3,142 research outputs found
Hereditary Polytopes
Every regular polytope has the remarkable property that it inherits all
symmetries of each of its facets. This property distinguishes a natural class
of polytopes which are called hereditary. Regular polytopes are by definition
hereditary, but the other polytopes in this class are interesting, have
possible applications in modeling of structures, and have not been previously
investigated. This paper establishes the basic theory of hereditary polytopes,
focussing on the analysis and construction of hereditary polytopes with highly
symmetric faces.Comment: Discrete Geometry and Applications (eds. R.Connelly and A.Ivic
Weiss), Fields Institute Communications, (23 pp, to appear
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
Combinatorially two-orbit convex polytopes
Any convex polytope whose combinatorial automorphism group has two orbits on
the flags is isomorphic to one whose group of Euclidean symmetries has two
orbits on the flags (equivalently, to one whose automorphism group and symmetry
group coincide.) Hence, a combinatorially two-orbit convex polytope is
isomorphic to one of a known finite list, all of which are 3-dimensional: the
cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic
triacontahedron. The same is true of combinatorially two-orbit normal
face-to-face tilings by convex polytopes.Comment: 20 page
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
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