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 Author Comment/Response Kai G. Gauer 02/22/02 10:54pm I'm wondering what that easiest approach would be to creating a graphing function (in version 3.0) which would plot a (subset of a) Penrose tiling, based on parameters such as: number of lines of length a and/or of b, and/or number of tiles of shape kite and or dart, or the respective ratios between the number of lengths of one size to another, and/or to number of shapes to one size compared with another category, for some FINITE sub-tiling (ie there might be holes within the subtiling, etc). Are there any suggestions of algorithms to implement to allow for : given k kites and d darts found within a tiling (or analagously, for numbers of lines of particular length, etc), that we can: (a) find a way to count the number of different ways of arranging them within a cut-out plane region (by cut-out region of penrose space, I basically mean a patch removed from the Penrose tiling as found above, but in this case, only the exterior border of the cut-out means anything - also, for later on, I would like to preserve the notion that when replacing the tiles back in, that they still abide with the normal touching rules) such that (k+d) (or such that k kites EXACTLY, and d darts exactly) fit back into the cut-out (hte first case might have k1=/=k kites and d1 darts whose are still adds up to (k+d) in total area). (b) consruct a table that lets me pick which one to plot, given a number of sufficiently specified parameters, and, how to tell the function how to plot the chosen cut-out, all filled up. (c) Take the cut-out if it is somewhere near a rectangle (or maybe even trapezoidal or other) shape on its borders, and glue the edges of the cut-out together to form a rolled up cylinder (or preferably, a torus) such that the edge rules are still abided with at the cut-out boundary when matching the unstretched cut-out region. (d) cut the cut-out (not necessarily at the same place, but whenever cutting out, at least staying only on lines of the Penrose tile matchings of two or more tiles), and count the total number of non-isomorphic cut-outs that can be produced by the original cut-out shape. A cut-out should always be able to fold out flat onto the plane. (e) Analyze for which values of (k+d) a torus can even be constructed, and how thin a torus can be made (relative to the unit length of the shortest type of penrose border such that the circumference of one or both of the parameters is kept to a minimum). Start with small (k+d) total tiles, and give a way of extending to higher numbers of total numbers of tiles. Generalize, if possible, to your favourite tiling, and then start tiling tori without the usual Mathematica ideas of just using best fit close enough to square boxes or trapezoids to do so. I'd prfer to just stick with plot to do the job, since we're only plotting straight lines of fixed length, and angles of fixed degree of opening up. I'm mainly stuck on the 1st step, however, of convincing Mathematica that there probably exists a nice quick and recursive algortihm that would paste out a random cut-out of size (d+k) to just start to work with. Also, the matching and cutting rules would be needed to added to this plot algorithm later on, so bear that in mind. Any ideas would be appreciated. I wonder what it might be like to play something like checkers on such a type of cut-out board, subjected to appropriate rule changes of move etc. I guess that I'm sort of sickof just watching games like Minesweeper and Life spin out their rules in a square or triangular or hexagonal tiling space as a subset of the plane (and not a torus or cylinder). URL: ,

 Subject (listing for 'Penrose Tilings (and other non-periodic Tilings)') Author Date Posted Penrose Tilings (and other non-periodic Tilings) Kai G. Gauer 02/22/02 10:54pm Re: Penrose Tilings (and other non-periodic Til... Casey Mann 02/25/02 10:45pm Re: Penrose Tilings (and other non-periodic Til... Kai G. Gauer 03/04/02 03:54am Re: Penrose Tilings (and other non-periodic Til... Casey Mann 03/05/02 2:24pm
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