In geometry, the Császár polyhedron (ˈt͡ʃaːsaːr) is a nonconvex toroidal polyhedron with 14 triangular faces.
This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph K_7 onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces.
The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals: every two vertices of the polygon are connected by an edge, so there is no line segment between two vertices that does not lie on the polyhedron boundary. That is, the vertices and edges of the Császár polyhedron form a complete graph.
The combinatorial description of this polyhedron has been described earlier by Möbius. Three additional different polyhedra of this type can be found in a paper by .
If the boundary of a polyhedron with v vertices forms a surface with h holes, in such a way that every pair of vertices is connected by an edge, it follows by some manipulation of the Euler characteristic that
This equation is satisfied for the tetrahedron with h = 0 and v = 4, and for the Császár polyhedron with h = 1 and v = 7. The next possible solution, h = 6 and v = 12, would correspond to a polyhedron with 44 faces and 66 edges, but it is not realizable as a polyhedron. It is not known whether such a polyhedron exists with a higher genus.
More generally, this equation can be satisfied only when v is congruent to 0, 3, 4, or 7 modulo 12.
The Császár polyhedron is named after Hungarian topologist Ákos Császár, who discovered it in 1949. The dual to the Császár polyhedron, the Szilassi polyhedron, was discovered later, in 1977, by Lajos Szilassi; it has 14 vertices, 21 edges, and seven hexagonal faces, each sharing an edge with every other face. Like the Császár polyhedron, the Szilassi polyhedron has the topology of a torus.
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In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by exactly two polygons, and at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link of the vertex.
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point.
For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with X is called the relaxation complexity rc(X). This parameter was introduced by Kaibel & Weltge (2015) and captures the ...
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EPFL2018
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We derive a new upper bound on the diameter of a polyhedron , where . The bound is polynomial in and the largest absolute value of a sub-determinant of , denoted by . More precisely, we show that the diameter of is bounded by . If is bounded, then we show ...