|
Semiregular Polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polytopes of the Hyperspaces'' which included a wider definition. Gosset's list In three-dimensional space and below, the terms ''semiregular polytope'' and '' uniform polytope'' have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the ''k''21 polytopes, where the rectified 5-cell is the special case of ''k'' = 0. These were all listed by Gosset, but a proof o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-demicube
In Five-dimensional space, five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with Alternation (geometry), alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only Semiregular polytope, semiregular 5-polytope (made of more than one type of regular Facet (geometry), facets), he called it a Semiregular polytope, 5-ic semi-regular. Emanuel Lodewijk Elte, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM5 for a 5-dimensional ''half measure'' polytope. Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol \left\ or . It exists in the semiregular k 21 polytope, k21 polytope family as 121 with the Gosset polytopes: Gosset 2 21 polytope, 221, Gosset 3 21 polytope, 321, and Gosset 4 21 polytope, 421. The graph formed by the vertices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HC P1-P3
HC, hc or H/C may refer to: Science, technology, and mathematics Medicine * Health Canada * Hemicrania continua * Hyperelastosis cutis or hereditary equine regional dermal asthenia Chemistry * Hemocyanin, a metalloprotein abbreviated Hc * HC smoke, a US military designation for Hexachloroethane * Homocapsaicin, a capsaicinoid *Hydrocarbon, a category of substances consisting only of hydrogen and carbon Other uses in science, technology, and mathematics * 74HC-series integrated circuits, a logic family of integrated circuits * Felix HC, a series of Romanian personal microcomputers produced by ICE Felix Bucharest and which were ZX Spectrum clones * '' Hemianthus callitrichoides'', a freshwater aquatic plant native to Cuba * + h.c., a notation used in mathematics and quantum physics Sports * Head Coach * Hors catégorie (French), used in cycle races to designate a climb that is "beyond categorization" * UCI .HC road cycling races (1.HC and 2.HC), the second tier of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the same must also be true within each lower-dimensional face of the polytope. In two dimensions (and for two-dimensional faces of higher-dimensional polytopes) a stronger definition is used: only the regular polygons are considered as uniform, disallowing polygons that alternate between two different lengths of edges. This is a generalization of the older category of semiregular polytope, ''semiregular'' polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional uniform tiling, tilings and higher di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2 21 Polytope
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 (mathematics), E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an Semiregular polytope, 6-ic semi-regular figure. It is also called the Schläfli polytope. Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 221. The rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the triangle face centers of the 221, and is the same as the rectified 122. These polytopes are a part of family of 39 convex uniform 6-polytope, uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-polytope
In geometry, a five-dimensional polytope (or 5-polytope or polyteron) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell. Definition 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. Furthermore, the following requirements must be met: # Each cell must join exactly two 4-faces. # Adjacent 4-faces are not in the same four-dimensional hyperplane. # The figure is not a compound of other figures which meet the requirements. Characteristics The topology of any given 5-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular E-polytope
In geometry, a uniform ''k''21 polytope is a polytope in ''k'' + 4 dimensions constructed from the En (Lie algebra), ''E''''n'' Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol ''k''21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the ''k''-node sequence. Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular polytope, regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the ''5-ic semiregular figure''. Family members The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of hyperbolic 9-space constructed of ∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 600-cell
In geometry, the Rectification (geometry), rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cell (mathematics), cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices. Containing the cell Hyperplane#Notes, realms of both the regular 120-cell and the regular 600-cell, it can be considered analogous to the polyhedron icosidodecahedron, which is a rectified icosahedron and rectified dodecahedron. The vertex figure of the rectified 600-cell is a uniform pentagonal prism. Semiregular polytope It is one of three Semiregular 4-polytopes#Semiregular polytopes, semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a ''octicosahedric'' for being made of octahedron and icosahedron cells. Emanuel Lodewijk Elte, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub 24-cell
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular Tetrahedron, tetrahedral and 24 Regular icosahedron, icosahedral cell (mathematics), cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra. Topologically, under its highest symmetry, [3+,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with Th symmetry), 24 regular tetrahedra, and 96 triangular pyramids. Semiregular polytope It is one of three Semiregular 4-polytopes#Semiregular polytopes, semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a ''tetricosahedric'' for be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism. Topologically, under its highest symmetry, ,3,3 there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedron-octahedron segmentochoron. The vertex figure of the ''rectified 5-cell'' is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends. Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5-cell is not self-dual bec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-polytope
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: Vertex (geometry), vertices, Edge (geometry), edges, Face (geometry), faces (polygons), and Cell (mathematics), cells (Polyhedron, 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 Convex uniform honeycomb, 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 polyhedral net, nets in 3-space. Definition A 4-polytope is a closed Four-dimensional space, four-dimensional figure. It compr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polyhedra
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form , where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. The regular polyhedra There are five convex regular polyhedra, known as the Platonic solids; four regular star polyhedra, the Kepler–Poinsot polyhedra; and five regular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
