4-polytopeIn geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron.
Convex polytopeA convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue.
PolytopeIn elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
Abstract polytopeIn mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a realization of an abstract polytope in some real N-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.
Regular 4-polytopeIn mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures.
Uniform 4-polytopeIn geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms.
5-polytopeIn geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell. A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron, and a 4-face is a 4-polytope.
Regular Polytopes (book)Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries. The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations.
6-polytopeIn 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 with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron. A 4-face is a polychoron, and a 5-face is a 5-polytope.
Regular polytopeIn mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube).
