In a mailing list discussion way back in March 1998, someone referred to the following web page, and said: "Here's one that needs the scrutiny of a knowledgeable mathematician." Here's my response, only edited to add some links.Here's a link: Gerald Hawkins' work on crop circles and their relationship to Euclidean geometry and diatonic ratios (You will need to get the page, to look at the diagrams.)(V) As the triangle changes shape the circles expand and contract to touch the sides, and the diagram generates the four crop circle theorems.Sample: Diagram and caption from Hawkins' pageOn the left of Hawkins' page is a set of five diagrams, labelled "Theorem I" to "Theorem V". Only theorem V seems to have any text associated with it, and this is an explanation of the way in which the diagram is generic, rather than a text which looks like a conventional theorem.This is the first problem: these do not appear to be "theorems" at all in the accepted sense. For example, Theorem II has an outer circle, inside which is inscribed an equilateral triangle (i.e. its corners touch the outer circle), inside which is inscribed an inner circle. That's it. Huh? Where's the theorem?A theorem is a "general result": a geometric theorem can be quite simple, and shown by a single diagram, but it needs a description of just what the general result is. A highly recommended book full of examples is the Penguin Dictionary of Curious and Interesting Geometry, by David Wells (a friend of mine, not that that's relevant). The theorem I proved the other day with the bubbles and menu on a dinner table is a good example.Anyway, never mind, let's try to press on. Back to the text.I can't resist pointing out that the first sentence contains a very obvious typo: "This document contains a substantial amount outstanding photographic evidence." Plainly this proves nothing about anything, though it is symptomatic of a general inability to spell (and I don't just mean "it[']s" - we are introduced to an Italian mathematician called Fabionacci [sic]), which in turn suggests a general lack of care.Anyway, the meat of this page... Several years ago, astronomer Gerald S. Hawkins, former Chairman of the astronomy department at Boston University, noticed that some of the most visually striking of the crop-circle patterns embodied geometric theorems that express specific numerical relationships among the areas of various circles, triangles, and other shapes making up the patterns (Science News: 2/1/92, p. 76). In one case, for example, an equilateral triangle fitted snugly between an outer and an inner circle. It turns out that the area of the outer circle is precisely four times that of the inner circle.ABC is an equilateral triangle, P is the mid-point of one side, so AB = 2 PB; angle BPO is 90 degrees. Triangles APB and BPO are similar (angles are the same), so BO = 2 PO. QED.Incidentally, by drawing the circles first, this diagram was extremely simple to generate, contrary to the flim-flam on the left.Sketch of proof that radii are in ratio 1:2Never mind whether the term "theorem" is appropriate, the result here seems to be that the area ratio is precisely 4:1. Or more simply, the radii are in the ratio 2:1. Indeed. To put it another way, if you make two circles, one inside the other and of half the radius, then an equilateral triangle fits between them. (Of course it does: from the centre of an equilateral triangle to a vertex is twice the distance to the mid-point of a side, as the figure on the right shows.)It's unclear to me whether a crop circle has appeared with the pattern of this "Theorem" including the triangle, or simply one in which two circles had radii in the ratio 1:2. Three other patterns also displayed exact numerical relationships, all of them involving a diatonic ratio, the simple whole-number ratios that determine a scale of musical notes. "These designs demonstrate the remarkable mathematical ability of their creators," Hawkins comments.Theorem I is a bit mysterious - it looks to me as though the figure is wrong; Theorems III and IV have two concentric circles with a square and a regular hexagon fitting between them. It's trivial in these cases to work out the ratio of the radii: root(2):1 and 2:root(3). But Mr Hawkins prefers to square these values, getting 2:1 and 4:3. Fine, though one has to ask why he squares them?(Presumably it couldn't simply be because that gives him what he wants?) And it's not clear to me quite what the "remarkable mathematical ability" is? Is it the ability to construct shapes with ruler and compasses? Hawkins found that he could use the principles of Euclidean geometry to prove four theorems derived from the relationships among the areas depicted in these patterns. ...The web page claims that the "fifth theorem" has been published by the American National Council of Teachers of Mathematics: I sent an email request for the written proof, but this turned into a wild goose chase. I've tried asking again. What is most surprising is that all geometries give diatonic (musical) ratios. Never before have geometric theorems been linked with music.The claim that XYZ has "never been thought of" is easy to make. One would not think though that people would make claims quite so easy to refute. On the shelf behind me is another highly recommended book:"The music of the spheres", by Jamie James, which tells us that geometric [not-quite-theorems, are they] have been being linked with music since, um, gosh, about 2500 years ago.On this page there are a couple of "glossary" links, one for "geometric theorems". This is headed "Definitions", and the first definition reads as follows: A Crop geometry is an unembellished diagram with rotational symmetry, which contains a theorem that can be proved by Euclidean logic and constructions, and generally leads to a pair or more of diatonic ratios.This strikes me as, at the very least, very sloppily written.Does it mean that the diagram includes a construction which can be used to prove some result -- of some generality? of no generality at all? He doesn't define "embellished". The other definitions are equally obscure. This is not mathematical writing.My opinion is (and this is actually unrelated to the wider issue of crop circles)that this page is close to vacuous. So what do I make of the bits I can't make much of? Perhaps all the stuff about bending stalks is where the Truth lies hidden? Should I go on to analyse the page this leads to about "The relationship between diatonic ratios, crop circles and the Society for Psychic Research"??(Posted to the PandA mailing list, 24 Mar 1998)Want to know more?This interview with Gerald Hawkins on a New Age website is quite funny.
Symmetry - Themes For An Imagina
Download Zip: https://tinourl.com/2vzYkn
2ff7e9595c
Comments