For substitution tilings, there are many ways to derive a tree structure from the rules themselves, and I think there are canonical encodings in the topological theory, see for example Lorenzo Sadun “Topology of Tiling Spaces”. The half-hex tiling, for example, has the topology of a Quadtree. You may have already seen that quadtrees are sometimes used to describe snowflake growth, for image compression, or possibly for image scrambling. I also searched OEIS, and found: https://oeis.org/search?q=Penrose+coordination+&language=english&go=Search There are three dissimilar entries for 5-fold coordination sequences, but only one 5-fold fixed point of the Penrose tiling. It alternates with period 2, between sun and star. This explains two of the three entries, what about the third?? https://oeis.org/A302176/a302176_1.png The vertex at n=2 appears to be illegal by Penrose’s matching rules, so I don’t know why the name says “Penrose tiling”. Maybe a comment should he added saying: “This is not a Penrose Tiling”? And if the rules don’t hold, how is the pattern expanded? The reference is paywalled, so that is not helping... —Brad
On Jan 6, 2020, at 3:02 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Has anybody considered coordination trees (as it were) for aperiodic tilings (sic --- prefer quasi-crystallographic!), such as planar Penrose rhombs and its generalisations to solid honeycombs?
I gather that these might be of interest to crystallographers, without understanding details of the applications.
WFL
On 1/6/20, Tom Karzes <karzes@sonic.net> wrote: Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed.
Tom
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