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...

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...

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

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

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...

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...

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...

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...

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...

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...

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

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

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...

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...

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

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

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...

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

every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets. From the point...

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...

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...

realized geometrically without self-crossings (rather than as an abstract polytope). More generally this equation can be satisfied precisely when f  is...

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...