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Tetrahedron
In geometry , a TETRAHEDRON (plural: TETRAHEDRA or TETRAHEDRONS), also known as a TRIANGULAR PYRAMID, is a polyhedron composed of four triangular faces , six straight edges , and four vertex corners . The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-SIMPLEX. The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets
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Tetrahedroid
In algebraic geometry, a TETRAHEDROID (or TéTRAéDROïDE) is a special kind of Kummer surface studied by Cayley (1846 ), with the property that the intersections with the faces of a fixed tetrahedron are given by two conics intersecting in four nodes. Tetrahedroids generalize Fresnel's wave surface . REFERENCES * Cayley, Arthur (1846), "Sur la surface des ondes", Journal des Mathématiques Pures et Appliquées, 11: 291–296, Collected papers vol 1 pages 302–305 * Hudson, R. W. H. T. (1990), Kummer's quartic surface, Cambridge Mathematical Library, Cambridge University Press , ISBN 978-0-521-39790-2 , MR 1097176 This algebraic geometry related article is a stub . You can help by expanding it . * v * t * e Retrieved from "https://en.wikipedia.org/w/index.php?title= Tetrahedroid additional terms may apply
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Tetrahedron (journal)
TETRAHEDRON is a weekly peer-reviewed scientific journal covering the field of organic chemistry . According to the Journal Citation Reports , the journal has a 2014 impact factor of 2.641. Tetrahedron and Elsevier
Elsevier
, its publisher, support an annual symposium. In 2010, complaints were raised over its high subscription cost. CONTENTS * 1 Notable papers * 2 See also * 3 References * 4 External links NOTABLE PAPERSAs of 22 June 2013 , the Web of Science lists ten papers from Tetrahedron that have more than 1000 citations. The four articles that have been cited more than 2000 times are: * Wiberg, K. B. (1968). "Application of pople-santry-segal CNDO method to the cyclopropylcarbinyl and cyclobutyl cation and to bicyclobutane". Tetrahedron. 24 (3): 1083. doi :10.1016/0040-4020(68)88057-3 . – cited 2228 times * Haasnoot, C. A. G.; de Leeuw, F. A. A. M.; Altona, C. (1980)
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Platonic Solid
In three-dimensional space , a PLATONIC SOLID is a regular , convex polyhedron . It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria: Tetrahedron
Tetrahedron
Cube
Cube
Octahedron
Octahedron
Dodecahedron
Dodecahedron
Icosahedron
Icosahedron
Four faces Six faces Eight faces Twelve faces Twenty faces(Animation ) (Animation ) (Animation ) (Animation ) (Animation ) Geometers have studied the mathematical beauty and symmetry of the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato
Plato
who hypothesized in his dialogue, the _Timaeus _, that the classical elements were made of these regular solids
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Euler Characteristic
In mathematics , and more specifically in algebraic topology and polyhedral combinatorics , the EULER CHARACTERISTIC (or EULER NUMBER, or EULER–POINCARé CHARACTERISTIC) is a topological invariant , a number that describes a topological space 's shape or structure regardless of the way it is bent. It is commonly denoted by {displaystyle chi } (Greek lower-case letter chi ). The Euler characteristic
Euler characteristic
was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids . Leonhard Euler
Leonhard Euler
, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic
Euler characteristic
arises from homology and, more abstractly, homological algebra
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Conway Polyhedron Notation
In geometry, CONWAY POLYHEDRON NOTATION, invented by John Horton Conway and promoted by George W. Hart , is used to describe polyhedra based on a seed polyhedron modified by various prefix operations . Conway and Hart extended the idea of using operators, like truncation defined by Kepler
Kepler
, to build related polyhedra of the same symmetry. The basic descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. For example tC represents a truncated cube , and taC, parsed as t(aC), is a truncated cuboctahedron . The simplest operator dual swaps vertex and face elements, like a dual cube is an octahedron: dC=O. Applied in a series, these operators allow many higher order polyhedra to be generated. A resulting polyhedron will have a fixed topology (vertices, edges, faces), while exact geometry is not constrained
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Schläfli Symbol
In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The Schläfli symbol
Schläfli symbol
is named after the 19th-century Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas
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Face Configuration
In geometry , a VERTEX CONFIGURATION is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.b.c" describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example, "3.5.3.5" 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 equivalent with different starting points, so 3.5.3.5 is the same as 5.3.5.3. The order is important, so 3.3.5.5 is different from 3.5.3.5
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Wythoff Symbol
In geometry , the WYTHOFF SYMBOL represents a Wythoff construction
Wythoff construction
of a uniform polyhedron or plane tiling, from a Schwarz triangle . It was first used by Coxeter , Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol
Wythoff symbol
consists of three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 4 2 {displaystyle 3 4 2} with Oh symmetry , and 2 4 2 {displaystyle 2 4 2} as a square prism with 2 colors and D4h symmetry , as well as 2 2 2 {displaystyle 2 2 2 } with 3 colors and D 2 h {displaystyle D_{2h}} symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra
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Coxeter Diagram
In geometry , a COXETER–DYNKIN DIAGRAM (or COXETER DIAGRAM, COXETER GRAPH) is a graph with numerically labeled edges (called BRANCHES) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes ). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet ) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge ). An unlabeled branch implicitly represents order-3. Each diagram represents a Coxeter group , and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed , while Coxeter diagrams are undirected ; secondly, Dynkin diagrams must satisfy an additional (crystallographic ) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6
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List Of Spherical Symmetry Groups
Spherical symmetry groups are also called point groups in three dimensions ; however, this article is limited to the finite symmetries . There are five fundamental symmetry classes which have triangular fundamental domains: dihedral , cyclic , tetrahedral , octahedral , and icosahedral symmetry. This article lists the groups by Schoenflies notation , Coxeter notation , orbifold notation , and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion . Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6
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Tetrahedral Symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4
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Uniform Polyhedron
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 congruent . Uniform polyhedra may be regular (if also face and edge transitive), quasi-regular (if edge transitive but not face transitive) or semi-regular (if neither edge nor face transitive). The faces and vertices need not be convex , so many of the uniform polyhedra are also star polyhedra . There are two infinite classes of uniform polyhedra together with 75 others
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Harold Scott MacDonald Coxeter
HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London but spent most of his adult life in Canada . He was always called Donald, from his third name MacDonald. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the _Canadian Music Journal_. He worked for 60 years at the University of Toronto and published twelve books . He was most noted for his work on regular polytopes and higher-dimensional geometries
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List Of Wenninger Polyhedron Models
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger . The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes. It contains the 75 nonprismatic uniform polyhedra , as well as 44 stellated forms of the convex regular and quasiregular polyhedra. Models listed here can be cited as "Wenninger Model Number N", or WN for brevity. The polyhedra are grouped in 5 tables: Regular (1–5), Semiregular (6–18), regular star polyhedra (20–22,41), Stellations and compounds (19–66), and uniform star polyhedra (67–119). The four regular star polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings
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Regular Polyhedron
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 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 {n, m}, 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, known as the Platonic solids . These are the: tetrahedron {3, 3}, cube {4, 3}, octahedron {3, 4}, dodecahedron {5, 3} and icosahedron {3, 5}. There are also four regular star polyhedra , making nine regular polyhedra in all
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