The surest way clearly is a correct visualization and a mere counting. But that would be hard both for huge numbers as well as in dimensions beyond 3D.

The best way I would recomment are incidence matrices. Incidence matrices not in the way those were used in the past, i.e. by using mere 0-1-matrices between just 2 adjacent subdimensions, but those of all subelements. More easy, when used in symmetrical equivalence grouping. There al symmetry equivalent elements of any dimension have to be listed as rows respectively as columns. If available as Dynkin sub-symbols those could be provided as clue in addition, but the possibility of being describable as Dynkin symbol is not necessary here.

Just to provide an easy example. Consider THO, the tesseractihemioctachoron. It is a faceting of HEX (hexadecachoron = aerochoron), but uses hemifacets. I cant see asuitable Dynkin graph representation for THO. But this doesnt matter at all with respect to its incidence matrix. so we consider:

- Code: Select all
`vertices | 8 | 6 | 12 | 4 3`

-----------+---+----+----+----

edges | 2 | 24 | 4 | 2 2

-----------+---+----+----+----

triangles | 3 | 3 | 32 | 1 1

-----------+---+----+----+----

tetrahedra | 4 | 6 | 4 | 8 *

octahedra | 6 | 12 | 8 | * 4

As you can see, I note as matrix element m_i,j (i<>j) the number of j-elements incident to an i-element. For additional overview I guide the view by additional dimensional groupings using horizontal resp. vertical bording lines. The entries of the diagonal blocks are kind of special. If used in the same way as given so far those just would be Kronecker-deltas: non-equivalent elements surely would not be (completely co-)incident, while equivalent ones surely are. But we need the absolute counts of elements under the action of the used symmetry additionally. Thus I have chosen to display those absolute counts in that diagonal instead. The non-diagonal elements of the diagonal blocks, as mentioned above should be zero. But in order to depict that they never can be full-dimensional coincident I use an asterix instead.

This type of incidence matrix has some interesting properties. Those can be used to get all those numbers in a consistent way. (In fact, this is the true reason to outline all this under the topic of consideration!)

Firstly, the sub-diagonal entries of the matrix rows represent the total counts of the element represented there, i.e. if the incidence matrix of that thingy would be set up, they would be its diagonal elements.

Next, the super-diagonal entries of the matrix rows represent the total counts of the -figure elemnts of the there represented elements, i.e. of the vertex figure, edge figure etc. (Wendys "around")

The diagonal elements, as already said, are the total counts of the polytope under consideration.

Any of the above 3 count numbers are subject to the Euler equation: -1 + sum of vertex counts - sum of edge counts + ... +- 1 = 0 (at least as long as no holes etc. have to be considered)

You would have: m_i,j * m_i,i = m_j,j * m_j,i (for any i>j) - Note, that this relation form, which is rather easy to remember, makes use of having those absolute counts within the diagonal!

That is, having a figure in consideration, generally a lot of those numbers are more or less obvious - or at least: more easy to get. Others are not so obvious - as your edge counts in this thread-topic. But all those useful relations could help to fiddle them out!

--- rk

PS: more could be found at:

http://bendwavy.org/klitzing/explain/incmat.htm