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Polyhedra of small order and their Hamiltonian properties
We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up to 12 vertices and up to 26 edges, simplical polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, and bipartite polyhedra with up to 22 vertices.The results of the enumeration were used to systematically search for certain minimal non-Hamiltonian polyhedra. In particular, the smallest polyhedra satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Basic Polyhedral Theory
This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of
Operations Research and Management Science) describing parts of the theory of
convex polyhedra that are particularly important for optimization. The topics
include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem,
faces of polyhedra, projections of polyhedra, integral polyhedra, total dual
integrality, and total unimodularity.Comment: 14 page
Minimal Bending Energies of Bilayer Polyhedra
Motivated by recent experiments on bilayer polyhedra composed of amphiphilic
molecules, we study the elastic bending energies of bilayer vesicles forming
polyhedral shapes. Allowing for segregation of excess amphiphiles along the
ridges of polyhedra, we find that bilayer polyhedra can indeed have lower
bending energies than spherical bilayer vesicles. However, our analysis also
implies that, contrary to what has been suggested on the basis of experiments,
the snub dodecahedron, rather than the icosahedron, generally represents the
energetically favorable shape of bilayer polyhedra
Triangular hyperbolic buildings
We construct triangular hyperbolic polyhedra whose links are generalized
4-gons. The universal cover of those polyhedra are hyperbolic buildings, which
appartments are hyperbolic planes tesselated by regular triangles with angles
. Moreover, the fundamental groups of the polyhedra acts simply
transitively on vertices of the buildings
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