In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only...

pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of) simplicial polytopes this...

line. A simplicial polytope is a polytope whose facets are all simplices (plural of simplex). For example, every polygon is a simplicial polytope. The Euler...

Simplicial complex Simplicial homology Simplicial set Spectrahedron Ternary plot Elte, E.L. (2006) [1912]. "IV. five dimensional semiregular polytope"...

called simplicial if each of its regions is a simplex, i.e. in an n-dimensional space each region has n+1 vertices. The dual of a simplicial polytope is called...

defines a polytope as a set of points that admits a simplicial decomposition. In this definition, a polytope is the union of finitely many simplices, with the...

1)-simplex. Simple polytopes are topologically dual to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or...

between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905...

face of another simplex of the complex. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Bridge...

simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained in this way. One important...

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional...

may be defined for simplicial polytopes and also in the spherical and hyperbolic geometries. In the special case when the polytope is a quadrilateral...

the boundary Δ(n,d) of the cyclic polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n...

which correspond to the vertices of the polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the...

version of barycentric subdivision, it is not necessary for the polytope to form a simplicial complex: it can have faces that are not simplices. This is the...

algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different...

decrease), there are higher-dimensional polytopes for which this is not true. For simplicial polytopes (polytopes in which every facet is a simplex), it...

meaning an n-dimensional analogue of a triangle Simplicial polytope, a polytope with all simplex facets Simplicial complex, a collection of simplicies Pascal's...

Coxeter 1973, p. 130, §7.6 The symmetry group of the general regular polytope; "simplicial subdivision". Coxeter 1973, pp. 70–71, Characteristic tetrahedra;...

it is that it operates on simplicial cones, and these become proper simplices with an additional constraint. The simplicial cones in question are the...

illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...

convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described...

restrictions on f-vectors of convex simplicial polytopes, to this more general setting. The face lattice of a convex polytope, consisting of its faces, together...

for simplicial polytopes: it follows in this case from a conjecture of Imre Bárány and László Lovász (1982) that every centrally symmetric simplicial polytope...

same bounds hold as well for convex polytopes that are not simplicial, as perturbing the vertices of such a polytope (and taking the convex hull of the...

same technique shows that in any higher dimension d, there exist simplicial polytopes with shortness exponent logd 2. Similarly, Plummer (1992) used the...

of triangles known as the simplex, and the polytopes with triangular facets known as the simplicial polytopes. Each triangle has many special points inside...

significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have the fewest...

Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29) Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory...

was used to characterize the numbers of faces in each dimension of simplicial polytopes. Every ring can be thought of as a monoid in Ab, the category of...