Polytopes in 4D

[Jonathan Post]

Frank:

you might start with George Hart's wonderful domain
http://www.georgehart.com/

Also:
"Uniform Polytopes in For Dimensions" "This is the world's only website that
tabulates all the convex uniform (i.e., Platonic and Archimedean)
polychora(that is, four-dimensional polytopes), and until Norman W.
Johnson's book
Uniform Polytopesis published by Cambridge University Press, it remains the
only place in the world where you can find this information!"
http://members.aol.com/Polycell/uniform.html

"Convex polytopes and enumeration"
Rodica Simion. Department of Mathematics, The George Washington University
linkinghub.elsevier.com/retrieve/pii/S0196885896905059
Advances in Applied Mathematics, Volume 18, Number 2, February 1997, pp.
149-180(32)

www.admin.ias.edu/ma/2004/program/Parkcity.pdf

The enumeration of four-dimensional polytopes. Discrete Mathematics archive
Volume 91 , Issue 1 (August 1991)

Some questions are related to the facial structure of polytopes, for
example, the enumeration of faces by dimension...
www.math.cornell.edu/People/Faculty/billera.html

Facets and Vertices of Metric Polytopes for n<10. (computation performed
using an orbitwise vertex-enumeration algorithm)
www.is.titech.ac.jp/~deza/metric.html

The complete enumeration of the $4$-polytopes and $3$-spheres with eight
vertices. Source: Pacific J. Math. 117, no. 1 (1985)
projecteuclid.org/Dienst/UI/1.0/Display/euclid.pjm/1102706924

Combinatorial face enumeration in convex polytopes.
Computational Geometry, 4:191-198, 1994. K. Fukuda.
Komei Fukuda's Homepage, McGill University, *...*
www.ifor.math.ethz.ch/~fukuda/polyfaq/node43.html

Combinatorial n-spheres and simplicial complexes are equivalent by stellar
subdivisions to the boundary of the (n+1) -simplex.
stinet.dtic.mil/oai/oai?&verb=getRecord&metadataPrefix=html&identifier=AD0689375


On 12/26/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> 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
>
> ________________________________________________________________________
> Check Out the new free AIM(R) Mail -- 2 GB of storage and
> industry-leading spam and email virus protection.
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20061226/f9544c97/attachment-0002.htm>