Polytopes in 4D

[franktaw at netscape.net]

http://www.research.att.com/~njas/sequences/A000944 is the number of 3D 
polytopes with n vertices (or equivalently, with n faces), and 
http://www.research.att.com/~njas/sequences/A002840 is the number with 
n edges.

What about the equivalent questions in 4D?  How many 4D polytopes are 
there with n verices (equivalently, n polyhedral faces); and how many 
with n edges (equivalently, n polygonal components)?  The minimums, of 
course, are 5 vertices and 10 edges for the 4D simplex, but beyond that 
I'm not sure how to proceed.

The generalization to 5D, etc., is obvious, but I think the problem in 
4D is hard enough to be getting on with.  I don't think there is a 
simple equivalent graph problem in 4D - I think you have to specify the 
polygonal components, not just the vertices and edges.  (In 3D, the 
problem is equivalent to finding 3-connected simple planar graphs.)

For the same reason, at least for the moment I am interested only in 
simple polytopes, not stellated ones.

Franklin T. Adams-Watters

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