6,213 research outputs found
The triangulations of the 3-sphere with up to 8 vertices
AbstractThe different combinatorial types of triangulations of the 3-sphere with up to 8 vertices are determined. Using similar methods we show that one cannot always preassign the shape of a facet of a 4-polytope
Uniform Infinite Planar Triangulations
The existence of the weak limit as n --> infinity of the uniform measure on
rooted triangulations of the sphere with n vertices is proved. Some properties
of the limit are studied. In particular, the limit is a probability measure on
random triangulations of the plane.Comment: 36 pages, 4 figures; Journal revised versio
Combinatorial 3-manifolds with 10 vertices
We give a complete enumeration of all combinatorial 3-manifolds with 10
vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as
well as 518 vertex-minimal triangulations of the sphere product
and 615 triangulations of the twisted sphere product S^2_\times_S^1.
All the 3-spheres with up to 10 vertices are shellable, but there are 29
vertex-minimal non-shellable 3-balls with 9 vertices.Comment: 9 pages, minor revisions, to appear in Beitr. Algebra Geo
Face pairing graphs and 3-manifold enumeration
The face pairing graph of a 3-manifold triangulation is a 4-valent graph
denoting which tetrahedron faces are identified with which others. We present a
series of properties that must be satisfied by the face pairing graph of a
closed minimal P^2-irreducible triangulation. In addition we present
constraints upon the combinatorial structure of such a triangulation that can
be deduced from its face pairing graph. These results are then applied to the
enumeration of closed minimal P^2-irreducible 3-manifold triangulations,
leading to a significant improvement in the performance of the enumeration
algorithm. Results are offered for both orientable and non-orientable
triangulations.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the
final theorem to the non-orientable case; v3: fixed a flaw in the proof of
the conical face lemm
There are 174 Subdivisions of the Hexahedron into Tetrahedra
This article answers an important theoretical question: How many different
subdivisions of the hexahedron into tetrahedra are there? It is well known that
the cube has five subdivisions into 6 tetrahedra and one subdivision into 5
tetrahedra. However, all hexahedra are not cubes and moving the vertex
positions increases the number of subdivisions. Recent hexahedral dominant
meshing methods try to take these configurations into account for combining
tetrahedra into hexahedra, but fail to enumerate them all: they use only a set
of 10 subdivisions among the 174 we found in this article.
The enumeration of these 174 subdivisions of the hexahedron into tetrahedra
is our combinatorial result. Each of the 174 subdivisions has between 5 and 15
tetrahedra and is actually a class of 2 to 48 equivalent instances which are
identical up to vertex relabeling. We further show that exactly 171 of these
subdivisions have a geometrical realization, i.e. there exist coordinates of
the eight hexahedron vertices in a three-dimensional space such that the
geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for
these configurations and show in particular subdivisions of hexahedra with 15
tetrahedra that have a strictly positive Jacobian
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