Sunday, March 08, 2009

Penrose tiles: five dimensions



Click the picture to zoom in.

I just rewrote a 1993 program of mine from Turbo Pascal to Mathematica 7 (and streamlined it). The quasiperiodic tiles due to Penrose can be obtained by projecting certain unit two-dimensional squares with integer coordinates of all vertices - those that belong to a certain "strip" - from a five-dimensional space into a two-dimensional subplane inside the strip (the plane is an eigenspace under the cyclic permutations of coordinates).

I knew the construction from my favorite diploma thesis advisor in Prague, Dr Miloš Zahradník, and it was included in our linear algebra textbook (EN). It was a lot of fun to prove that this aperiodic tiling covers the whole plane even though it is composed out of two types of diamonds only (with internal angles 72° and 108°; or 36° and 144°, and with the uniform length edges).
Download the Mathematica notebook
... (or PDF preview)
The Mathematica version of the program seems to be much more comprehensible than the Turbo Pascal edition.


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