A hexahedron plural hexahedra is a polyhedron with six faces. A Regular polyhedron regular hexahedron, with all its faces Square geometry square , is a cube . There are many kinds of hexahedra, some topologically similar to the cube and some not. Three are briefly examined below class wikitable colspan 5 Parallelogram faced Image Parallelepiped.svg 100px BR Parallelepiped BR Three pairs of BR parallelogram s Image Rhombohedron.svg 100px BR Rhombohedron BR Three pairs of BR rhombus rhombi Image Trigonal trapezohedron.png 100px BR Trigonal trapezohedron BR congruent rhombus rhombi Image Cuboid.png 100px BR Cuboid BR Three pairs of BR rectangles Image Hexahedron.png 100px BR Cube BR Square geometry square colspan 5 Others Image Pentagonal pyramid.png 100px BR Pentagonal pyramid BR pentagon and triangle s Image Triangular bipyramid.png 100px BR Triangular dipyramid BR triangle s Image Usech kvadrat piramid.png 100px BR Quadrilateral frustum BR apex truncated BR square pyramid Topologically distinct hexahedra There are seven topologically distinct convex hexahedra, ref http www.numericana.com data polycount.htm Counting polyhedra ref one of which exists in two mirror image forms. Two polyhedra are topologically distinct if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces. An example of each type is depicted below, along with the number of sides on each of the faces and the numbers of vertices and edges. class wikitable style background white width 100 valign top width 25 center image Hexahedron1.svg center Parallelepiped Faces 4,4,4,4,4,4 8 vertices 12 edges width 25 center image Hexahedron2.svg center Pentagonal pyramid . Faces 5,3,3,3,3,3 6 vertices 10 edges width 25 center image Hexahedron3.svg center Faces 5,4,4,3,3,3 ... fr Hexa dre ko it Esaedro la Hexahedron nl Zesvlak ja no Heksaeder pt Hexaedro sq Heksaedri ... more details
Uniform polyhedra db Uniform dual polyhedron stat table ctCO In geometry , the tetradyakis hexahedron is a nonconvex Isohedral figure isohedral polyhedron . It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices. It is the Dual polyhedron dual of the uniform star polyhedron uniform cubitruncated cuboctahedron . References Citation last1 Wenninger first1 Magnus author1 link Magnus Wenninger title Dual Models publisher Cambridge University Press isbn 978 0 521 54325 5 id MathSciNet id 730208 year 1983 p. 92 External links mathworld urlname TetradyakisHexahedron title Tetradyakis hexahedron http gratrix.net polyhedra uniform summary http gratrix.net Uniform polyhedra and duals Category Polyhedra Geometry stub ... more details
Uniform polyhedra db Uniform polyhedron stat table stH In geometry , the stellated truncated hexahedron or quasitruncated hexahedron is a uniform star polyhedron , indexed as U sub 19 sub . It is represented by Schl fli symbol t sub 0,1 sub 4 3,3 . Uniform polyhedra db Polyhedra word description stH Note that stellated truncated hexahedron is not a true stellation of the truncated hexahedron its convex core is nonuniform. how about a picture? Related polyhedra It shares the vertex arrangement with three other uniform polyhedron uniform polyhedra the convex rhombicuboctahedron , the small rhombihexahedron , and the small cubicuboctahedron . class wikitable width 400 style vertical align top text align center Image Small rhombicuboctahedron.png 100px BR Rhombicuboctahedron Image Small cubicuboctahedron.png 100px BR Small cubicuboctahedron Image Small rhombihexahedron.png 100px BR Small rhombihexahedron Image Stellated truncated hexahedron.png 100px BR Stellated truncated hexahedron See also List of uniform polyhedra External links mathworld urlname StellatedTruncatedHexahedron title Stellated truncated hexahedron Category Polyhedra Polyhedron stub eo Steligita senpintigita kubo fr Hexa dre tronqu toil ja ... more details
Summary Symmetry Group T sub d sub or 332 on the sphere Tetrahedral reflective symmetry . Yellow triangle is fundamental domain. Numbers are the reflection symmetry order at each node. This full figure also represents the edges of the polyhedron V4.6.6 tetrakis hexahedron expanded onto the surface of a sphere. Licensing PD self date October 2006 ... more details
Multiple issues context April 2009 unreferenced April 2009 orphan April 2009 wikify December 2010 If the regular polyhedra are considered as skeletal 3 dimensional figures with flexible vertices and unidimensional edges, two of them. the hexadron cube and the dodecahedron exhibit properties different from the other three in that they can be folded or collapsed The hexahedron first into a double square then a line. and the dodecahedron into a triangle. This can be demonstrated by constructing them out of drinking straws and twistable inserts. References Reflist DEFAULTSORT Skeletal Regular Polyhedron Category Geometry math stub ... more details
Uniform polyhedra db Uniform polyhedron stat table lrH In geometry , the small rhombihexahedron is a nonconvex uniform polyhedron , indexed as U sub 18 sub . It has 18 faces 12 squares and 6 octagons , 48 edges, and 24 vertices. Its vertex figure is a quadrilateral More quadrilaterals crossed quadrilateral . Related polyhedra This polyhedron shares the vertex arrangement with the stellated truncated hexahedron . It additionally shares its edge arrangement with the convex rhombicuboctahedron having 12 square faces in common and with the small cubicuboctahedron having the octagonal faces in common . class wikitable width 400 style vertical align top text align center Image Small rhombicuboctahedron.png 100px BR Rhombicuboctahedron Image Small cubicuboctahedron.png 100px BR Small cubicuboctahedron Image Small rhombihexahedron.png 100px BR Small rhombihexahedron Image Stellated truncated hexahedron.png 100px BR Stellated truncated hexahedron It may be constructed as the exclusive or of three octagonal prism s. External links mathworld2 urlname SmallRhombihexahedron title Small rhombihexahedron urlname2 UniformPolyhedron title2 Uniform polyhedron Category Polyhedra Polyhedron stub eo Malgranda rombo sesedro fr Petit rhombihexa dre ja ... more details
Uniform polyhedra db Uniform dual polyhedron stat table stH In geometry , the great triakis octahedron is the dual of the stellated truncated hexahedron U19 . It has 24 intersecting Triangle Types of triangles isosceles triangle faces. Because of the existence of this polyhedron, the Catalan solid known as the triakis octahedron is occasionally referred to as the small triakis octahedron. It is a stellation of the deltoidal icositetrahedron . References Citation last1 Wenninger first1 Magnus author1 link Magnus Wenninger title Dual Models publisher Cambridge University Press isbn 978 0 521 54325 5 id MathSciNet id 730208 year 1983 External links http mathworld.wolfram.com GreatTriakisOctahedron.html Mathworld Great Triakis Octahedron http mathworld.wolfram.com SmallTriakisOctahedron.html Mathworld Small Triakis Octahedron http gratrix.net polyhedra uniform summary Uniform polyhedra and duals Category Polyhedra geometry stub ... more details
border 1 bgcolor ffffff cellpadding 5 align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 Compound of five stellated truncated cubes align center colspan 2 Image UC58 5 quasitruncated hexahedra.png 200px bgcolor e7dcc3 Type Uniform polyhedron compound Uniform compound bgcolor e7dcc3 Index UC sub 58 sub bgcolor e7dcc3 Polyhedra 5 stellated truncated hexahedron stellated truncated cubes bgcolor e7dcc3 Faces 40 triangles , 30 octagram octagrams bgcolor e7dcc3 Edges 180 bgcolor e7dcc3 Vertices 120 bgcolor e7dcc3 Symmetry group Icosahedral symmetry icosahedral I sub h sub bgcolor e7dcc3 Subgroup restricting to one constituent Tetrahedral symmetry pyritohedral T sub h sub This uniform polyhedron compound is a composition of 5 stellated truncated hexahedron stellated truncated cubes , formed by star truncating each of the cubes in the compound of five cubes compound of 5 cubes . Cartesian coordinates Cartesian coordinates for the vertices of this compound are all the cyclic permutations of 2 &radic 2 , &radic 2, 2 &radic 2 , sup 1 sup sup 1 sup &radic 2 , 2 1 &radic 2 1, sup 2 sup sup 1 sup &radic 2 , sup 2 sup &radic 2 1 &radic 2 , sup 2 sup &radic 2 , sup 2 sup &radic 2 &radic 2 , sup 1 sup , 2 1 sup 1 sup &radic 2 where 1 &radic 5 2 is the golden ratio sometimes written . References citation first John last Skilling title Uniform Compounds of Uniform Polyhedra journal Mathematical Proceedings of the Cambridge Philosophical Society volume 79 pages 447 457 year 1976 doi 10.1017 S0305004100052440 id MR 0397554 . Category Polyhedral compounds polyhedron stub eo Kombina o de 5 steligitaj senpintigitaj kuboj ... more details
wiktionary tetrakis Tetrakis may refer to Mathematics Tetrakis hexahedron , an Archimedean dual solid, or a Catalan solid Tetrakis square tiling , a tiling of the Euclidean plane Chemistry Tetrakis acetonitrile copper I hexafluorophosphate , a copper I coordination complex with the formula Cu CH sub 3 sub CN sub 4 sub PF sub 6 sub Tetrakis dimethylamino titanium IV TDMAT is a compound with the formula C sub 8 sub H sub 24 sub N sub 4 sub Ti Tetrakis hydroxymethyl phosphonium chloride THPC is a phosphonium salt with the chemical formula CH sub 2 sub OH sub 4 sub P Cl Tetrakis triphenylphosphine palladium 0 , the chemical compound Pd P C sub 6 sub H sub 5 sub sub 3 sub sub 4 sub Tetrakis triphenylphosphine platinum 0 , the chemical compound with the formula Pt P C sub 6 sub H sub 5 sub sub 3 sub sub 4 sub disambig ... more details
border 1 bgcolor ffffff cellpadding 5 align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 Tetrakis cuboctahedron align center colspan 2 Image Tetrakis cuboctahedron.png 240px Tetrakis cuboctahedron bgcolor e7dcc3 Type Conway polyhedron notation Conway polyhedron bgcolor e7dcc3 Faces 32 triangles 2 types bgcolor e7dcc3 Edges 48 2 types bgcolor e7dcc3 Vertices 18 2 types bgcolor e7dcc3 Vertex configuration s 6 3 sup 5 sup BR 12 3 sup 6 sup bgcolor e7dcc3 Symmetry group Octahedral symmetry Octahedral O sub h sub bgcolor e7dcc3 Dual polyhedron Truncated rhombic dodecahedron bgcolor e7dcc3 Properties convex set convex Image Tetrakis cuboctahedron net.png 250px thumb net polyhedron Net The tetrakis cuboctahedron is a convex set convex polyhedron with 32 triangular Face geometry faces , 48 Edge geometry edges , and 18 Vertex geometry vertices . It a dual of the truncated rhombic dodecahedron . Its name comes from a topological construction from the cuboctahedron with the Conway kis operator kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained the 6 order 4 vertices at a different distance from the center as the other 12. Related polyhedra It can also be topologically constructed from the octahedron , dividing each triangular face into 4 triangles by adding mid edge vertices An Conway ortho operator ortho operation . From this construction, all 80 triangles will be equilateral. This polyhedron can be confused with a slightly smaller Catalan solid , the tetrakis hexahedron , which has only 24 triangles, 32 edges, and 14 vertices. gallery File Tetrakis cuboctahedron on octahedron.png Octahedron with edges bisected and faces divided into subtriangles of the tetrakis cuboctahedron File Cuboctahedron.png Cuboctahedron File Tetrakishexahedron.jpg Tetrakis hexahedron File Octahemioctahedron.png The nonconvex octahemioctahedron l ... more details
Semireg polyhedra db Semireg polyhedron stat table tC In geometry , the truncated cube , or truncated hexahedron , is an Archimedean solid . It has 14 regular faces 6 octagon al and 8 triangle geometry triangular , 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and math 2 sqrt 2 math . Area and volume The area A and the volume V of a truncated cube of edge length a are math A 2 6 6 sqrt 2 sqrt 3 a 2 approx 32.4346644a 2 math math V frac 1 3 21 14 sqrt 2 a 3 approx 13.5996633a 3. math Cartesian coordinates class wikitable Orthographic projection s File 3 cube t01.svg 120px File 3 cube t01 B2.svg 120px The following Cartesian coordinates define the vertices of a Truncation geometry truncated hexahedron centered at the origin with edge length 2 , 1, 1 , 1, , 1 , 1, 1, where math sqrt2 1 math Related polyhedra The truncated cube can be seen as a cube with its corners Truncation geometry truncated , as shown in this truncation sequence class wikitable Image Uniform polyhedron 43 t0.png 100px BR Cube Image Uniform polyhedron 43 t01.png 100px BR Truncated cube Image Uniform polyhedron 43 t1.png 100px BR cuboctahedron Image Uniform polyhedron 43 t12.png 100px BR Truncated octahedron Image Uniform polyhedron 43 t2.png 100px BR Octahedron It shares the vertex arrangement with three nonconvex uniform polyhedron nonconvex uniform polyhedra class wikitable width 400 style vertical align top text align center Image Truncated hexahedron.png 100px BR Truncated cube Image Uniform great rhombicuboctahedron.png 100px BR Nonconvex great rhombicuboctahedron Image Great cubicuboctahedron.png 100px BR Great cubicuboctahedron Image Great rhombihexahedron.png 100px BR Great rhombihexahedron A cube can be Alternation geometry alternately truncated producing tetrahedral symmetry , with 6 hexagonal faces, and 4 triangles at the truncated vertices. File Alternate truncated cube.png 100px See also Image Truncatedhe ... more details
center 14 align center isosceles triangle BR V3.8.8 align center O sub h sub tetrakis hexahedron BR or disdyakis hexahedron or hexakis tetrahedron Image tetrakis cube.png 80px Tetrakis hexahedron Image tetrakishexahedron.jpg 80px Tetrakis hexahedron br small image tetrakishexahedron.gif Animation ... more details
File Sikhote alin.jpg thumb Neumann bands Hexahedrites are a type of iron meteorite . They are composed almost exclusively of the nickel iron alloy kamacite and are lower in nickel content than the octahedrite s. ref Vagn F. Buchwald, Handbook of Iron Meteorites . University of California Press, 1975. ref The nickel concentration in hexahedrites is always below 5.8 and only rarely below 5.3 . ref John T. Wasson, Meteorites Their Record of Early Solar System History . W. H. Freeman, 1985. ref The name comes from the cubic i.e. hexahedron structure of the kamacite crystal. After etching, hexahedrites do not display a Widmanst tten pattern , but they often do show Neumann line s parallel lines that cross each other at various angles, and are indicative of impact shock on the parent body. These lines are named after Johann G. Neumann , who discovered them in 1848. ref J. G. Burke, Cosmic Debris Meteorites in History . University of California Press, 1986. ref Chemical classification Concentrations of trace elements germanium , gallium and iridium are used to separate the iron meteorite s into chemical classes, which correspond to separate asteroid parent bodies. Chemical classes that include hexahedrites are ref John T. Wasson, Meteorites Classification and Properties . Springer Verlag, 1974. ref IIAB iron meteorite s includes also some octahedrite s IIG iron meteorite s includes also some octahedrite s References references astronomy stub Category Meteorite types de Hexaedrit sl Heksaedrit zh ... more details
In geometry , a simplicial polytope is a d polytope whose facet mathematics facets are all Simplex simplices . For example, a simplicial polyhedron contains only triangular faces ref Polyhedra, Peter R. Cromwell, 1997. p.341 ref and corresponds via Steinitz s theorem to a maximal planar graph . They are topologically Dual polytope dual to simple polytope s. Polytopes which are both simple and simplicial are either simplices or two dimensional polygons . Examples Simplicial polyhedra include Bipyramid s Gyroelongated dipyramid s Deltahedron Deltahedra Equilateral triangles Platonic solid Platonic tetrahedron , octahedron , icosahedron Johnson solid s triangular dipyramid , gyroelongated square dipyramid , triaugmented triangular prism , snub disphenoid Catalan solid s triakis tetrahedron , triakis octahedron , tetrakis hexahedron , disdyakis dodecahedron , triakis icosahedron , pentakis dodecahedron , disdyakis triacontahedron Simplicial tilings Regular triangular tiling Laves tiling s tetrakis square tiling , triakis triangular tiling , bisected hexagonal tiling Simplicial 4 polytope s include convex regular 4 polytope Pentachoron 4 simplex , 16 cell , 600 cell Dual convex uniform honeycomb s Disphenoid tetrahedral honeycomb Dual of cantitruncated cubic honeycomb Dual of omnitruncated cubic honeycomb Dual of cantitruncated alternated cubic honeycomb Simplicial higher polytope families simplex cross polytope Orthoplex Notes reflist References cite book last Cromwell first Peter R. title Polyhedra publisher Cambridge University Press date 1997 isbn 0521664055 url http books.google.com books?id OJowej1QWpoC&lpg PP1&dq Polyhedra&pg PP1 v onepage&q &f false See also Simplicial complex Delaunay triangulation Geometry stub Category Euclidean geometry ... more details
Image TwoCells.png thumb A trivial example of a Corner point grid with only two cells. In geometry , a corner point grid is a tessellation an Euclidean space Euclidean 3D volume where the base cell has 6 Face geometry faces hexahedron . A set of straight lines defined by their end points define the pillars of the corner point grid. The pillars have a lexiographical ordering that determines neighbouring pillars. On each pillar, a constant number of nodes corner points is defined. A corner point cell is now the volume between 4 neighbouring pillars and two neighbouring points on each pillar. Each cell can be identified by integer coordinates math i,j,k math , where the math k math coordinate runs along the pillars, and math i math and math j math span each layer. The cells are ordered naturally, where the index math i math runs the fastest and math k math the slowest. In the special case of all pillars being vertical, the top and bottom face of each corner point cell are described by Bilinear interpolation bilinear surfaces and the side faces are plane geometry plane s. Corner point grids are supported by most reservoir simulation software, and has become an industry standard. Degeneracy A main feature of the format is the ability to define erosion surfaces in geological modelling , effectively done by collapsing nodes along each pillar. This means that the corner point cells degenerate and may have less than 6 faces. For the corner point grids non neighboring connections are supported, meaning that grid cells that are not neighboring in ijk space can be defined as neighboring. This feature allows for representation of faults with significant throw displacement. Moreover, the neighboring grid cells do not need to have matching cell faces just overlap . Category Tiling Category Geometry Category Petroleum production geometry stub ... more details
Uniform polyhedra db Uniform polyhedron stat table lCCO In geometry , the small cubicuboctahedron is a uniform star polyhedron , indexed as U sub 13 sub . It has 20 faces 8 triangles, 6 squares, and 6 octagons , 48 edges, and 24 vertices. Its vertex figure is a quadrilateral More quadrilaterals crossed quadrilateral . Related polyhedra It shares the vertex arrangement with the stellated truncated hexahedron . It additionally shares its edge arrangement with the rhombicuboctahedron having the triangular faces and 6 square faces in common , and with the small rhombihexahedron having the octagonal faces in common . class wikitable width 400 style vertical align top text align center Image Small rhombicuboctahedron.png 100px BR Rhombicuboctahedron Image Small cubicuboctahedron.png 100px BR Small cubicuboctahedron Image Small rhombihexahedron.png 100px BR Small rhombihexahedron Image Stellated truncated hexahedron.png 100px BR Stellated truncated hexahedron Related tilings File Uniform tiling 443 t01.png thumb The t sub 0,1 sub 4, 3, 3 tiling is the tiling on the universal cover of the small cubicuboctahedron. br small Yellow and red reversed in this tiling, compared to polyhedron. small As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3 topologically it is a surface of genus 3 , and thus can be interpreted as a polyhedral Immersion mathematics immersion of a genus 3 polyhedral surface. Stated alternatively, it corresponds to a uniform tiling of this surface. In the language of abstract polytope s, the SCCO is a faithful realization of this abstract toroidal polyhedron, meaning that it is a nondegenerate polyhedron and that they have the same symmetry group every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space it is a uniform tiling, and the SCCO is a uniform polyhedron . Higher genus surfaces genus 2 or greater admit a metric of negative constant curvature by t ... more details
has a different Wythoff symbol . class wikitable align center Name Regular hexahedron Square Prism geometry ... ly faced hexahedron No No No The cube is a part of a sequence of rhombic polyhedra and tilings with n,3 Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares ... more details
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