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
Infobox face uniform tiling Image File Order 3 heptakis heptagonal tiling.png Type Uniform tilings in hyperbolic plane Dual semiregular hyperbolic tiling Face List triangle Right triangle Edge Count Infinite Vertex Count Infinite Symmetry Group 732 Face Type V4.6.14 Dual Great rhombitriheptagonal tiling Property List face transitive In geometry , the order 3 bisected heptagonal tiling is a semiregular dual Uniform tilings in hyperbolic plane tiling of the hyperbolic plane . It is constructed by congruent right triangle s with 4, 6, and 14 triangles meeting at each vertex. The image shows a Poincar disk model projection of the hyperbolic plane. It is labeled V4.6.14 because each right triangle face has three types of vertices one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the dual tessellation of the great rhombitriheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex. Naming An alternative name is 3 7 kisrhombille by John Horton Conway Conway , seeing it as a 3 7 rhombic tiling, divided by a Conway kis operator kis operator, adding a center point to each rhombus, and dividing into four triangles File Order73 qreg rhombic til.png 160px 3 7 rhombic tiling rhombille Related polyhedra and tilings It is topologically related to a polyhedra sequence see Bisected hexagonal tiling Related polyhedra and tilings discussion . This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and are the reflection domains for the 2,3, n triangle group s for the heptagonal tiling, the important 2,3,7 triangle group . See also the Uniform tilings in hyperbolic plane The .5B7.2C3.5D .287 3 2.29 group family uniform tilings of the hyperbolic plane with 2,3,7 symmetry . class wikitable image tetrakishexahedron.jpg 120px BR Tetrakis Hexahedron V4.6.6 image Disdyakisdodecahedron.jpg 120px BR Disdyakis dodecahedron V4.6.8 image Disd ... more details
Infobox Book See Wikipedia WikiProject Novels or Wikipedia WikiProject Books name Here Come the Blobbies title orig translator image File here come blobbies cover.jpg 200px Cover image caption First edition cover author Jorge Antonio Tello Aliaga illustrator Jorge Antonio Tello Aliaga cover artist country United States language English language English series genre Children s literature publisher Pers Publishing pub date 28 July 2003 english pub date media type Print Hardcover and Audio CD pages 40 isbn 9781932179323 preceded by followed by Here Come the Blobbies is a children s picture book written and illustrated by Jorge Antonio Tello Aliaga. The Blobbies are cute creatures that can blobbiemorph into many different shapes. Storyline The Blobbies come from Blobbieworld, where every grain of sand, every gust of wind, and every drop of water can feel, talk and smile. It is ruled by the Elemental Blobbies Blobbie Fire, Blobbie Earth, Blobbie Water, Blobbie Air, and Blobbie Void. Seven new Blobbies in the colors of the rainbow, called Blobbie Colors, are created by Blobbie Fire and given strict orders not to wake up the eldest, Blobbie Void. They live and play until Blobbie Violet becomes bored with the simple shapes of Blobbieworld. It persuades Blobbie Indigo to help it go back into space and wake up Blobbie Void, whom they believe will teach them new shapes. Unfortunately as soon as they wake it up it turns into a portal that lets in evil creatures called Hexicones who attack with freezing beams of darkness. To save the Blobbie Colors, Blobbie Void transports them to an obscure planet called Earth. While there, the Blobbies learn many new and advanced shapes while reading books in a library. Eventually they decide to return and take on the Hexacones. Blobbie Void brings them back and they discover that the Hexacone King is carrying the evil Hexahedron Crystal, the source of their power. Blobbie Green sacrifices himself by turning into a Venus Flytrap and swallowing ... more details
with 14 vertices and 24 faces, a tetrakis hexahedron , formed by attaching a low square pyramid ... hexahedron. A different form of the tetrakis hexahedron, formed by using taller pyramids ... of the compound of three octahedra, has the same combinatorial structure as the tetrakis hexahedron ... more details
center Name Regular hexahedron Square Prism geometry prism Cuboid Trigonal trapezohedron align center ... No quadrilateral ly faced hexahedron No No No The vertices of a cube can be grouped into two groups ... group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. class wikitable ... valign top File Rhombic hexahedron spherical.png 100px BR Cube File Rhombic dodecahedron spherical.png ... 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 is one of a family of uniform polyhedra related to the cube and regular octahedron. class wikitable width 640 Norman Johnson mathematician Johnson name Parent Truncated Rectified Bitruncated BR tr. dual Birectified BR dual Cantellated Omnitruncated BR small Cantitruncated small Snub rowspan 2 Extended BR Schl fli symbol math begin Bmatrix 4 , 3 end Bmatrix math math t begin Bmatrix 4 , 3 end Bmatrix math math begin Bmatrix 4 3 end Bmatrix math math t begin Bmatrix 3 , 4 end Bmatrix math math begin Bmatrix 3 , 4 end Bmatrix math math r begin Bmatrix 4 3 end Bmatrix math math t beg ... more details
and tilings with n ,3 Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where ... V3.6.3.6 V3.7.3.7 V3.8.3.8 align center valign top File Rhombic hexahedron spherical.png 100px BR Cube ... more details
math sqrt 3 ,s 2 math math left frac pi 6 sqrt 3 right frac 1 3 approx 0.671 math cube hexahedron Image hexahedron.jpg 50px Hexahedron cube math ,s 3 math math 6 ,s 2 math math left frac pi 6 right frac ... more details
In geometry and polyhedral combinatorics , the Kleetope of a polyhedron or higher dimensional convex polytope math P is another polyhedron or polytope math P sup K sup formed by replacing each facet geometry facet of math P with a shallow Pyramid geometry pyramid . ref harvs last Gr nbaum year 1963 year2 1967 txt . ref Kleetopes are named after Victor Klee . ref citation first Joseph last Malkevitch url http www.ams.org samplings feature column fcarc klee title People Making a Difference publisher American Mathematical Society . ref Examples The triakis tetrahedron is the Kleetope of a tetrahedron , the triakis octahedron is the Kleetope of an octahedron , and the triakis icosahedron is the Kleetope of an icosahedron . In each of these cases the Kleetope is formed by adding a triangular pyramid to each face of the original polyhedron. John Horton Conway Conway generalizes Kepler s kis prefix as this same Conway kis operator kis operator . class wikitable Kleetopes of the Platonic solid s File Triakistetrahedron.jpg 150px BR The triakis tetrahedron , the Kleetope of the tetrahedron . File Tetrakishexahedron.jpg 150px BR The tetrakis hexahedron , the Kleetope of the cube . File Triakisoctahedron.jpg 150px BR The triakis octahedron , the Kleetope of the octahedron . File Pentakisdodecahedron.jpg 150px BR The pentakis dodecahedron , the Kleetope of the dodecahedron . File Triakisicosahedron.jpg 150px BR The triakis icosahedron , the Kleetope of the icosahedron . The tetrakis hexahedron is the Kleetope of the cube , formed by adding a square pyramid to each of its faces, and the pentakis dodecahedron is the Kleetope of the dodecahedron , formed by adding a pentagonal pyramid to each face of the dodecahedron. class wikitable Some other convex Kleetopes File Disdyakisdodecahedron.jpg 240px BR The disdyakis dodecahedron , the Kleetope of the rhombic dodecahedron . File Disdyakistriacontahedron.jpg 240px BR The disdyakis triacontahedron , the Kleetope of the rhombic triacont ... more details
Infobox Polyhedron with net Image File Square pyramid.png Polyhedron Type Johnson solid Johnson br triangular hebesphenorotunda J sub 92 sub J sub 1 sub pentagonal pyramid J sub 2 sub Face List 4 triangle s br 1 Square geometry square Edge Count 8 Vertex Count 5 Symmetry Group C sub 4v sub , 4 , 44 Vertex List 4 3 sup 2 sup .4 br 3 sup 4 sup Dual self Property List convex set convex Net Image File Square pyramid net.svg In geometry , a square pyramid is a Pyramid geometry pyramid having a square geometry square base. If the Apex geometry apex is perpendicularly above the center of the square, it will have C sub 4v sub symmetry. Johnson solid J1 If the sides are all equilateral triangle s, the pyramid is one of the Johnson solid s J sub 1 sub . The 92 Johnson solids were named and described by Norman Johnson mathematician Norman Johnson in 1966. The Johnson square pyramid can be characterized by a single edge length parameter a . The height H from the midpoint of the square to the apex , the surface area A including all five faces , and the volume V of such a pyramid are math H frac 1 sqrt 2 a math math A 1 sqrt 3 a 2 math math V frac sqrt 2 6 a 3. math Other square pyramids Other square pyramids have isosceles triangle sides. For square pyramids in general, with base length l and height h , the surface area and volume are math A l 2 l sqrt l 2 2h 2 math math V frac 1 3 l 2h. math Related polyhedra class wikitable width 480 Image Square bipyramid.png 160px Image Tetrakishexahedron.jpg 160px File Usech kvadrat piramid.png 160px align center A regular octahedron can be considered a square bipyramid , i.e. two Johnson square pyramids connected base to base. The tetrakis hexahedron can be constructed from a cube with short square pyramids added to each face. Square frustum is a square pyramid with the apex truncated. Dual polyhedron The square pyramid is topologically a self dual polyhedron . The dual edge lengths are different due to the polar reciprocation . class wiki ... more details
Cleanup date October 2008 Image Modular Origami.jpg thumb right Examples of modular origami made up of Sonobe units an augmented icosahedron and an augmented octahedron, which require 30 and 12 units, respectively. The Sonobe module is a unit used to build modular origami , created by Mitsunobu Sonobe. Each individual unit is folded from a square sheet of paper, of which only one face is visible in the finished module many ornamented variants of the plain Sonobe unit that expose both sides of the paper have been designed. The Sonobe unit has the shape of a parallelogram with 45 and 135 degrees angles, divided by creases into two diagonal tabs at the ends and two corresponding pockets within the inscribed center square. The system can build a wide range of three dimensional geometric forms by docking these tabs into the pockets of adjacent units. The most popular intermediate model is the augmented icosahedron , shown at right. It requires 30 units to build. Three interconnected Sonobe units will form an open bottomed triangular pyramid with a right angle apex equivalent to the corner of a cube and three tab pocket flaps protruding from the base. This particularly suits polyhedra that have equilateral triangular faces Sonobe modules can replace each notional edge of the original deltahedron by the central diagonal fold of one unit and each equilateral triangle with a right angle pyramid consisting of one half each of three units, without dangling flaps. The pyramids can be made to point inwards assembly is more difficult but some cases of encroaching can be obviously prevented. The simplest shape made of these pyramids, often called Toshie s Jewel , named after origami enthusiast Toshie Takahama , is a three unit hexahedron built around the notional scaffold of a flat equilateral triangle two faces , three edges the protruding tab pocket flaps are simply reconnected on the underside, resulting in two triangular pyramids joined at the base. Building pyramids on a tetr ... more details
The dihedral angle s for the edge transitive polyhedra are class wikitable align center Picture Name Schl fli symbol Schl fli BR symbol Vertex configuration Vertex Face BR configuration exact dihedral angle BR radians approximate BR dihedral angle BR degrees align center colspan 6 Platonic solid s regular convex align center Image Tetrahedron.png 30px align left Tetrahedron 3,3 3 sup 3 sup arccos 1 3 70.53 align center Image Hexahedron.png 30px align left Hexahedron or Cube geometry Cube 4,3 4 sup 3 sup 2 90 align center Image Octahedron.png 30px align left Octahedron 3,4 3 sup 4 sup &minus arccos 1 3 109.47 align center Image Dodecahedron.png 30px align left Dodecahedron 5,3 5 sup 3 sup &minus arctan 2 116.56 align center Image Icosahedron.png 30px align left Icosahedron 3,5 3 sup 5 sup &minus arccos &radic 5 3 138.19 align center colspan 6 Kepler Poinsot solid s regular nonconvex align center Image Small stellated dodecahedron.png 30px align left Small stellated dodecahedron 5 2,5 5 2 sup 5 sup &minus arctan 2 116.56 align center Image Great dodecahedron.png 30px align left Great dodecahedron 5,5 2 5 sup 5 2 sup arctan 2 63.435 align center Image Great stellated dodecahedron.png 30px align left Great stellated dodecahedron 5 2,3 5 2 sup 3 sup arctan 2 63.435 align center Image Great icosahedron.png 30px align left Great icosahedron 3,5 2 3 sup 5 2 sup arcsin 2 3 41.810 align center colspan 6 Polyhedron Archimedean solid Quasiregular solids Rectification geometry Rectified regular align center Image Uniform polyhedron 33 t1.png 30px align left Tetratetrahedron math begin Bmatrix 3 3 end Bmatrix math 3.3.3.3 &minus arccos 1 3 109.47 align center Image Cuboctahedron.png 30px align left Cuboctahedron math begin Bmatrix 3 4 end Bmatrix math 3.4.3.4 &minus arccos 1 sqrt 3 125.264 align center Image Icosidodecahedron.png 30px align left Icosidodecahedron math begin Bmatrix 3 5 end Bmatrix math 3.5.3.5 math pi arccos sqrt frac 5 2 sqrt 5 15 math 142.623 alig ... more details
Hexahedron flake A hexahedron, or cube, flake defined in the same way as the Sierpinski tetrahedron ... frac log 20 log 3 math 2.7268. Another hexahedron flake can be produced in a manner similar to the Vicsek ... more details
other uses In geometry , a cuboid is a solid figure bounded by six faces, forming a convex polyhedron . There are two competing but incompatible definitions of a cuboid in mathematical literature. In the more general definition of a cuboid, the only additional requirement is that these six faces each be a quadrilateral , and that the undirected graph formed by the vertices and edges of the polyhedron should be graph isomorphism isomorphic to the graph of a cube . ref citation title Polytopes and Symmetry first Stewart Alexander last Robertson publisher Cambridge University Press year 1984 isbn 978 0 521 27739 6 page 75 ref Alternatively, the word cuboid is sometimes used to refer to a shape of this type in which each of the faces is a rectangle and so each pair of adjacent faces meets in a right angle this more restrictive type of cuboid is also known as a right cuboid , rectangular box , rectangular hexahedron , right rectangular prism , or rectangular parallelepiped . ref citation title Elements of Synthetic Solid Geometry first Nathan Fellowes last Dupuis publisher Macmillan year 1893 page 53 ref General cuboids By Euler characteristic Euler s formula the number of faces F , vertices V , and edges E of any convex polyhedron are related by the formula F V E 2 . In the case of a cuboid this gives 6 8 12 2 that is, like a cube, a cuboid has 6 Face geometry faces , 8 vertex geometry vertices , and 12 edges. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum the shape formed by truncation of the apex of a square pyramid . Rectangular cuboid cellpadding 5 style background fff float right margin left 10px width 250px class wikitable style background e7dcc3 colspan 2 Rectangular Cuboid colspan 2 Image Cuboid.png 240px Rectangular cuboid style background aeae74 Type Prism geometry Prism style background e7dcc3 Faces 6 rectangle s style background e7dcc3 Edges 12 style background e7dcc3 Vertices 8 style background e7dcc3 ... more details
border 1 bgcolor ffffff cellpadding 5 align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 Parallelepiped align center colspan 2 Image Parallelepiped.svg 240px Rhombohedron bgcolor e7dcc3 Type Prism geometry Prism bgcolor e7dcc3 Faces 6 parallelogram s bgcolor e7dcc3 Edges 12 bgcolor e7dcc3 Vertices 8 bgcolor e7dcc3 List of spherical symmetry groups Symmetry group Cyclic symmetries C sub i sub , 2 sup sup ,2 sup sup , 1× bgcolor e7dcc3 Properties convex In geometry , a parallelepiped is a three dimensional figure formed by six parallelogram s. The term rhomboid is also sometimes used with this meaning. By analogy, it relates to a parallelogram just as a cube relates to a square geometry square . In Euclidean geometry , its definition encompasses all four concepts i.e., parallelepiped , parallelogram , cube , and square . In this context of affine geometry , in which angles are not differentiated, its definition admits only parallelograms and parallelepipeds . Three equivalent definitions of parallelepiped are a polyhedron with six faces hexahedron , each of which is a parallelogram, a hexahedron with three pairs of parallel faces, and a prism geometry prism of which the base is a parallelogram . The rectangular cuboid six rectangular faces , cube six square geometry square faces , and the rhombohedron six rhombus faces are all specific cases of parallelepiped. Parallelepiped is now usually IPAc en icon p r l l p p d , IPAc en p r l l p a p d , or IPAc en p d traditionally it was IPAc en p r l l p p d respell PARR lel EP i ped ref Oxford English Dictionary 1904 Webster s Second International 1947 ref in accordance with its etymology in Ancient Greek Greek , a body having parallel planes . Parallelepipeds are a subclass of the prismatoid s. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four paral ... more details
truncated hexahedron , the small rhombihexahedron having the triangular faces and six square ... br Stellated truncated hexahedron In the arts Image Pacioli.jpg thumb 300px Rhombicuboctahedron in top ... more details
, hexahedron, octahedron, dodecahedron and icosahedron are present in it. Moreover, an attempt ... Platonic bodies tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron are present ... more details
Image hexahedron.jpg 100px BR Hexahedron BR Cube BR 3& 124 2 4 BR W3, U06, K11, C18 BR V 8,E 12,F 6 6 4 br 2 Image Truncatedhexahedron.jpg 100px BR Truncated cube Truncated hexahedron BR 2 3& 124 4 ... Group O sub h sub Group I sub h sub Group I sub h sub Image Stellated truncated hexahedron vertfig.png ... truncated hexahedron.png 100px BR Stellated truncated hexahedron BR Quasitruncated hexahedron BR ... more details
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