This property is even stronger. Not only do I just have to look a distance proportional to the diameter to find another example, but those examples are arranged quasiperiodically, like a tiling on a larger scale. I recommend looking at the textbook model of a 1D quasicrystal, obtained by selecting points on an infinite square lattice that fall within the strip spanned by one square and having irrational slope. These tilings are comprised of sequences of line segments of two lengths. As a "local pattern” you can take a particular finite subsequence that appears in one quasicrystal. Using the strip construction it’s easy to identify all the other instances of that same subsequence, and you find that they form another 1D quasicrystal (whose density, in particular, is especially easy to compute). In higher dimensions you have to be careful because the theorem is not true unless you avoid special cases. Here it’s best to think of not just one tiling but tilings (plural) as set-valued functions. For example, 2D Penrose is really P(z), a set of points (tile vertices) parameterized by a point z in R^2. If we exclude a countable set in R^2 (the projection of a lattice in R^4), then all the P(z) are equivalent by translation. Several years ago I wrote a paper where I looked at the best behaved P(z) possible, where the vertices are analytic functions of z. The exceptional z are branch points that, not surprisingly, are connected with the symmetry of the quasicrystal. I still think this is the best approach for understanding those exceptional patterns. http://front.math.ucdavis.edu/9904.5016 On May 20, 2014, at 8:18 AM, James Propp <jamespropp@gmail.com> wrote:
Can anyone provide a reference for Penrose's "local isomorphism theorem", asserting that every Penrose tiling looks like every other?
I seem to recall hearing a strong version of this stated once, to the effect that if you have a pattern of diameter n in one Penrose tiling, and you look for that pattern in a second Penrose tiling, you won't have to travel further than distance 2n.
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