## 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).