Polytopes in 4D

[Jonathan Post]

Sorry, accidently omitted the best such book since Coxeter, available in a
new edition, with many tables useful for OEIS.  I borrowed a copy from
caltech's library, couldn't bear to return it for some time, hope to buy a
copy one of these days.

Grünbaum, Branko <http://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaum>, *Convex
polytopes*, New York & London:
Springer<http://en.wikipedia.org/wiki/Springer-Verlag>,
c2003. ISBN 0-387-00424-6<http://en.wikipedia.org/w/index.php?title=Special:Booksources&isbn=0387004246>.
(2nd edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.)

On 12/26/06, Jonathan Post <jvospost3 at gmail.com> wrote:
>
> 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<http://www.is.titech.ac.jp/%7Edeza/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<http://www.ifor.math.ethz.ch/%7Efukuda/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
> > <http://www.research.att.com/%7Enjas/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<http://www.research.att.com/%7Enjas/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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> >
>
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