|
4-polytopes
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, 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. Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be ''cut and unfolded'' as nets in 3-space. Definition A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid. The boundary of the 120-cell is composed of 120 dodecahedral cell (mathematics), cells with 4 meeting at each vertex. Together they form 720 pentagonal faces, 1200 edges, and 600 vertices. It is the 4-Four-dimensional space#Dimensional analogy, dimensional analogue of the regular dodecahedron, since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the ''dodecaplex'' has 120 dodecahedral facets, with 3 around each edge. Its dual polytope is the 600-cell. Geometry The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons and above). As the sixth and largest regular convex 4-polytope, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral Cell (geometry), cells. The 600-cell's boundary is composed of 600 Tetrahedron, tetrahedral Cell (mathematics), cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-Four-dimensional space#Dimensional analogy, dimensional analogue of the icosahedron, since it has five Tetrahedron, tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. Its dual polytope is the 120-cell. Geometry The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius). It can be deconstructed into twenty-five overla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
24-cell
In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, Octacube (sculpture), octacube, hyper-diamond or polyoctahedron, being constructed of Octahedron, octahedral Cell (geometry), cells. The boundary of the 24-cell is composed of 24 octahedron, octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual polyhedron, self-dual. The 24-cell and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius. The 24-cell does not have a regular analogue in three dimensions or any other number of dimensions, either below or above. It is the only one of the six convex regular 4-polytopes which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
