I just sent Brad Klee a copy of the article that A302176 is based on, and I'll be happy to send a copy to anyone else who is interested. This has to do with a coordination sequence for a vertex in a certain Penrose tiling Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Jan 6, 2020 at 5:53 PM Brad Klee <bradklee@gmail.com> wrote:
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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