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Ditrigonal Polyhedron
In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal. Ditrigonal vertex figures There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.Har'El, 1993 The three uniform star polyhedron with Wythoff symbol of the form 3 , ''p'' ''q'' or , ''p'' ''q'' are ditrigonal, at least if ''p'' and ''q'' are not 2. Each polyhedron includes two types of faces, being of triangles, pentagons, or pentagrams. Their vertex configurations are of the form ''p''.''q''.''p''.''q''.''p''.''q'' or (''p''.''q'')3 with a symmetry of order 3. Here, term ditrigonal refers to a hexagon having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ''ditrigonal'' means "having two sets of 3 angles"). Mathworld (retrieved 10 June 2016) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Configuration
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector, vertex sequence. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book ''Mathematical Models (Cundy and Rollett), Mathematical Models''.Laughlin (2014), p. 16 For uniform polyhedron, uniform polyhedra, there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chirality (mathematics), Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) For example, "" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equival ... [...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] |
Great Complex Icosidodecahedron
In geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ..., the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, but are considered as two overlapping edges as topological polyhedron. It can be Wythoff construction, constructed from a number of different vertex figures. As a compound The great complex icosidodecahedron can be considered a compound polyhedron, compound of the small stellated dodecahedron, , and great icosahedron, , sharing the same vertices and edges, while the second is hidden, being completely contained inside the first. See also *Small complex icosidodecahedron References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Complex Icosidodecahedron
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron. A small complex icosidodecahedron can be constructed from a number of different vertex figures. A very similar figure emerges as a geometrical truncation of the great stellated dodecahedron, where the pentagram faces become doubly-wound pentagons ( --> ), making the internal pentagonal planes, and the three meeting at each vertex become triangles, making the external triangular planes. As a compound The small complex icosidodecahedron can be seen as a compound of the icosahedron and the great dodecahedron where all vertices are precise and edges coincide. The small complex icosidodecahedron resembles an icosahedron, because the great dodecahedron is comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Ditrigonal Dodecacronic Hexecontahedron
In geometry, the great ditrigonal dodecacronic hexecontahedron (or great lanceal trisicosahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great ditrigonal dodecicosidodecahedron. Its faces are kites A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have .... Part of each kite lies inside the solid, hence is invisible in solid models. Proportions Kite faces have two angles of \arccos(\frac-\frac\sqrt)\approx 98.183\,872\,491\,81^, one of \arccos(-\frac+\frac\sqrt)\approx 112.296\,452\,073\,54^ and one of \arccos(-\frac+\frac\sqrt)\approx 51.335\,802\,942\,83^. Its dihedral angles equal \arccos()\approx 127.686\,523\,427\,48^. The ratio between the lengths of the long edges and the short ones equals \frac\approx 1.917\,288\,176\,70. References * p. 62 E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Ditrigonal Dodecacronic Hexecontahedron
In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ... small ditrigonal dodecicosidodecahedron. It is visually identical to the small dodecicosacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models. Proportions Faces have two angles of \arccos(\frac+\frac\sqrt)\approx 12.661\,078\,804\,43^, one of \arccos(-\frac-\frac\sqrt)\approx 116.996\,396\,851\,70^ and one of 360^-\arccos(-\frac-\frac\sqrt)\approx 217.681\,445\,539\,45^. Its dihedral angles equal \arccos()\approx 146.230\,659\,755\,53^. The ratio between the lengths of the long and short edges is \frac\approx 1.110\,008\,944\,41. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Ditrigonal Dodecicosidodecahedron
In geometry, the great ditrigonal dodecicosidodecahedron (or great dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U42. It has 44 faces (20 triangles, 12 pentagons, and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is an isosceles trapezoid. Related polyhedra It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great icosicosidodecahedron (having the triangular and pentagonal faces in common) and the great dodecicosahedron (having the decagrammic faces in common). See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are ... References External links * Uniform polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Ditrigonal Dodecicosidodecahedron
In geometry, the small ditrigonal dodecicosidodecahedron (or small dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U43. It has 44 faces (20 triangles, 12 pentagrams and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small icosicosidodecahedron (having the triangular and pentagrammic faces in common) and the small dodecicosahedron (having the decagonal faces in common). See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are ... References External links * Uniform polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Ditrigonal Icosidodecahedron Cd
Great may refer to: Descriptions or measurements * Great, a relative measurement in physical space, see Size * Greatness, being divine, majestic, superior, majestic, or transcendent People * List of people known as "the Great" * Artel Great (born 1981), American actor * Great Osobor (born 2002), Spanish-born British basketball player Other uses * ''Great'' (1975 film), a British animated short about Isambard Kingdom Brunel * ''Great'' (2013 film), a German short film * Great (supermarket), a supermarket in Hong Kong * GReAT, Graph Rewriting and Transformation, a Model Transformation Language * Gang Resistance Education and Training, or GREAT, a school-based and police officer-instructed program * Global Research and Analysis Team (GReAT), a cybersecurity team at Kaspersky Lab *'' Great!'', a 2018 EP by Momoland *Great! TV, British TV channel group * ''The Great'' (TV series), an American comedy-drama See also * * * * * The Great (other) The Great is the moniker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ditrigonal Dodecadodecahedron Cd
In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal. Ditrigonal vertex figures There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.Har'El, 1993 The three uniform star polyhedron with Wythoff symbol of the form 3 , ''p'' ''q'' or , ''p'' ''q'' are ditrigonal, at least if ''p'' and ''q'' are not 2. Each polyhedron includes two types of faces, being of triangles, pentagons, or pentagrams. Their vertex configurations are of the form ''p''.''q''.''p''.''q''.''p''.''q'' or (''p''.''q'')3 with a symmetry of order 3. Here, term ditrigonal refers to a hexagon having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ''ditrigonal'' means "having two sets of 3 angles"). Mathworld (retrieved 10 June 2016) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Ditrigonal Icosidodecahedron Cd
Small means of insignificant size. Small may also refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (journal), ''Small'' (journal), a nano-science publication * HTML element#Presentation, <small>, an HTML element that defines smaller text Arts and entertainment Fictional characters * Small, in the British children's show Big & Small Other uses * Small (surname) * List of people known as the Small * "Small", a song from the album ''The Cosmos Rocks'' by Queen + Paul Rodgers See also * Smal (other) * Smalls (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
