28,116 research outputs found
Spherical Tiling by 12 Congruent Pentagons
The tilings of the 2-dimensional sphere by congruent triangles have been
extensively studied, and the edge-to-edge tilings have been completely
classified. However, not much is known about the tilings by other congruent
polygons. In this paper, we classify the simplest case, which is the
edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We
find one major class allowing two independent continuous parameters and four
classes of isolated examples. The classification is done by first separately
classifying the combinatorial, edge length, and angle aspects, and then
combining the respective classifications together.Comment: 53 pages, 40 figures, spherical geometr
Half domination arrangements in regular and semi-regular tessellation type graphs
We study the problem of half-domination sets of vertices in vertex transitive
infinite graphs generated by regular or semi-regular tessellations of the
plane. In some cases, the results obtained are sharp and in the rest, we show
upper bounds for the average densities of vertices in half-domination sets.Comment: 14 pages, 12 figure
Enumerating Polytropes
Polytropes are both ordinary and tropical polytopes. We show that tropical
types of polytropes in are in bijection with cones of a
certain Gr\"{o}bner fan in restricted
to a small cone called the polytrope region. These in turn are indexed by
compatible sets of bipartite and triangle binomials. Geometrically, on the
polytrope region, is the refinement of two fans: the fan of
linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite
binomial fan. This gives two algorithms for enumerating tropical types of
polytropes: one via a general Gr\"obner fan software such as \textsf{gfan}, and
another via checking compatibility of systems of bipartite and triangle
binomials. We use these algorithms to compute types of full-dimensional
polytropes for , and maximal polytropes for .Comment: Improved exposition, fixed error in reporting the number maximal
polytropes for , fixed error in definition of bipartite binomial
Inflations of ideal triangulations
Starting with an ideal triangulation of the interior of a compact 3-manifold
M with boundary, no component of which is a 2-sphere, we provide a
construction, called an inflation of the ideal triangulation, to obtain a
strongly related triangulations of M itself. Besides a step-by-step algorithm
for such a construction, we provide examples of an inflation of the
two-tetrahedra ideal triangulation of the complement of the figure-eight knot
in the 3-sphere, giving a minimal triangulation, having ten tetrahedra, of the
figure-eight knot exterior. As another example, we provide an inflation of the
one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven
tetrahedra, of a nonorientable compact 3-manifold with Klein bottle boundary.
Several applications of inflations are discussed.Comment: 48 pages, 45 figure
Flag arrangements and triangulations of products of simplices
We investigate the line arrangement that results from intersecting d complete
flags in C^n. We give a combinatorial description of the matroid T_{n,d} that
keeps track of the linear dependence relations among these lines. We prove that
the bases of the matroid T_{n,3} characterize the triangles with holes which
can be tiled with unit rhombi. More generally, we provide evidence for a
conjectural connection between the matroid T_{n,d}, the triangulations of the
product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d
tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and
effective criterion to ensure the vanishing of many Schubert structure
constants in the flag manifold, and a new perspective on Billey and Vakil's
method for computing the non-vanishing ones.Comment: 39 pages, 12 figures, best viewed in colo
The Average-Case Area of Heilbronn-Type Triangles
From among triangles with vertices chosen from points in
the unit square, let be the one with the smallest area, and let be the
area of . Heilbronn's triangle problem asks for the maximum value assumed by
over all choices of points. We consider the average-case: If the
points are chosen independently and at random (with a uniform distribution),
then there exist positive constants and such that for all large enough values of , where is the expectation of
. Moreover, , with probability close to one. Our proof
uses the incompressibility method based on Kolmogorov complexity; it actually
determines the area of the smallest triangle for an arrangement in ``general
position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a
roll, {\em New Scientist}, November 6, 1999, 44--4
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