2,542 research outputs found
Tropical Convexity
The notions of convexity and convex polytopes are introduced in the setting
of tropical geometry. Combinatorial types of tropical polytopes are shown to be
in bijection with regular triangulations of products of two simplices.
Applications to phylogenetic trees are discussed.
Theorem 29 and Corollary 30 in the paper, relating tropical polytopes to
injective hulls, are incorrect. See the erratum at
http://www.math.uiuc.edu/documenta/vol-09/vol-09-eng.html .Comment: 20 pages, 6 figure
Combinatorial Space Tiling
The present article studies combinatorial tilings of Euclidean or spherical
spaces by polytopes, serving two main purposes: first, to survey some of the
main developments in combinatorial space tiling; and second, to highlight some
new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science
Regular Incidence Complexes, Polytopes, and C-Groups
Regular incidence complexes are combinatorial incidence structures
generalizing regular convex polytopes, regular complex polytopes, various types
of incidence geometries, and many other highly symmetric objects. The special
case of abstract regular polytopes has been well-studied. The paper describes
the combinatorial structure of a regular incidence complex in terms of a system
of distinguished generating subgroups of its automorphism group or a
flag-transitive subgroup. Then the groups admitting a flag-transitive action on
an incidence complex are characterized as generalized string C-groups. Further,
extensions of regular incidence complexes are studied, and certain incidence
complexes particularly close to abstract polytopes, called abstract polytope
complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder,
A. Deza, and A. Ivic Weiss (eds), Springe
Supersymmetry and Polytopes
We make an imaginative comparison between the Minimal Supersymmetric Standard
Model and the 24-cell polytope in four dimensions, the Octacube.Comment: Presented to the Workshop on Geometry and Physics: Supersymmetry.
Bilbao, Spain. May 200
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