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 subtiling
(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
cutout plane region (by cutout 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 cutout
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 cutout (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 cutout, all filled up.
(c) Take the cutout if it is somewhere near a rectangle (or maybe even trapezoidal or other) shape on its
borders, and glue the edges of the cutout together to form a rolled up cylinder (or preferably, a torus) such
that the edge rules are still abided with at the cutout boundary when matching the unstretched cutout region.
(d) cut the cutout (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 nonisomorphic cutouts that can be
produced by the original cutout shape. A cutout 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 cutout 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
cutout 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).
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