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Noble Polyhedron
A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same). They were first studied in any depth by Edmund Hess and Max Brückner in the late 19th century, and later by Branko Grünbaum. Classes of noble polyhedra There are several main classes of noble polyhedra: * Regular polyhedra, that is, the five Platonic solids and the four Kepler–Poinsot polyhedra. * Disphenoid tetrahedra. * Crown polyhedra, also known as stephanoid polyhedra. * A variety of miscellaneous examples, e.g. the stellated icosahedra D and H, or their duals. It is not known whether there are finitely many, and if so how many might remain to be discovered. If we allow some of Grünbaum's stranger constructions as polyhedra, then we have two more infinite series of toroids (besides the crown polyhedra mentioned above): * Wreath polyhedra. These have triangular faces in coplanar pairs which share an edge. * V-faced polyhedra. These have vertices in coincident pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Fifty-Nine Icosahedra
''The Fifty-Nine Icosahedra'' is a book written and illustrated by Harold Scott MacDonald Coxeter, H. S. M. Coxeter, Patrick du Val, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic regular icosahedron, icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto in 1938, a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell. Authors' contributions Miller's rules Although J. C. P. Miller, Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie. His contribution is immortalised in his set of rules for defining which stellation forms should be considered "properly significant and distinct": :''(i) The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.'' :''(ii) All parts composing t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Scott MacDonald 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 Coxete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Espoo, Finland
Espoo (, ; ) is a city in Finland. It is located to the west of the capital, Helsinki, in southern Uusimaa. The population is approximately . It is the most populous Municipalities of Finland, municipality in Finland. Espoo is part of the Helsinki Metropolitan Area, which has approximately million inhabitants. Espoo is on the northern shore of the Gulf of Finland and borders Helsinki, Vantaa, Kirkkonummi, Vihti and Nurmijärvi. The city includes the Enclave and exclave, enclave of Kauniainen. Espoo covers an area of . Espoo is a bilingual municipality with Finnish language, Finnish and Swedish language, Swedish as its official languages. The population consists of Finnish speakers, Swedish speakers, and speakers of other languages, well above the national average. Espoo was settled in the Prehistory, Prehistoric Era, with evidence of human settlements dating back 8,000 years. However, the population disappeared during the early Iron Age. During the Middle Ages, Early Mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helsinki, Finland
Helsinki () is the capital and most populous city in Finland. It is on the shore of the Gulf of Finland and is the seat of southern Finland's Uusimaa region. About people live in the municipality, with million in the capital region and million in the metropolitan area. As the most populous urban area in Finland, it is the country's most significant centre for politics, education, finance, culture, and research. Helsinki is north of Tallinn, Estonia, east of Stockholm, Sweden, and west of Saint Petersburg, Russia. Helsinki has significant historical connections with these three cities. Together with the cities of Espoo, Vantaa and Kauniainen—and surrounding commuter towns, including the neighbouring municipality of Sipoo to the east—Helsinki forms a metropolitan area. This area is often considered Finland's only metropolis and is the world's northernmost metropolitan area with over one million inhabitants. Additionally, it is the northernmost capital of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron
In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform polyhedra may be Regular polyhedron, regular (if also Isohedral figure, face- and Isotoxal figure, edge-transitive), Quasiregular polyhedron, quasi-regular (if also edge-transitive but not face-transitive), or Semiregular polyhedron, semi-regular (if neither edge- nor face-transitive). The faces and vertices don't need to be Convex polyhedron, convex, so many of the uniform polyhedra are also Star polyhedron, star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra. They are 2 infinite classes of Prism (geometry), prisms and antiprisms, the convex polyhedrons as in 5 Platonic solids and 13 Archimedean solids—2 Quasiregular polyhedron, quasiregular and 11 Semiregular polyhedron, semiregular&m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. Kepler first named it in Latin as ''cubus simus'' in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol s \scriptstyle\begin 4 \\ 3 \end, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol t \scriptstyle\begin 4 \\ 3 \end. Construction The snub cube can be generated by taking the six faces of the cube, pulling them outward so they no longer touch, then giving them each a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with equilateral triangles. The snub cube may also be constructed from a rhombicuboctahedron. It started by twisting its square face (in blue), allowing its triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: * In three-dimensional geometry, some authors call a facet of a polyhedron any polygon whose corners are vertices of the polyhedron, including polygons that are not ''Face (geometry), faces''. To ''facetting, facet'' a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to ''stellation'' and may also be applied to higher-dimensional polytopes. * In polyhedral combinatorics and in the general theory of polytopes, a Face (geometry), face that has dimension ''n'' − 1 (an (''n'' − 1)-face or hyperface) is called a Face (geometry)#Facet, facet. In this terminology, every facet is a face. * A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex.. For (boundary complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Final Stellation Of The Icosahedron
In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, Simple polyhedron, simple, and star polyhedron. Background Johannes Kepler in his ''Harmonices Mundi'' applied the stellation process, recognizing the small stellated dodecahedron and great stellated dodecahedron as regular polyhedra. However, Louis Poinsot in 1809 rediscovered two more, the great icosahedron and great dodecahedron. This was proved by Augustin-Louis Cauchy in 1812 that there are only four regular star polyhedrons, known as the Kepler–Poinsot polyhedron. extended the stellation theory bey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Polyhedron
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topology (Mathematics), topological Genus (mathematics), genus () of 1 or greater. Notable examples include the Császár polyhedron, Császár and Szilassi polyhedron, Szilassi polyhedra. Variations in definition Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by exactly two polygons, and at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link (geometry), link of the vertex. For toroidal polyhedra, this manifold is an orientability, orientable surface. Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus. In this area, it is important to distinguish embedding, embedded toroidal polyhedra, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isohedral Figure
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not merely Congruence (geometry), congruent but must be ''transitive'', i.e. must lie within the same ''symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by Translation (geometry), translations, Rotation (mathematics), rotations, and/or Reflection (mathematics), reflections that maps onto . For this reason, Convex polytope, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an Parity (mathematics), even number of faces. The Dual polyhedron, dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
