I recently determined some limits on the number of edges and faces of the CRF

polychora vertex figures.

There is a program named plantri (

http://cs.anu.edu.au/~bdm/plantri/) with

which you can enumerate all polyhedra in three dimensions. With the qualifiers

of plantri you can limit the number of edges of the faces and the number of

edges of the polyhedra, the degree of the vertices and so on.

If you compute all the solid angles of the Johnson polyhedra you find that the

maximal number of faces is 22 for the regular tetrahedron. However, we know

that no more than 20 tetrahedra fit together in 4d to form an icosahedron. The

maximal number of vertices is much more difficult to determine. It could in

principle go up to 30=(12-2)*3, if all vertices are simple. However, in the

end I obtained 13, and my believe is that it is 12 namely the icosahedron.

What other limits do we have? The dihedral angles of the Johnson polyhedra

determine the maximal degree of the vertices, i.e. the number of faces that

can meet at a vertex. The minimal diehdral angle is 31.7175 from the

pentagonal cupola which would lead to a degree of 11. But since the vertex is

asymmetric, the maximal number of tiles has to be even, namely 10. But we find

that there is no polyhedron which fits into the gap between two pentagonal

cupolas. Thus this vertex does not exist. I have enumerated all possible edge

configurations and found further 10-degree vertices, but it was possible to

show that none of them exists. In the end the maximal degree was found to be 9

for 4 configurations which I could not eliminate.

How else could one reduce the possible vertex figures? Well by computing the

total solid angle: The smallest triangle has solid angle 00.4387, the

quadrangle 0.08774, the pentagon 0.173235 in units of the surface of a

sphere. Thus #3*0.04387+#4*0.08774+#5*0.173235 < 1.0.

So the number of possible vertex figures of a certain type is lower than given

in the following table:

#vert 4 5 6 7 8 9 10 11 12 13

1 2 7 33 249 2473 29846 394498 5528006 79919023 <- sum

#face

4 1 1

5 2 1 1

6 7 1 2 2 2

7 25 2 7 9 5 2

8 149 2 11 39 55 35 7

9 944 8 71 248 379 235 3

10 ^ 5 76 590 1976 2930 389

11 | 38 748 5290 16401 11684 3

12 sum 14 558 8309 50226 109398 409

13 219 7776 91966 449409 15892

14 50 4442 106558 1008926 213923

15 1404 78684 1400693 1296312

16 233 36528 1282828 4157926

17 9714 780953 7698651

18 1249 306470 8609942

19 70454 5875223

20 7595 2384890

Obviously, this estimate is completely useless. But it shows that it is

impossible by starting from the combinatorical vertex figure to determine the

allowed vertex figures in 4d. Because each number means only a type! There are

more than 100 vertex figures of type (4,4), 44 of type (5,5), 32 of tpye (6,6)

and maybe several hundred of the biypyramidral type (5,6) with

(#vert,#faces). I have determined the latter but not yet fully analysed them.

Although I could show that there are no infinite series for the bipyramidal

case (an the triangular pyramidal case (6,5), there seem to be vertex figures

with polygons with up to 169 edges!

To my opinion, it would be interesting to list the combinatorial content of all vertex figures of all known CRF polyhedral. I started such work, but I have not yet finished the segmentochora. Compared to the above list one finds that there is a lot of space at the top. The maxmium vertex figure is still the icosaedron with 12 vertices and 20 faces.