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

manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig...

Note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form {a,...

chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope.: 22–25  For abstract polytopes of rank 2, this means that: A) the...

In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular...

regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. Only counting polytopes of...

24-cell, 120-cell, 600-cell Abstract regular polytope 11-cell, 57-cell Schläfli–Hess 4-polytope (Regular star 4-polytope) Icosahedral 120-cell, Small...

incidence complexes to create the modern idea of an abstract polyhedron (as an abstract 3-polytope), notably presented by McMullen and Schulte. Polyhedra...

In the study of abstract polytopes, a chiral polytope is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms...

face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive...

A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes...

geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets. An abstract n-polytope is a partially ordered set P...

In abstract geometry, a hemitesseract is an abstract, regular 4-polytope, containing half the cells of a tesseract, existing in real projective space...

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure...

understood as distinct realizations of the same abstract section. A vertex figure of an n-polytope is an (n−1)-polytope. For example, a vertex figure of a polyhedron...

segment. Not only does this fit in better with modern theories of abstract polytopes, but it also more closely copies the way in which Poinsot (1809) created...

antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated digon, t{2} is a square, {4}. An alternated...

In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra...

In mathematics, the 11-cell is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices...

57-cell (pentacontaheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has...

In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement of convexity is relaxed. One precise combinatorial...

polyhedron, like this cube. Every edge is shared by three or more faces in a 4-polytope, as seen in this projection of a tesseract. In geometry, an edge is a particular...

and abstract polytopes. The following are some of the aspects of polytopes studied in discrete geometry: Polyhedral combinatorics Lattice polytopes Ehrhart...

Applications. Elsevier Science. Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)...

of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the regular and uniform polytopes, the...

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

polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying...

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...

will not be connected and will have a "*" table entry. Every polytope, and abstract polytope has a Hasse diagram expressing these connectivities, which...

combinatorics of the tiling (this is a general way of obtaining an abstract polytope from a tiling) – the vertices, edges, and faces of the polyhedron...