d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges (also d facets). The vertex figure of a simple d-polytope...

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...

maximal planar graph. They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional...

convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for...

definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled...

is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly...

mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely...

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In...

{\displaystyle n} edges meet at v {\displaystyle v} (that is, it is a simple polytope), and there are integer vectors parallel to these edges forming a Z...

geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after...

or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces. A polygon vertex xi of a simple polygon...

6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E7. This polytope is the vertex figure for a uniform tessellation of 7-dimensional...

deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the...

polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes....

etc. Every simple 0/1-polytope is a Cartesian product of 0/1 simplexes. Ziegler, Günter M. (2000). "Lectures on 0/1-polytopes". Polytopes—combinatorics...

root vectors of the simple Lie group E8, this polytope is sometimes referred to as the E8 root polytope. The vertices of this polytope can also be obtained...

n-polytope elements are doubled from the (n − 1)-polytope elements and then creating new elements from the next lower element. Take an n-polytope with...

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...

omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each...

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means...

six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. A 6-polytope is a closed six-dimensional figure...

uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete...

regular polytopes. Coxeter lists a few of these in his book Regular Polytopes. McMullen added six in his paper New Regular Compounds of 4-Polytopes. Self-duals:...

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension...

single plane. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons...

122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named...

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known...

graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected...

labels it the γ4 polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope. The Oxford English...

5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets...