A hexahedron plural hexahedra is any polyhedron with six Face geometry faces , although usually implies the cube as a Regular polyhedron regular hexahedron with all its faces Square geometry square , and three squares around each Vertex geometry vertex . 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. class wikitable width 700 colspan 6 Quadrilateral ly faced hexahedra br 4 sup 6 sup faces, 12 edges, 8 vertices align center 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 Image Usech kvadrat piramid.png 100px br Quadrilateral frustum br apex truncated br square pyramid align center colspan 6 Others align center Image Hexahedron2.svg 100px br Pentagonal pyramid br 5.3 sup 5 sup Faces BR 10 E, 6 V Image Hexahedron5.svg 100px br Triangular dipyramid br 3 sup 6 sup Faces BR 9 E, 5 V File Hexahedron3.svg 100px br 5.4.4.3.3.3 Faces br 11 E, 7 V File Hexahedron4.svg 100px br 5.5.4.4.3.3 Faces br 12 E, 8 V image Hexahedron6.svg 100px br 4.4.4.4.3.3 Faces br 11 E, 7 V image Hexahedron7.svg 75px image Hexahedron7a.svg 75px br Tetragonal antiwedge. Chiral &ndash exists in left handed and right handed mirror image forms. br 4.4.3.3.3.3 Faces br 10 E, 6 V There are three further topologically ... de Hexaeder es Hexaedro eo Sesedro eu Hexaedro fr Hexa dre gl Hexaedro ko it Esaedro la Hexahedron ... 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 polyhedron 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 , and Coxeter Dynkin diagram , CDD node 1 4 rat d3 node 1 3 node . It is sometimes called quasitruncated hexahedron because it is related to the truncated cube , CDD node 1 4 node 1 3 node , except that the square faces become inverted into 8 3 octagrams. 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
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 an antiparallelogram . 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 polyhedron 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 s 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 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 mr 0397554 issue 3 . 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
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 Category Meteorite types meteoroid stub 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 Tessellation Category Geometry Category Petroleum production geometry stub ... 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 scriptstyle 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 Orthogonal projections The truncated cube has five special orthogonal projection s, centered, on a vertex, on two types of edges, and two types of faces triangles, and octagons. The last two correspond to the B sub 2 sub and A sub 2 sub Coxeter plane s. class wikitable width 640 Orthogonal projections Centered by Vertex Edge br 3 8 Edge br 8 8 Face br Octagon Face br Triangle Image File Cube t01 v.png 100px File Cube t01 e38.png 100px File Cube t01 e88.png 100px File 3 cube t01 B2.svg 100px File 3 cube t01.svg 100px align center Projective BR symmetry 2 2 2 4 6 Cartesian coordinates 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 scriptstyle 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 polyhedra class wikitable width 400 style vertical align top text align center Image Truncated hexahedron.png 10 ... more details
8 triangle s br 6 Square geometry square s 24 12 O sub h sub truncated cube br or truncated hexahedron BR 3.8.8 Image truncatedhexahedron.jpg 60px Truncated hexahedron br small image truncatedhexahedron.gif Animation small Image truncated hexahedron.png 80px Image truncated hexahedron flat.svg 80px ... 8 hexagons br 6 octagons 72 48 O sub h sub snub cube br or snub hexahedron br or snub cuboctahedron br 2 chirality mathematics chiral forms BR 3.3.3.3.4 Image snubhexahedronccw.jpg 60px Snub hexahedron Ccw br small image snubhexahedronccw.gif Animation small br Image snubhexahedroncw.jpg 60px Snub hexahedron ... 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 small cubicuboctahedron 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 small cubicuboctahedron is a uniform polyhedron . Higher genus surfaces genus 2 or greater admit a metr ... more details
In geometry , a prismatoid is a polyhedron where all vertices lie in two parallel planes. If both planes have the same number of vertex geometry vertices , and the lateral faces are either parallelograms or trapezoids, it is called a prismoid . If the areas of the two parallel faces are A sub 1 sub and A sub 3 sub , the cross sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A sub 2 sub , and the height the distance between the two parallel faces is h, then the volume of the prismatoid is given by math V frac h A 1 4A 2 A 3 6 math This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson s rule , since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height. Prismatoid families Image Pentagonal pyramid.png 100px Image Pentagonal prism.png 100px Image Pentagonal antiprism.png 100px Image Pentagrammic crossed antiprism.png 100px Image Pentagonal cupola.png 100px Image Pentagonal frustum.svg 100px Families of prismatoids include Pyramid geometry Pyramids , where one plane contains only a single point Wedge geometry Wedges , where one plane contains only two points Prism geometry Prisms , where the polygons in each plane are congruent and joined by rectangles or parallelograms Antiprism s, where the polygons in each plane are congruent and joined by an alternating strip of triangles Crossed antiprism s cupola geometry Cupolas , where the polygon in one plane contains twice as many points as the other and is joined to it by alternating triangles and rectangles Frustum Frusta obtained by Truncation geometry truncation of a pyramid Quadrilateral faced Hexahedron hexahedral prismatoids Parallelepiped s six parallelogram faces Rhombohedron s six rhombus faces Trigonal trapezohedron Trigonal trapezohedra six congruent rhombus faces Cuboid s six rectangular faces frustum Quadrilateral frusta an Ape ... more details
Merge to List of geometric shapes discuss Talk List of geometric shapes Merge proposal date August 2010 Following is a list of some mathematics mathematically well defined shape s. See also list of geometric shapes , list of polygons, polyhedra and polytopes , and list of curves . 0D with no surface Point geometry point 1D with 0D surface interval mathematics interval Line geometry line 2D with 1D surface B zier curve As Bt sup n sup 0 s 1 0 t 1 s t 1, A sup n sup , A sup n &minus 1 sup B , ..., B sup n sup are vectors circle x sup 2 sup y sup 2 sup r sup 2 sup ellipse parabola hyperbola Plane mathematics plane polygon chiliagon decagon enneagon googolgon hectagon heptagon hendecagon hexagon myriagon octagon pentagon quadrilateral triangle 3D with 1D surface helix x sin z y cos z 3D with 2D surface B zier triangle As Bt Cu sup n sup 0 s 1 0 t 1 0 u 1 s t u 1, A sup p sup B sup q sup C sup r sup vectors if p q r n and p , q , r are nonnegative integers cylinder geometry cylinder hyperplane m bius strip platonic solid dodecahedron hexahedron cube icosahedron octahedron tetrahedron torus doughnut quadric cone geometry cone cylinder geometry cylinder ellipsoid spheroid sphere hyperboloid paraboloid sphere 4D with 3D surface polychoron hecatonicosachoron hexacosichoron hexadecachoron icositetrachoron pentachoron simplex tesseract spherical cone 5D with 4D surfaces regular 5 polytopes 5 dimensional simplex 5 dimensional cross polytope 5 dimensional hypercube 5 measure polytope Fractal s Apollonian gasket Cantor set Dragon curve Koch snowflake L vy C curve Lyapunov fractal Mandelbrot set Sierpinski carpet Peano curve Sierpinski triangle See also List of mathematical topics Periodic table of shapes The Periodic table of mathematical shapes Category Mathematics related lists Shapes ... more details
Semireg dual polyhedra db Semireg dual polyhedron stat table dtC In geometry , a triakis octahedron is an Archimedean solid Archimedean dual solid, or a Catalan solid . Its dual is the truncated cube . It can be seen as an octahedron with triangular pyramid s added to each face that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron , or, more fully, trigonal trisoctahedron . Both names reflect the fact that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron , a different polyhedron with three quadrilateral faces for every face of an octahedron. This convex polyhedron is topologically similar to the concave stellated octahedron . They have the same face connectivity, but the vertices are in different relative distances from the center. It is also called the small triakis octahedron , so as to differentiate it from the great triakis octahedron , the dual of the stellated truncated hexahedron . If its shorter edges have length 1, its surface area and volume are math A 3 sqrt 7 4 sqrt 2 math math V frac 1 2 3 2 sqrt 2 . math Cultural references A triakis octahedron is a vital element in the plot of cult author Hugh Cook science fiction author Hugh Cook s novel Chronicles of an Age of Darkness The Wishstone and the Wonderworkers The Wishstone and the Wonderworkers . References The Geometrical Foundation of Natural Structure book Section 3 9 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 The thirteen semiregular convex polyhedra and their duals, Page 17, Triakisoctahedron The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman Strass, ISBN 978 1 56881 220 5 http www.akpeters.com product.asp?ProdCode 2205 Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis oc ... more details
Uniform polyhedra db Uniform polyhedron stat table ThH In geometry , the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron , indexed as U sub 4 sub . It has 6 vertices and 12 edges, and 7 faces 4 triangular and 3 square. Its vertex figure is a antiparallelogram crossed quadrilateral . It has Coxeter Dynkin diagram of CDD node 1 d3 rat d2 node 3 node 1 . It is the only non prismatic uniform polyhedron with an odd number of faces. It is a hemipolyhedron . The hemi part of the name means some of the faces form a group with half as many members as some regular polyhedron here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube hence hemihexahedron . Hemi faces are also oriented in the same direction as the regular polyhedron s faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular . The half as many characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four right triangle s, with two visible from each side. Related polyhedra It has the same vertices and edges as the regular octahedron . It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces passing through the centre of the polyhedron. class wikitable width 400 style vertical align top text align center Image octahedron.png 100px BR Octahedron Image tetrahemihexahedron.png 100px BR Tetrahemihexahedron The dual figure is the tetrahemihexacron . It is covering space 2 covered by the cuboctahedron , ref name richter Harv Richter ref which accordingly has the same abstract polytope abstract vertex figure 2 triangles and two squares 3.4.3.4 and twice the vertices, edges, and faces. class wikitable width 400 style vertical align top text align center Image cuboctahedron.png 100px br Cuboctahedron Image tetrahemihexahedro ... more details
Infobox face uniform tiling Image File Tiling Dual Semiregular V4 8 8 Tetrakis Square.svg Type List of uniform tilings Dual semiregular tiling Face List triangle 45 45 90 triangle Symmetry Group 442 Face Type V4.8.8 Dual Truncated square tiling Property List face transitive In geometry , the tetrakis square tiling is a tiling of the Euclidean plane. It is square tiling with each square divided into four triangles from the center point, forming an infinite arrangement of lines . John Horton Conway Conway calls it a kisquadrille ref John H. Conway, Heidi Burgiel, Chaim Goodman Strass, The Symmetries of Things 2008, ISBN 978 1 56881 220 5 http www.akpeters.com product.asp?ProdCode 2205 Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table ref , represented by a Conway kis operator kis operation that adds a center point and triangles to replace the faces of a square tiling quadrille . It is also called the Union Jack lattice because of the similitude of its unit cell to the Union Jack UK flag . It is labeled V4.8.8 because each isosceles triangle face has two types of vertices one with 4 triangles, and two with 8 triangles. Dual tiling It is the dual tessellation of the truncated square tiling which has one square and two octagons at each vertex. ref MathWorld urlname DualTessellation title Dual tessellation ref File P1 dual.png 320px Related polyhedra and tilings It is topologically related to the polyhedron tetrakis hexahedron , V4.6.6 image Tetrakishexahedron.jpg 100px The symmetry type is with the coloring cmm a primitive cell is 8 triangles, a fundamental domain 2 triangles 1 2 for each color with the dark triangles in black and the light ones in white p4g a primitive cell is 8 triangles, a fundamental domain 1 triangle 1 2 each for black and white with the edges in black and the interiors in white p4m a primitive cell is 2 triangles, a fundamental domain 1 2 See also Tilings of regular polygons List of uniform tilings Notes reflist Reference ... more details
Mesh generation is the practice of generating a polygon al or polyhedron polyhedral polygon mesh mesh that approximates a geometric domain. The term grid generation is often used interchangeably. Typical uses are for rendering to a computer screen or for physical simulation such as finite element analysis or computational fluid dynamics . The input model form can vary greatly but common sources are CAD , NURBS , B rep and STL file format . The field is highly interdisciplinary, with contributions found in mathematics , computer science , and engineering . Three dimensional meshes created for finite element analysis need to consist of tetrahedron tetrahedra , pyramid geometry pyramid s, prism geometry prism s or hexahedron hexahedra . Those used for the finite volume method can consist of arbitrary polyhedron polyhedra . Those used for finite difference method s usually need to consist of piecewise structured arrays of hexahedra known as multi block structured meshes. See also Delaunay triangulation Fortune s algorithm Polygon mesh Regular grid Ruppert s algorithm Tessellation Unstructured grid Stretched grid method References citation last Edelsbrunner first Herbert authorlink Herbert Edelsbrunner isbn 9780521793094 publisher Cambridge University Press title Geometry and Topology for Mesh Generation year 2001 postscript none . citation last1 Frey first1 Pascal Jean last2 George first2 Paul Louis isbn 9781903398005 publisher Hermes Science title Mesh Generation Application to Finite Elements year 2000 postscript none . Citation author P. Smith and S. S. Sritharan year 1988 title Theory of Harmonic Grid Generation journal Complex Variables volume 10 pages 359 369. url http www.nps.edu Academics Schools GSEAS SRI R3.pdf postscript none Citation doi 10.1080 00036819208840072 author S. S. Sritharan year 1992 title Theory of Harmonic Grid Generation II journal Applicable Analysis volume 44 issue 1 pages 127 149. postscript none citation last1 Thompson first1 J. F. authorl ... more details
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