[Note: I think the ">>" lines were written by Laura Helen, the ">" lines by Arthur Rubin, and then the unmarked lines by Arne Halvorsen; this was all one post, and not in the customary order. -- djr] ============================================================================== From: adh@cx.dnv.no (Arne Dehli Halvorsen) Newsgroups: sci.math Subject: Re: penrose tiles - and worse! Date: Wed, 23 Nov 1994 09:47:39 In article a_rubin@dsg4.dse.beckman.com (Arthur Rubin) writes: >In Laura Helen writes: Couple of additional comments: It is rather difficult to tile the plane with Penrose tiles from scratch, and for some time it was doubted that one could find strictly local rules that would enable automatic tiling of the plane. This, and the fact that icosahedral-symmetry crystals have been found, was taken (by Penrose) to mean that there had to be non-local, as yet unidentified laws of physics. (Roger Penrose: The Emperors' New Mind) Local rules have now been found, as far as I know. (Sorry, no reference) To make up a large tiled area, it is easier to start out with a recursive subdivision algorithm - given tiles A, B of size X, you can replace each A with n copies of A of size Y, m copies of B of size Y, and similar for each B. I think it is possible to make tile sets for any "near-symmetry" : 7, 9, 111, (but I can't prove it). Typical tile sets for 5-symmetry are: 1: kite and dart. (Take a parallellogram, angles A:72, B:108, C:72, D:108. Draw lines from B and from D at 36 degrees, so that they meet at E, forming angles of 216 degrees on one side, 144 on the other. Now you have two shapes. Make a few dozen copies and assemble them so they never form a parallellogram again. I *think* that is the right way. Anyway, certain sides are allowed to meet, others not. ). D-------------C /\ / / \ / / E. / / . / A-------------B 2 Two parallellograms - the one from point 1, and another with angles 36, 144, 36, 144. Same thing holds - you need to identify the rules - which sides get to meet which. Another thing - a recent issue of New Scientist showed a *worse* non-periodic tiling, in that no rotational symmetry could be found - the angles in the tiling are incommensurate (nice word) with 360 degrees (or 2pi rad). Subdivision algorithm is: B /| / | / | /---| /|\ | / | \ | / | \| A-------| - | C As well as a line from A to C, which I can't find a good ASCII art symbol for. This divides a triangle of sides 1, 2, sqrt(5) into 5 similar triangles, two of which are reflected. Could anyone tell me what a crystal based on this one would be like? Could it be as strong as or stronger than a crystal, but with an amorphous diffraction pattern? Grateful for info, Arne D Halvorsen >>I'm curious about Penrose tiles. They're a way of tiling the plane. I>>think 2 types of tiles are used. Each copy of a given tile type has the >>same environment, just as in a crystallographic space group. And, there >>is 5-fold symmetry in the tiling. However, the tiling is nonperiodic -- >>there are no space groups with 5-fold symmetry. I'm not sure if there >>is a "tiling" of 3D space with 5-fold symmetry in all 3 dimensions. >Correct. >>Crystals have actually been found which are arranged in Penrose tile >>patterns. They have diffraction patterns with 5-fold symmetry, something >>that will make most crystallographers faint. They're some sort of >>metal compounds. I'm not sure if these crystals are formed out of >>2D layers of Penrose tiles stacked up, or if they have some 5-fold >>symmetry in 3 dimensions. >I'm curious also. I've seen some of the papers, but I don't remember. >>What's the math behind Penrose tiles? Is there anything analogous to a >>space group? >The only thing that I see analogous to a "space group" is that any local >configuration of the tiles occurs infinitely often in the pattern, so you >could look as the "space group" as the limit (normally considered to be >empty) of the local isomorphism "groups", >-- >Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea >216-5888@mcimail.com 70707.453@compuserve.com arubin@pro-sol.cts.com (personal) >My opinions are my own, and do not represent those of my employer. >This space intentionally left blank.