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In my work on lattice point enumeration of polytopes, I stumbled upon the following sequence: \begin{eqnarray} 1, 120, 579, 1600, 3405, 6216, 10255, 15744, 22905, 31960, 43131, ... \end{eqnarray} which counts the Structured great rhombicosidodecahedral numbers (A100145) by the formula \begin{eqnarray} a(n)=\tfrac{1}{6} (222 n^3-312 n^2+96 n). \end{eqnarray} Such numbers fall into the category of figurate numbers, which count the number of points in a sequence of similar discrete geometric shapes. For example, the triangular and square numbers bear their names because they count the dots arranged in a sequence of triangular $(1,3,6,10,...)$ and square $(1,4,9,16,...)$ configurations. One generalizes these to higher dimensional regular polyhedral numbers like tetrahedral (A000292) or dodecahedral (A006566) numbers, for instance. These numbers are always enumerated by $\mathbb{Q}$-polynomials of degree $n$, where $n$ is the dimension of the polyhedron.

For the sequence above, the author gives the following description:

Structured polyhedral numbers are a type of figurate polyhedral numbers. Structurate polyhedra differ from regular figurate polyhedra by having appropriate figurate polygonal faces at any iteration, i.e. a regular truncated octahedron, n=2, would have 7 points on its hexagonal faces, whereas a structured truncated octahedron, n=2, would have 6 points - just as a hexagon, n=2, would have. Like regular figurate polygons, structured polyhedra seem to originate at a vertex and since many polyhedra have different vertices (a pentagonal diamond has 2 "polar" vertices with 5 adjacent vertices and 5 "equatorial" vertices with 4 adjacent vertices), these polyhedra have multiple structured number sequences, dependent on the "vertex structures" which are each equal to the one vertex itself plus its adjacent vertices. For polystructurate polyhedra the notation, structured polyhedra (vertex structure x) is used to differentiate between alternate vertices, where VS stands for vertex structure.

At first read, this doesn't make any sense. I thought the regular truncated octahedron had 6 vertices at each hexagonal face, not 7 as the author claims. (I know that this sequence isn't bogus because I can generate it in a completely different context, that of computing the cohomology and geometric genera in a singularity theory problem.)

Can anyone make sense of this and help me understand the difference between regular and structured polyhedra?

Update (4-1-11): I emailed the author of the entry on OEIS and never heard back from him. I think the responsibility now lies with us to figure this out.

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The most important distinctions to understand what the author meant in his description by "structured" figurate polyhedra as opposed to "regular" is between vertices and points and between "from an edge or a vertex" and "centered".

You wrote: ''I thought the regular truncated octahedron had 6 vertices at each hexagonal face, not 7 as the author claims.''

Precisely one of the simplest examples in 2D is the hexagonal numbers. A hexagon has 6 vertices, but when you produce a figurate number diagram of it, it has :

  • 1, 6, 12, 18, ... (A008458) points if you fill only the edges,

  • 1, 6, 15, 28, ... (A000384) from greek tradition if you grow hexagons as embracing smaller ones starting from a vertex (see illustration of classical hexagonal numbers or

  • 1, 7, 19, 37, 61, ... (A003215) points or circles or dots if you try to fill uniformly the greater hexagon by centered smaller ones. You could call this arrangement "regular" because the surface of the polygon is uniformly covered by points.

In his series of sequences in the OEIS, the author decided to use "structured" faces as opposed to "regular" (he should perhaps have said "centered" as in "centered hexagonal numbers"). There is no difference between "regular" and "structured" for triangular and square faces (because each growth step covers the new surface regularly), but there are for hexagonal ones (and there is a lot of troubles with pentagons).

It explains the comment of the author that there can be several sequences for the same basic geometric shape in certain cases, depending on the arrangement of the growth vertices of reference on different faces.


PS: I am one of the editors of the OEIS and I invite you to submit any additions, correction, comments, links and references to any of the 43 sequences James Record added to the encyclopedia. We are are particularly fond of alternate interpretations of sequences and links to the mathematical research literature.

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I'd followed as much as you described, trying to work specifically with the truncated tetrahedron, but I couldn't figure out what to do with the points in the interior of the solid to get the numbers for the "structured truncated tetrahedron" sequence. –  Isaac Mar 31 '11 at 15:26
    
I suppose the points appear only on the boundary. Am I incorrect? –  user02138 Mar 31 '11 at 22:56
    
Your question is in fact two questions: boundary of faces ? boundary of the polyhedron ? –  ogerard Apr 1 '11 at 6:59
    
(following of comment above) In fact, the two are possible independently and I have first to check what James Record did. As I wrote I believe that at least, the faces of his structured polyhedra contained the "structural" points (traces of smaller polygons). He may have counted in inner "structural" points (traces of smaller polyhedron) -- unfolding them in 4D would make things clearer. Whatever his choice (I will come back about that), it will certainly suggests new sequences to be added to the OEIS and a lot of clarification. –  ogerard Apr 1 '11 at 7:09
    
@Ogerard: Can you construct one sequence (as an example) of structured polyhedra from your post? It would be illuminating to see it worked out. –  user02138 Apr 1 '11 at 23:26

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