1,990 research outputs found
Fat 4-polytopes and fatter 3-spheres
We introduce the fatness parameter of a 4-dimensional polytope P, defined as
\phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in
4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes
bounded?
We describe and analyze a hyperbolic geometry construction that produces
4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of
2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite
covering spaces of surfaces, we show that fatness is not bounded for the more
general class of strongly regular CW decompositions of the 3-sphere.Comment: 12 pages, 12 figures. This version has minor changes proposed by the
second refere
The -Construction for Lattices, Spheres and Polytopes
We describe and analyze a new construction that produces new Eulerian
lattices from old ones. It specializes to a construction that produces new
strongly regular cellular spheres (whose face lattices are Eulerian). The
construction does not always specialize to convex polytopes; however, in a
number of cases where we can realize it, it produces interesting classes of
polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple
4-polytopes, as requested by Eppstein, Kuperberg and Ziegler. We also construct
for each an infinite family of -simplicial 2-simple
-polytopes, thus solving a problem of Gr\"unbaum.Comment: 21 pages, many figure
Six topics on inscribable polytopes
Inscribability of polytopes is a classic subject but also a lively research
area nowadays. We illustrate this with a selection of well-known results and
recent developments on six particular topics related to inscribable polytopes.
Along the way we collect a list of (new and old) open questions.Comment: 11 page
Non-projectability of polytope skeleta
We investigate necessary conditions for the existence of projections of
polytopes that preserve full k-skeleta. More precisely, given the combinatorics
of a polytope and the dimension e of the target space, what are obstructions to
the existence of a geometric realization of a polytope with the given
combinatorial type such that a linear projection to e-space strictly preserves
the k-skeleton. Building on the work of Sanyal (2009), we develop a general
framework to calculate obstructions to the existence of such realizations using
topological combinatorics. Our obstructions take the form of graph colorings
and linear integer programs. We focus on polytopes of product type and
calculate the obstructions for products of polygons, products of simplices, and
wedge products of polytopes. Our results show the limitations of constructions
for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge
product surfaces of R\"orig & Ziegler (2009) and complement their results.Comment: 18 pages, 2 figure
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