OEIS/Tilings: Difference between revisions

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Created page with "===New format=== On May 02 2020 Brian Galbach wrote: Here's a new file with a very compact and simple code for each tiling. It has several sections separated by semicolons,..."
 
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===New format===
===Generating functions of coordination sequences ===
On May 02 2020 Brian Galbach wrote:
* Brian Galebach, ''[https://oeis.org/A250120/a250120.html k-uniform tilings (k <= 6) and their A-numbers]''
 
* Chaim Goodman-Strauss and N. J. A. Sloane, ''[https://doi.org/10.1107/S2053273318014481 A Coloring Book Approach to Finding Coordination Sequences]'', Acta Cryst. A75 (2019), 121-134, also [http://NeilSloane.com/doc/Cairo_final.pdf on NJAS's home page]. Also [http://arxiv.org/abs/1803.08530 arXiv:1803.08530].
Here's a new file with a very compact and simple code for each tiling. It has several sections separated by semicolons, which I will explain below.
* M.[ichael P.] Benson, ''Growth series of finite extensions of '''Z'''n are rational'', Invent. Math. 73 (1983), no. 2, 251–269. MR 714092
 
* Branko Grunbaum and Geoffrey C. Shephard, ''[https://www.jstor.org/stable/2689529?seq=4#metadata_info_tab_contents Tilings by Regular Polygons]'', Mathematics Magazine, Vol. 50, No. 5 (Nov., 1977), pp. 227-247
Each code starts with a single capital letter representing the uniformity of the tiling.  A is 1-uniform, B is 2-uniform, etc.
* Sean A. Irvine, ''[https://oeis.org/A008000/a008000_1.pdf Generating Functions for Coordination Sequences of Zeolites after Grosse-Kunstleve, Brunner, and Sloane]'' (with coefficients of many g.f.s)
 
* R. W. Grosse-Kunstleve, G. O. Brunner, and N. J. A. Sloane, [https://oeis.org/A005897/a005897.html ''Algebraic description of coordination sequences and exact topological densities for zeolites'']. Acta Cryst., A52:879–889, 1996.
Next, we have a letter representing the vertex type for each vertex class in the tiling. These are as follows:
* Ralf W. Grosse-Kunstleve, ''[https://oeis.org/A005897/a005897.html Zeolites, Frameworks, Coordination Sequences & Encyclopedia of Integer Sequences]'' (a005897.html), 1996
A: 3.3.3.3.3.3
* [https://commons.wikimedia.org/wiki/Wallpaper_group_diagrams Wallpaper group diagrams]
B: 3.3.3.3.6
* cf. also '''[[OEIS/coors|coors]]'''
C: 3.3.3.4.4
D: 3.3.4.3.4
E: 3.3.4.12
F: 3.3.6.6
G: 3.4.3.12
H: 3.4.4.6
I: 3.4.6.4
J: 3.6.3.6
K: 3.12.12
L: 4.4.4.4
M: 4.6.12
N: 4.8.8
O: 6.6.6
 
Following the vertex type designations, we have alternating sections:
 
The first section describes the classes connected to class A. The number of connections must of course be the same as the number of polygons in class A. And each connected class must be from A to the nth letter of the alphabet, where our tiling is n-uniform. (Important: Do not confuse the vertex type, which is A to O, with the vertex class, which is A, B, or C in a 3-uniform tiling, for example.)
 
The second section describes the orientation of each of those vertices connected to class A. The letter, whether it is uppercase or lowercase, gives us the edge of the connected vertex that is connected to the current edge. A or a is the first edge, B or b is the second edge, etc. And uppercase means that the direction of rotation of the connected vertex is the same as the current vertex, while lowercase means that the direction of rotation is reversed.
 
Then we repeat the above two sections for class B, class C, etc.
 
That's it.  Let's do an example just so this is clear. I'll pick a line pretty much at random. Okay, this one:
C;FJO;BABC;BBAA;AABB;CADC;ACC;DCB
 
So we have a 3-uniform tiling (C).  The three vertex classes have the vertex types 3.3.6.6, 3.6.3.6, and 6.6.6 (FJO).
 
Now let's look at vertex class A. It has connected to it vertex classes B, A, B, and C.  And the edges connected from each of those vertices are edge 2, edge 2, edge 1, and edge 1. And since these are all uppercase letters, there is no reversing of the direction of rotation relative to the direction of our central vertex.
 
Class B has connected to it vertex classes A, A, B, and B. And the connected edges are edge 3, edge 1, edge 4, and edge 3, and no direction reversal.
 
Finally, class C has connected to it vertex classes A, C, and C.  And the connected edges from each of those vertices are edge 4, edge 3, and edge 2.

Latest revision as of 20:39, 27 May 2020

Generating functions of coordination sequences

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