that Leylines used to be real but lost statistical significance as urban life spread!)

Any effective way of dealing with the issue would to plot megalithic sites and establish their dates. Then having created a data set which can belooked at progressively (i.e. there might have been a particular period when ley lines were in vogue . . .) and theories tested. At present we are far short of that.

Many of the proponents of Ley lines display unclear thought and are using wooly categories. But at least they more vague abstractions are usually recognised as having distinct weaknesses. However I sometimes think that the abuse of statistics as dispalyed here (and countless other places) should be recognised a pseudoscience, all the more dangerous because of the pretensions of those who produce it.

And by the way The Anome, just because somebody disagrees with you does not mean that they don't understand what you are saying.Harry Potter 19:02 18 Jul 2003 (UTC)

Oh dear. I can see that you have again deleted the illustrations that you find so embarrassing. Thanks for digging out Watkins' definitions: they seem if anything a bit looser than the ones I have been using: 50m over 100 km is a slope of 1/2000, whereas 1/4 of one degree is 1/1440.

You again misunderstand the bit about drawing lines through town centres. But computer checking of data sets is the way to go. I confidently expect you to argue this to the bitter end, no matter how many examples of chance-generated alignments you are given, from whatever data set. But testing is the way to go: I stand by the skeptical position: I'm willing to let the data settle the argument. -- The Anome

By the way, I'm having a bit of difficulty with Watkins' remark "if only three accidently placed points are on the sheet, the chance of a three point alignment is 1 in 720."

I want to be able to make this mathematically precise.

The problem is in defining the probability distribution of the points on the plane. If the region of interest is infinite, strange things happen. See http://www.mathpages.com/home/kmath299.htm for more on this. If the region of interest is finite, things like the shape of the region will play a part (consider a long, thin, rectangular map with an aspect ratio of 100:1 for a reductio ad absurdam argument: then consider the case of a map with a 4:3 ratio). -- The Anome 12:17 23 Jul 2003 (UTC)

Some (very rough) random-number-based checking suggests that to get odds of 720:1 with a square map requires a maximum angular deviation of the third point from the longest side of (roughly) 0.0026 rad, which is about 0.14 degrees, so the overall size of the allowed angular span is about 0.28 degrees. This approximately matches Watkins' calculations. Choosing a non-square map has the expected bias effects. However, this angular definition is much less mathematically tractable than the simple line-and-distance criterion. -- The Anome 14:06 23 Jul 2003 (UTC)

Watkins 720:1 is based on selecting an origin and a marker and using these to define opposite rays 1/4 degrees wide, making a total of 1/2 degree which divided by 360 gives 720:1. So from this I would conclude that this models a acircular piece of paper centred on the origin. What I think is important is that Watkins lays stress on fieldwork with line of sight, backed up by map work. Since Watkins day others have taken Ley lines in directions clearly at odds with his modest proposal, particularly with the post UFO stuff. (therer was a comic strip called Slaine where they had ships moving along in the air powered by the leylines which they travelled on, which I am sure you'ld find amusing. Harry Potter 04:09 24 Jul 2003 (UTC)

I've just be leafing throu Thom's Megalithic Sites in Britain, 1967 the appendix includes a pice On calculating the azimuth of a line from the co-ordinates of the two ends where he offers a correction of lsin(lat) where l is the longtitude and lat is the lattitude He claims that noe error greater than a quarter of a minute of an arc will be introduced. The book has a great deal of statistical information which I will read. I'm going to have to get rid of that stupid diagram - it is such an insult to the likes of Thom who took great care in making observations of 300 megalithic sites using theodolites and an understanding of mathematics.Harry Potter 17:57 26 Jul 2003 (UTC)

Unfortunately Stan isn't keeping up with the debate here. As we have moved onto alignments based on angles from a point of origin, the illustration is completely irrelevant. If Stan wants to produce a relevant diagram that would be most welcome, but why continue with something which does not do what it says, i.e. it does not simulate Ley lines as described by Watkins.Harry Potter 23:31 26 Jul 2003 (UTC)

What's the problem? Surely it's not the case that all ley lines have to pass through a single point? There will be different points of origin for different ley lines, right? Evercat 23:34 26 Jul 2003 (UTC)

Of course they may have different origins, but each one is defined as apir of opposed rays issuing from its own origin, whereas the so-called simulation discusses narrow oblongs considerd as lines, clearly something quite different.Harry Potter 11:56 27 Jul 2003 (UTC)

A standard approach in pseudo-science is to keep twisting the definition hoping to escape the flaws of the previous one. The diagram demolishes the whole mystique of ley lines, so of course the effort will be to come up with excuses as to why it's irrelevant. Rays vs oblongs doesn't matter if the tolerances are set right. But I'll wait a few days, see if there's input from other scientific types. Where's RK when you need him? Stan 12:58 27 Jul 2003 (UTC)

What's the problem? Who is twisting the definition? You have abandonned any attempt to grasp what Watkins was talking about. T Yes all sorts of people have come along and tried to rearrange what Watkins was talking about, followed by a whole load of equally vacuous statisticians. The diagram demolishes nothing. as to the mystique of ley lines, where is this. You do not wish to engage in a real discussion of megalithic alignment, but instead, like the person who always took a monkey with him for company, you enjoy the company of wild-eyed ufologists because you feel that you can present yourself as reasonable when looked at in their company. Oblongs are very different from rays when considered from the point of view of postulating real people standing in certain sites making measurements as opposed to modern people sitting at home with a pile of OS maps, rulers and highly sharpened double HH pencils. As for tolerances, they have to be set very high to cope with your sort of pseudoscience all the more invidious for its scientific pretensions. At least John Michel in his earlier books makes clear that what he is speaking about maybe a type of human fancy, even if this is put aside following and to accentuate his market success with an uncritical readership.Harry Potter 14:09 27 Jul 2003 (UTC)

A line is a line is a line. Ley lines are "alignments" - ie the objects are "in a line". There's no difficulty here. Evercat 14:15 27 Jul 2003 (UTC)

A line is a mathematical abstraction, which can be made in several different ways. Within Euclidean geometryt as a line there is no difference between taking an origin and an angle to create polar co-ordinates to using cartesian co-ordinates. However, in the statistical analysis these are two quite distinct ways of operating, which will produce different results and indeed operate from different perceptual basis. It appears to me that some sort of grasp of semiotics is necessary if peoples valued contribution to wikipedia are to be as useful as they could be.

Let me quote a little bit from Gerald Hawkins' Stonehenge Decoded:

There are agreat many numbers and alignments at Stonehenge, and the numbers and lines never cease to fascinate people. Even the most rational of the Age of reasonbers, Samuel Johnson' observed most carefully the crosslines as he walked. And one of the most notorious 'marvels' of modern France is the fact that Paris is so aligned that on Napoleon's Birthday - August 15 - the sunas seen from the Champs Elysee, sets in the centre of the Arc de triomphe. Actually that supposed marvel is a good example of an apparently extraordinary and speculation-worthy circumstance that is in fact not very remarkable. Let us examine the situation closely: what are the chances of a simple coincidence? First we find that the Arc is so wide that the sun sets in it for a two week period; that reduces the odds from 365 to 26 to 1. Then we must note that the sun also sets in the Arc during a two week period in April; the odds fall to 13 to 1. Then it must be admitted that sunrise on the same day would be equally phenomenal; reduce the odds to 6 1/2 to 1. Then we may suppose that sunset or sunrise on the day of Napoleon's death would be equally notable; 3 1/4 to 1. And what if the birthday sun rose or set through some similar great arch or other Napoleon relic? and so forth. The Napoleonic sunset clearly has no significance. I think that any good coincidentalist could find just as marvellous Napoleonic magic at Stonehenge; perhaps the moon rose on the line from the centre of Stonehenge passing over the battlefield of Waterloo, on the morbning of the battle there. What if it did?

The numbers game is nothing but a game if played without purpose and method. But there can be good result if speculation is implemented properly.

Statistics can used to create models which give an indication of the probability of the chance creation of specific situations. But here they arguments and illustration have proved to be as specious as much of the claims about ley lines - only with the added annoyance of a bogus claim to skepticism and scientific foundation. This is pseudoscience. It does not help with an account of ley lines. What is needed is close textual analysis and semiotic differentiation so that the question of megalithic alignment can be given some sensible analysis rather than being the football of ufologists and so-called skeptics, who are just as much given to head-banging irrationalism as each other. Maybe there should also be some space for other interpretations, on this page or elsewhere, as these elements of human behaviour can also be usefully included in the wikipedia. However to suggest that the controversial diagram does anythinmg other than illustrate the poor quality of the debate is to show an inability or refusal to grasp some of the fundametal issues.Harry Potter 15:28 27 Jul 2003 (UTC)

I have restored the picture, once again. It shows that alignments naturally exist between random points. It shows why people will find such lines, no matter what method they use. Evercat 17:57, 5 Aug 2003 (UTC)

Perhaps Evercat or others of his persuasion might care to show what relevance it has to anything. Ok then lets put it on the Alignments of random points page.Harry Potter 21:08, 5 Aug 2003 (UTC)

You know perfectly well why it's relevant. One of the skeptical claims in the article is that ley lines just are alignments of random points, those points being the sites of landmarks. Evercat 21:13, 5 Aug 2003 (UTC)

What I know is that for a null hypothesis it is necessary to have some actual results and the random set created to compare with it. This is not what is happening. Instead a picture is thrown up out of the blue unrelated to any actual claim for Ley lines to exist and without the theortical basis of it either. This is not statistics. Perhaps it is important to use the Alignments of random points to help sort this one out.Harry Potter 21:28, 5 Aug 2003 (UTC)

This is a rather more sensible argument than most of the above. Your complaint, if I understand you correctly, is that we need something to compare the diagram with - ie are there more "ley lines" in the world than there should be?

Still, the image is just there to show that there are bound to be some. Obviously believers in ley lines will say there are more than can be explained by chance. I'll add a sentence to that effect in the article, if you wish.

I would like to see the comments of The Anome on this matter... Evercat 00:35, 6 Aug 2003 (UTC)

I think it is not so much what beleivers or non-beleivers in Ley lines say as, but what the reality is. Whereas Watkins had quite a down to earth approach, this has not been followed by other Ley-lineologists who produce quite specious argunments, only to be met by so-called sceptics, whose arguments are equally specious. What can we conclude from all of this:

1 The discussion of what Ley lines are or might be has changed in time. Broadly I would say there is the pre-Watkins era (up to the twenties), the Watkins era (which is noted by the formation of the Old Straight Track Club, providing an institution which could draw in people and document their activity. Although this usrvived his death it did peter out in the forties. Then there was the renewed interest from ufologists, which then got rejigged through John Michel into the geomancy school. Thus I think there are four different uses of the term arising from different periods.

2 Watkins was instrumental in helping develop a new approach to megalithic structures which took account of Ley lines. However by the time that scientists with access to computers got involved in the sixties (here I am thinking of Alexander Thom and Geoffrey Hawkins, the term ley line had already become too associated with UFOlogy that they did not want to use it. Their work was associated with a major shift in thinking as regards trhe level of sophistication of the megalithic builders which had previously been under-estimated. Some of the later claims may appear to over estimate the potential of the megalithic builders, but when they rely on extra human intervention, perhaps this is just an other form of underestimation.

3 It would be good if a clear account of the probablistic model was given. The model "proves" nothing, it only offers a comparison with chance produced events which anyone can then pursue or not pursue as they think fit. Such a choice has ontological considerations which cannot be swept aside by so-called sceptics, in that signification is a human activity in which human beings are continually engaging. Caveat emptor maybe, but anything beyond that is to substitute scientism for science.Harry Potter 19:58, 6 Aug 2003 (UTC)

One problem I see with the idea that we should "compare actual areas with the random projections" is that there will naturally be areas where there are more ley lines than would be expected by chance (and there will also be areas where there are less than expected by chance) - in order to do this sort of comparison between the actual world and what would be expected, wouldn't it be necessary to compare not just areas where ley lines are claimed, but very large swathes of countryside? ie to be significant, such a comparison can't just focus on a few small areas... (this is just a practical problem, though, right?) Evercat 19:52, 21 Aug 2003 (UTC)

Saying the above, I don't know if it helps or hurts your position - on the one hand it may mean the "skeptical" position is difficult to falsify (which is bad for it) but on the other hand, it means that even if there are areas where there seem to be too many leys, this doesn't really prove anything. Hmm. Evercat 20:02, 21 Aug 2003 (UTC)

Well. I'm not so much interested in defending a position as contributing towards a process whereby we are able to think about what is going on more clearly. One problem with the random model is that there is no reason to think that megalithic sites are located randomly. In fact it is quite clear that they are grouped more intensively in certain areas. Another issue is the timescale - i.e. what is to be gained by flattening out a period of say 1,000 years and treating it as one moment. If we look at the complex around Avebury, West Kennet, The Sanctuary and Silbury Hill we have an awful lot of ancient sites built and developed in different peiods. Speculation is based on different ideas of what is considered likely. So one person may want to assume all the sites were developed as part of an evolving single plan, whereas someone else might be happier thinking some of the monuments were built by people who felt the previous stuff was irrelevant to them, or had been squeeezed out of significant ceremonies. This does not mean that the speculation is useless, but rather limited. In the end decent fieldwork can gather the information which can then be subjected to profound analysis. This is precisely what happened in modern astronomy. Tycho Brahe collated a mass observations without developing much in the way of a theory. But it was upon his broad shoulders that Johannes Kepler could climb in order to develop his theory of celestial motion. In turn Isaac Newton climbed the human pyramid to stand upon Kepler's shoulders . . . and so things progress.Harry Potter 21:19, 21 Aug 2003 (UTC)

I doubt there'll ever be such a level of interest in this topic, so debate is likely to be less profound than you'd like... Evercat 21:32, 21 Aug 2003 (UTC)

Well if the level of debate continues as it has done in the last half of the twentieth century, then I think there will be quite a lot. However I think it will move forward where it is not dominated by either 'believers' or 'skeptics' but by people who wish to understand how and why people built such monuments.Harry Potter 22:58, 21 Aug 2003 (UTC)

Editing the following 'graph into Handwaving is irrelevant and disruptive. If it is useful anywhere, figure out where by considering in the context of Ley Lines.

Handwaving is also used to describe the activity of surveyors when signalling to each other when aligning their theodolites. Some people say that this goes back to neolithic times when Dodmen alledgedly organised prehistoric sites into Ley Lines. --Jerzy 16:33, 2003 Oct 28 (UTC)

Removed from ("The Sceptical approach"): Proponents of ley lines maintain that there are specific claims about specific alignments for which the alignment of random points is not the appropriate statistical model. Without a concrete example this doesn't tell us anything very much (other than that proponents don't like mathematical counter arguments.) Andy G 17:30, 11 Jun 2004 (UTC)

a set of points, chosen from a given set of landmark points, all of which lie within at least an arc of 1/4 degree.

Um, should that be "at most an arc of 1/4 degree"? Josh Cherry 15:38, 17 July 2005 (UTC)

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