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Great Icosahedral 120-cell
In geometry, the great icosahedral 120-cell, great polyicosahedron or great faceting, faceted 600-cell is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arrangement as the great stellated 120-cell, and grand stellated 120-cell, and face arrangement of the grand 600-cell. See also * List of regular polytopes * Convex regular 4-polytope * Kepler-Poinsot solids - regular star polyhedron * Star polygon - regular star polygons References * Edmund Hess, (1883) ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder' *Coxeter, H. S. M. Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . * John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 26, Regular Star-polytopes, pp. 404–408) * External ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ortho Solid 014-uniform Polychoron 3p5-t0
Ortho- is a Greek prefix meaning “straight”, “upright”, “right” or “correct”. Ortho may refer to: * Ortho, Belgium, a village in the Belgian province of Luxembourg Science * List of commonly used taxonomic affixes (ortho-) * Arene substitution patterns, two substituents that occupy adjacent positions on an aromatic ring * Chlordane, an organochlorine compound that was used as a pesticide Mathematics * Orthogonal, a synonym for perpendicular * Orthonormal, the property that a collection of vectors are mutually perpendicular and each of unit magnitude * Orthodrome, a synonym for great circle, a geodesic on the sphere * Orthographic projection, a parallel projection onto a perpendicular plane Medicine * Orthomyxovirus, a family of viruses to which influenza belongs * Orthodontics, a specialty of dentistry concerned with the study and treatment of malocclusions * Orthopedic, the study of the musculoskeletal system * Ortho-DOT, a psychedelic drug * Ortho-cept and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face Arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equal distance and angles from a center point. Two polytopes share the same ''vertex arrangement'' if they share the same 0-skeleton. A group of polytopes that shares a vertex arrangement is called an ''army''. Vertex arrangement The same set of vertices can be connected by edges in different ways. For example, the ''pentagon'' and ''pentagram'' have the same ''vertex arrangement'', while the second connects alternate vertices. A ''vertex arrangement'' is often described by the convex hull polytope which contains it. For example, the regular ''pentagram'' can be said to have a (regular) ''pentagonal vertex arrangement''. Infinite tilings can also share common ''vertex arrangements''. For example, this triangular lattice of points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including ''The Fifty-Nine Icosahedra'' (1938) and ''Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmund Hess
Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes. Publications *''Über die zugleich gleicheckigen und gleichflächigen Polyeder.'' In: Sitzungsberichte der Gesellschaft zur Beförderung der gesamten Naturwissenschaften zu Marburg (1876). *''Über einige merkwürdige, nicht convexe Polyeder.'' In: Sitzungsberichte der Gesellschaft zur Beförderung der gesamten Naturwissenschaften zu Marburg (1877), S. 1–13. *''Über vier Archimedeische Polyeder höherer Art.'' Kassel 1878. *''Combinationsgestalten höherer Art.'' In: Sitzungsberichte der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg (1879), Nr. 9, S. 99–103. *''Vergleichung der Volumina verschiedener Gruppen von Polyedern, deren Oberfläche denselben Werth hat.'' In: Sitzungsberichte der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg (1879), Nr. 9, S. 103–112. *''Einleitung in die Lehre von d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can arise through truncation operations on regular simple or star polygons. Branko Grünbaum identified two primary usages of this terminology by Johannes Kepler, one corresponding to the regular star polygons with List of self-intersecting polygons, intersecting edges that do not generate new vertices, and the other one to the isotoxal Concave polygon, concave simple polygons.Grünbaum & Shephard (1987). Tilings and Patterns. Section 2.5 Polygram (geometry), Polygrams include polygons like the pentagram, but also compound figures like the hexagram. One definition of a ''star polygon'', used in turtle graphics, is a polygon having ''q'' ≥ 2 Turn (geometry), turns (''q'' is called the turning number or Density (polygon), density), like in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polyhedron
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon, nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains. Mathematical studies of star polyhedra are usually concerned with regular polyhedron, regular, Uniform polyhedron, uniform polyhedra, or the Dual polyhedron, duals of the uniform polyhedra. All these stars are of the self-intersecting kind. Self-intersecting star polyhedra Regular star polyhedra The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting Face (geometry), faces, or self-intersecting vertex figures. There are four List of regular polytopes and compounds#Three dimensions 2, regular star polyhedra, known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Regular 4-polytope
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. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. History The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: ''F'' − ''E'' + ''V'' 2). That excludes cells and vertex figures such as the great dodecahedron and small stellated dodecahedron . Edmund Hess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. 1-polytopes There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol , or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. 2-polytopes (polygons) The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand 600-cell-ortho-4gon
Grand may refer to: People with the name * Grand (surname) * Grand L. Bush (born 1955), American actor Places * Grand, Oklahoma, USA * Grand, Vosges, village and commune in France with Gallo-Roman amphitheatre * Grand County (other), several places * Grand Geyser, Upper Geyser Basin of Yellowstone, USA * Le Grand, California, USA; census-designated place * Mount Grand, Brockville, New Zealand Arts, entertainment, and media * ''Grand'' (Erin McKeown album), 2003 * "Grand" (Kane Brown song), 2022 * ''Grand'' (Matt and Kim album), 2009 * ''Grand'' (magazine), a lifestyle magazine related to related to grandparents * ''Grand'' (TV series), American sitcom, 1990 * Grand Production, Serbian record label company Other uses * Great Recycling and Northern Development Canal, also known as GRAND Canal * Grand (slang), one thousand units of currency * Giant Radio Array for Neutrino Detection, also known as GRAND See also * * * Grand Hotel (other) * Grand sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
