What is a Structured Polyhedron?

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.