In geometry, a 4-polytope 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.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.
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 particular, all its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.
In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are dimensions of 2 (polygon) or higher.
In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.
In geometry, an apeirogon or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.
In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.
In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.
In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons.
In geometry of 4 dimensions or higher, a double pyramid, duopyramid, or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term duopyramid was used by George Olshevsky, as the dual of a duoprism.
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a cylinder.
In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron. They have either skew regular faces or skew regular vertex figures.
