Talk:Alignments of random points

Comments
Perhaps this page could be developed positively to deal with a realistic appraisal of the question of the alignement of random points.Harry Potter 21:19, 5 Aug 2003 (UTC)

needs derivation
In the article: "An estimate of the probability of alignments existing by chance",

In the second paragraph the statement: "The probability that the point is "near enough" to the line is roughly w/d."

What happens if w > d? Probability > 1?


 * Think in terms of areas, and assume a uniform distribution. Assume the area is a non-pathological shape (square, round, or some other compact shape, say). Also remember that this is an order-of-magnitude computation, hence the word "roughly". And yes, if the line tolerance is wide enough, the probability will be 1, since the tolerance zones of all possible lines passing through the area will cover then entire area. I'll toss in some words to clarify. -- The Anome 23:18, May 8, 2005 (UTC)

Uniform distribution
"...estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points."

I thought the idea was that the points are distributed randomly, not uniformly. Davilla 20:43, 29 August 2006 (UTC)


 * The random points are distributed uniformly, i.e. they occur with equal likelyhood everywhere, as opposed to being clustered in one corner. linas 14:22, 14 September 2006 (UTC)

Original Research
Davilla 20:45, 29 August 2006 (UTC)


 * I'm removing the OR tag as there is no explanation as to why you think its OR. It looks plausible to me. linas 14:22, 14 September 2006 (UTC)

Possibly relevant paper?
This looks like it might be relevant: Alignments in Two-Dimensional Random Sets of Points David G. Kendall and Wilfrid S. Kendall Advances in Applied Probability Vol. 12, No. 2 (Jun., 1980), pp. 380-424 Published by: Applied Probability Trust Article Stable URL: http://www.jstor.org/stable/1426603 -- 80.168.164.171 (talk) 18:08, 23 February 2013 (UTC)


 * Cool. I've added it to the article. -- The Anome (talk) 10:58, 8 May 2014 (UTC)

More relevant stuff...

 * "Landscape alignments among 21 natural features and 61 Anasazi great kiva sites on the Southern Colorado Plateau: a comparison with random patterns"
 * The expected number of alignments in a random set of points
 * Point pattern analysis with Past
 * Hammer, Ø. 2009. "New methods for the statistical detection of point alignments." Computers & Geosciences 35:659 - 666. doi:10.1016/j.cageo.2008.03.012
 * "Comprehensive Strip Based Lineament Detection Method (COSBALID) from point-like features: a GIS approach" doi:10.1016/j.cageo.2003.09.004
 * "Numerical recognition of alignments in monogenetic volcanic areas: Examples from the Michoacán-Guanajuato Volcanic Field in Mexico and Calatrava in Spain"

Lineament is a nice word used in the geological sense to refer to linear features.

-- The Anome (talk) 23:57, 14 May 2014 (UTC)

This appears to have been an influential book in the ley line / chance alignment debate:


 * Nigel Pennick and Paul Devereux, "Lines on the Landscape: Ley Lines and Other Linear Enigmas", Robert Hale Ltd; 1st edition (May 1989) ISBN 070903704X

-- The Anome (talk) 11:26, 15 May 2014 (UTC)

Here's a paper by Alfred Watkins:


 * Alfred Watkins, “The proof of ancient track alignment” Journal of the Antiquarian Association of the British Isles, 2, 65–71 (June 1931) http://www.cantab.net/users/michael.behrend/repubs/bg_pioneers/pages/watkins.html

and a Google Books page for a relevant New Scientist magazine article can be found by Googling, thus:


 * https://www.google.com/search?q=a+statistical+model+for+ley+lines+new+scientist+30+december+1982&ie=utf-8&oe=utf-8


 * with this being a better citation:

-- The Anome (talk) 11:50, 15 May 2014 (UTC)


 * See also this amusing article by Ben Goldacre: "Did aliens help to line up Woolworths stores?", The Guardian, 16 January 2010. MFlet1 (talk) 16:42, 17 March 2015 (UTC)


 * MFlet1 Very funny. Not as funny, we have Tom Brooks (writer) which more or less takes him seriously. Dougweller (talk) 17:12, 17 March 2015 (UTC)

Similarity to the Bible Code phenomenon
In my opinion the Torah Codes seem to be essentially based on the same sort of cognitive phenomenon as ley lines: the power of combinatorial explosion in creating apparently impossible coincidences overwhelms intuition, leading to conviction that something deep and mysterious has been found. -- The Anome (talk) 23:48, 14 May 2014 (UTC)

Graphs
For non-math experts, the equations are hard to visualize. Could they be graphed to show the relationship between n and k? 198.96.2.93 (talk) 15:22, 27 February 2020 (UTC)

Lead
The problem of "Alignements of random points" is not even defined! The article must start by defining it first. I cannot even manage to understand what this article is about at all. — Preceding unsigned comment added by 207.134.139.130 (talk) 15:32, 21 March 2018 (UTC)


 * I rewrote the first few sentences. McKay (talk) 05:36, 10 March 2023 (UTC)

Furstenberg set conjecture
The Furstenberg set conjecture sounds like it would be relevant to this:


 * https://www.quantamagazine.org/mathematicians-cross-the-line-to-get-to-the-point-20230925/
 * https://isp.page/news/the-furstenberg-set-conjecture-a-longstanding-mathematical-puzzle-finally-solved/
 * https://arxiv.org/abs/2308.08819

&mdash; The Anome (talk) 14:26, 29 September 2023 (UTC)

Simplification
Unless I'm reading it wrongly, surely


 * $$ \mu = \frac { \pi n \left( n-1 \right) \left( n-2 \right)

\cdots \left( n - \left( k-1 \right) \right) } {k \left( k-2 \right) !} \left( \frac {w}{L} \right)^{k-2} \left( {\frac {d}{L}} \right)^{k} $$

(from the cited source here) can be written as


 * $$ \mu = \pi \, \frac { n! }

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

which is surely easier to understand, and I'm a bit baffled why it wasn't written that way in the source (note that the source's 2p is the same as this article's w). Or maybe I'm confused. Can anyone check my working? &mdash; The Anome (talk) 15:28, 29 September 2023 (UTC)
 * Both formulations are certainly correct, and your version is certainly more concise. But some people consider that ellipses are simpler to understand than a quotient of two factorials. Also your formulation complicates computation, as implying a much larger numerator. I you want a concise formulation, you could use the binomial coefficient and write
 * $$ \mu = \pi \,(k-1)\,\binom{n}{k} \left( \frac {w}{L} \right)^{k-2}

\left( {\frac {d}{L}} \right)^{k} .$$
 * D.Lazard (talk) 11:59, 30 September 2023 (UTC)


 * I like the binomial coefficient version. - CRGreathouse (t | c) 02:07, 1 October 2023 (UTC)