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, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes.
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 four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.
In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation of the regular tesseract.
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell.
In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation of the regular 120-cell.
In four-dimensional geometry, a runcinated 120-cell is a convex uniform 4-polytope, being a runcination of the regular 120-cell.
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.
In mathematics, a regular 4-polytope or regular polychoron 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.
The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.
In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.
