There's actually quite a lot of info on the wiki page: http://en.wikipedia.org/wiki/Penrose_tiling The first set of tiles he found, "P1", has 6 tile types (consisting of three different pentagon types, i.e. with difference edge-matching rules, and three other shapes): http://upload.wikimedia.org/wikipedia/commons/8/8c/Penrose_Tiling_%28P1%29.s... The different pentagon edge-matching rules are indicated by different colors, but to meet the strict definition, these actually need to be enforced by adding bumps to the edges to prevent disallowed combinations. Tom Thane Plambeck writes:
In 1984 or so, somewhere in the UK, perhaps Nottingham University, or maybe Leeds, I attended a lecture by Roger Penrose in which he explained how he first found his aperiodic tilings.
He essentially started with Robert Munafo's two diagrams, but then, just as Tom Karzes suggests below, he went on to describe how he struggled to reorganize/regroup the iteration of the diagrams to obtain only finitely many *sized* tile types. The first "success" involved 5 or 6 tiles I think, one of which he called the "witch's hat."
I think the "kite and dart" (ie, two tile) tilings came later.
I hesitate to send this because I'm sure others know this history much better than I do.
On Sun, Sep 5, 2010 at 9:41 PM, Tom Karzes <karzes@sonic.net> wrote:
Robert, I believe the distinction being made is this: For Ammann tilings, Penrose tilings, etc., the substitution rules, when applied properly, always result in a fixed number of different tiles. For example, with a Penrose tiling, you never need more than 2 tile types, regardless or how deeply you carry out the expansion. Or to put it another way, you can generate arbitrarily large tilings without ever needing more than 2 different tile types. This is because the tile sizes can always be made to "sync up", with as many old sizes being eliminated as new sizes being introduced.
For your rules to qualify, I believe you would need to be able to provide some number n, which is the maximum number of tile types (taking size into account) ever needed to genererate arbitrarily large tilings. For your substitution rules, it appears that n is infinite.
Tom
Robert Munafo writes: > As far as I can tell, my tiling is no different in that respect from > the Ammann tilings, for example. (See > http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a3) So, > like I said above, I want some book or website that explains how and > why they are different.
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