3D analogue of A124278

Jonathan Post jvospost3 at gmail.com
Sun Oct 21 01:10:37 CEST 2007

I just used the old Math tools: paper, protractor, ruler, scissors,
Scotch tape.  I easily built an integer tetrahedron with two right
angle triangle faces.

Start with a (3,4,5) triangle.  Tape that to a (5, 12, 13) triangle
along the common edge of length 5.

Now, tape a (non-right) (4,13,14) triangle along the common edge of
length 4 with the (3,4,5) and a common edge of length 13 with the
(5,12,13).  Now we see the 4th triangle (trickier to tape the missing
trinagle in place, unless maybe one uses something more rigid than my
paper, perhaps cover stock or cardboard): it is a (3,12,14).

The edge lengths, sorted, are nicely 3,4,5,12,13,14.

It should be fairly straightforward (as the constraints to avoid
degeneracy are obvious) to compute a list of such possible integer
tetrahedra, ordered by the shortest edge, or by the sum of the lengths
of 6 edges.

It is slightly trickier to do this for the 2 topologically different
pentahedra, the 7 topologically distinct hexahedra, and so forth.
Surely Mathematica in the right hands, with a better brain than mine,
can do so?  This would, it seems to me, make several prtty sequences
from elementary Solid Geometry.

On 10/20/07, Jonathan Post <jvospost3 at gmail.com> wrote:
> 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.
> >

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