This isn't really a tiling: you can take nearly any shape and "tile" the plane with it in the way you describe--given any unfilled region, just scale down your shape until it fits and recurse. See the Apollonian gasket, for example. On Sun, Sep 5, 2010 at 1:52 PM, Robert Munafo <mrob27@gmail.com> wrote:
I have a recursive substitution-rule to tile the plane, which seems simple enough but nevertheless does not seem to be discussed in the normal places (like the tilings encyclopedia, http://tilings.math.uni-bielefeld.de/substitution_rules). Google searches haven't helped either (example: an Image search with the keywords: tiling regular pentagon triangle).
The rules for my tiling can be seen here:
http://mrob.com/pub/math/images/penta-tiling.jpg
I consider this a "simple" and "obvious" tiling because there are only two rules, and each rule tries to maximize the size of the pentagon(s) on the right-hand side. However, it is "non-simple" in the sense that, with repeated applications of the substitution rules, increasingly many different sizes of triangles and pentagons are produced.
- Robert
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