98 research outputs found
Extremal properties for dissections of convex 3-polytopes
A dissection of a convex d-polytope is a partition of the polytope into
d-simplices whose vertices are among the vertices of the polytope.
Triangulations are dissections that have the additional property that the set
of all its simplices forms a simplicial complex. The size of a dissection is
the number of d-simplices it contains. This paper compares triangulations of
maximal size with dissections of maximal size. We also exhibit lower and upper
bounds for the size of dissections of a 3-polytope and analyze extremal size
triangulations for specific non-simplicial polytopes: prisms, antiprisms,
Archimedean solids, and combinatorial d-cubes.Comment: 19 page
The flip-graph of the 4-dimensional cube is connected
Flip-graph connectedness is established here for the vertex set of the
4-dimensional cube. It is found as a consequence that this vertex set has 92
487 256 triangulations, partitioned into 247 451 symmetry classes.Comment: 20 pages, 3 figures, revised proofs and notation
Counting a black hole in Lorentzian product triangulations
We take a step toward a nonperturbative gravitational path integral for
black-hole geometries by deriving an expression for the expansion rate of null
geodesic congruences in the approach of causal dynamical triangulations. We
propose to use the integrated expansion rate in building a quantum horizon
finder in the sum over spacetime geometries. It takes the form of a counting
formula for various types of discrete building blocks which differ in how they
focus and defocus light rays. In the course of the derivation, we introduce the
concept of a Lorentzian dynamical triangulation of product type, whose
applicability goes beyond that of describing black-hole configurations.Comment: 42 pages, 11 figure
The Flip-Graph of the 4-Dimensional Cube is Connected
Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92,487,256 triangulations, partitioned into 247,451 symmetry classe
Lower Bounds for Simplicial Covers and Triangulations of Cubes
We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their exterior faces
- …