# 3D analogue of A124278

Jonathan Post jvospost3 at gmail.com
Sat Oct 20 22:46:32 CEST 2007

```Did we ever discuss what polyhedra have every face a polygon from
A124278? Or A124287? Or which polyhedra have Hamiltonian paths with
nondecreasing edge lengths?

I know that there is a literature on polyhedra all of whose edges are
of integer lengths.  Doesn't that intersect the puzzles behind the two
sequences cited, or the 3-D generalizations on which I so vaguely
speculate?

To begin with, we know how many (not necessarily regular or uniform)
convex polyhedra there are with F faces, from tetrahedra on up.

1 Tetrahedron, 2 Pentahedra, 7 Hexahedra, 34( ?) Heptahedra, 257
Octahedra, 2606 topologically distinct convex Nonahedra (corresponding
to the 2606 nonisomorphic nonahedral graphs), at least 7 Decahedra
(Mathworld doesn't say how many total). Is 1, 2, 7, 34, 257, 2606 a
sequence of interest? A000944 is from the same question, but
enumerated by vertex count.

Or should we restrict ourselves to convex polyhedra, and ignore such
gems as the Szilassi polyhedron,  a concave heptahedron that contains
a hole?

See, for instance, the table linking to examples for small F at
http://mathworld.wolfram.com/Polyhedron.html

Which of these can have all edge lengths integral?  Even without
making restrictions, it's already a hard problem.

On 1/19/07, Jonathan Post <jvospost3 at gmail.com> wrote:
> What is the 3D analogue of A124278?  We might describe it as:
> "Triangle of the number of nondegenerate k-polyhedra having total
> edge-length n and whose edge lengths are nondecreasing on each face."
> or equivalently: "Triangle of the "number of nondegenerate k-faced
> polyhedra having total edge-length n and whose faces are polygons with
> nondecreasing sides."
>
> A completely different 3D analogue of A124278 would be: "Triangle of
> the number of nondegenerate k-polyhedra having total area n."
>
> Proper phrasing of these definition, and corresponding formulae, would
> generalize first to 4D analogues, and then higher dimensions.
>

```