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.
GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Facet (geometry)In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: In three-dimensional geometry, a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face. To facet a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higher-dimensional polytopes.
Line segmentIn geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ).
Hypercube graphIn graph theory, the hypercube graph Q_n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q_3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q_n has 2^n vertices, 2^n – 1n edges, and is a regular graph with n edges touching each vertex. The hypercube graph Q_n may also be constructed by creating a vertex for each subset of an n-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each n-digit binary number, with two vertices adjacent when their binary representations differ in a single digit.
Steinitz's theoremIn polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.
Vertex (geometry)In geometry, a vertex (: vertices or vertexes) is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place.
N-skeletonIn mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace X_n that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n. In other words, given an inductive definition of a complex, the n-skeleton is obtained by stopping at the n-th step. These subspaces increase with n. The 0-skeleton is a discrete space, and the 1-skeleton a topological graph.
Face (geometry)In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane tile.
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.
