Alternate names Cantellated hexeract Small rhombated hexeract acronym srox Jonathan Bowers ref Klitzing ... names Bicantellated hexeract Small birhombated hexeract acronym saborx Jonathan Bowers ref Klitzing ... names Cantitruncated hexeract Great rhombihexeract acronym grox Jonathan Bowers ref Klitzing, o3o3o3x3x4x ... Bicantitruncated hexeract Great birhombihexeract acronym gaborx Jonathan Bowers ref Klitzing, o3o3x3x3x4o ... , including the regular 6 cube or 6 orthoplex . Hexeract family Notes reflist References Harold Scott ... more details
class wikitable align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 6 cube BR Hexeract bgcolor ffffff align center colspan 2 File 6 cube graph.svg 280px BR Orthogonal projection BR inside Petrie polygon BR Orange vertices are doubled, and the center yellow has 4 vertices bgcolor e7dcc3 Type Regular 6 polytope bgcolor e7dcc3 Family hypercube bgcolor e7dcc3 Schl fli symbol 4,3 sup 4 sup bgcolor e7dcc3 Coxeter Dynkin diagram CDD node 1 4 node 3 node 3 node 3 node 3 node bgcolor e7dcc3 5 faces 12 5 cube 4,3,3,3 File 5 cube graph.svg 25px bgcolor e7dcc3 4 faces 60 tesseract 4,3,3 File 4 cube graph.svg 25px bgcolor e7dcc3 Cells 160 cube 4,3 File 3 cube graph.svg 25px bgcolor e7dcc3 Faces 240 square geometry 4 File 2 cube.svg 25px bgcolor e7dcc3 Edges 192 bgcolor e7dcc3 Vertices 64 bgcolor e7dcc3 Vertex figure 5 simplex bgcolor e7dcc3 Petrie polygon dodecagon bgcolor e7dcc3 Coxeter group B sub 6 sub , 3 sup 4 sup ,4 bgcolor e7dcc3 Dual 6 orthoplex File 6 orthoplex.svg 25px bgcolor e7dcc3 Properties Convex polytope convex In geometry , a 6 cube is a six dimension al hypercube with 64 Vertex geometry vertices , 192 Edge geometry edge s, 240 square Face geometry faces , 160 cubic Cell mathematics cells , 60 tesseract hypercell 4 faces , and 12 5 cube 5 faces. It has Schl fli symbol 4,3 sup 4 sup , being composed of 3 5 cube s around each 4 face. It can be called a hexeract , a portmanteau of tesseract the 4 cube with hex for six dimensions in Greek language Greek . It can also be called a regular dodeca 6 tope or dodecapeton , being a 6 polytope 6 dimensional polytope constructed from 12 regular Facet geometry facet s. Related polytopes It is a part of an infinite ... 6Cube QuasiCrystal.jpg 320px br Hexeract Quasicrystal structure orthographically projected br to 3D ... generated from the B sub 6 sub Coxeter plane , including the regular 6 cube or 6 orthoplex . Hexeract ... Garrett Jones Polytopes Category 6 polytopes Geometry stub cs Hexerakt es Hexeract it Esseratto ru ... more details
, or hexeract . Construction There are two Coxeter group s associated with the 6 orthoplex, one regular polytope regular , Dual polytope dual of the hexeract with the C sub 6 sub or 4,3,3,3,3 Coxeter ... the regular 6 cube or 6 orthoplex. Hexeract family References Harold Scott MacDonald Coxeter H.S.M. ... more details
class wikitable align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 PAGENAME bgcolor ffffff align center colspan 2 no image bgcolor e7dcc3 Type List of regular polytopes Tessellations of Euclidean Space Regular 6 space honeycomb bgcolor e7dcc3 Family Hypercube honeycomb bgcolor e7dcc3 Schl fli symbol 4,3,3,3,3,4 BR 4,3,3,3,3 sup 1,1 sup bgcolor e7dcc3 Coxeter Dynkin diagram s CDD node 1 4 node 3 node 3 node 3 node 3 node 4 node BR CDD node 1 3 node 3 node 3 node 3 node split1 nodes BR CDD node 1 4 node 3 node 3 node 3 node 3 node 4 node 1 BR CDD node 1 4 node 3 node 4 node 2 node 1 4 node 3 node 4 node bgcolor e7dcc3 6 face type hexeract 4,3,3,3,3 bgcolor e7dcc3 5 face type penteract 4,3,3,3 bgcolor e7dcc3 4 face type tesseract 4,3,3 bgcolor e7dcc3 Cell type cube 4,3 bgcolor e7dcc3 Face type square geometry 4 bgcolor e7dcc3 Face figure cube 4,3 BR octahedron bgcolor e7dcc3 Edge figure 8 tesseract 4,3,3 BR 16 cell bgcolor e7dcc3 Vertex figure 64 hexeract 4,3,3,3,3 BR hexacross bgcolor e7dcc3 Coxeter group math tilde C 6 math , 4,3,3,3,3,4 BR math tilde B 6 math , 4,3,3,3,3 sup 1,1 sup bgcolor e7dcc3 Dual Self dual polytope self dual bgcolor e7dcc3 Properties vertex transitive , edge transitive , face transitive , cell transitive The 6 cube honeycomb or hexeractic honeycomb is the only regular space filling tessellation or honeycomb geometry honeycomb in Euclidean 6 space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3 space. There are many different Wythoff construction s of this honeycomb. The most symmetric form is Regular polytope regular , with Schl fli symbol 4,3,3,3,3,4 . Another form has two alternating 6 cube facets like a checkerboard with Schl fli symbol 4,3,3,3,3 sup 1,1 sup . The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schl fli symbol &infin sup 6 sup . See also Tesseractic tetracomb Penteractic pentacomb List of regular polytopes References ... more details
Properties Convex polytope convex Alternate names Truncated hexeract Acronym tox Jonathan Bowers ... hexeract Acronym botox Jonathan Bowers ref Klitzing, o3o3o3x3x4o botox ref Construction and coordinates ... Properties Convex polytope convex Alternate names Tritruncated hexeract Acronym xog Jonathan Bowers ... from the B sub 6 sub Coxeter plane , including the regular 6 cube or 6 orthoplex . Hexeract family Notes ... more details
names Rectified hexeract acronym rax Jonathan Bowers Construction The rectified 6 cube may be constructed ... Alternate names Birectified hexeract acronym brox Jonathan Bowers Construction The birectified 6 cube ... . Hexeract family Notes reflist References Harold Scott MacDonald Coxeter H.S.M. Coxeter H.S.M. Coxeter ... more details
class wikitable align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 7 cube BR Hepteract bgcolor ffffff align center colspan 2 File 7 cube t0.svg 280px BR Orthogonal projection BR inside Petrie polygon BR The central orange vertex is doubled bgcolor e7dcc3 Type Regular 7 polytope bgcolor e7dcc3 Family hypercube bgcolor e7dcc3 Schl fli symbol 4,3 sup 5 sup bgcolor e7dcc3 Coxeter Dynkin diagram CDD node 1 4 node 3 node 3 node 3 node 3 node 3 node bgcolor e7dcc3 6 faces 14 hexeract 4,3 sup 4 sup File 6 cube graph.svg 25px bgcolor e7dcc3 5 faces 84 penteract 4,3 sup 3 sup File 5 cube graph.svg 25px bgcolor e7dcc3 4 faces 280 tesseract 4,3,3 File 4 cube graph.svg 25px bgcolor e7dcc3 Cells 560 cube 4,3 File 3 cube graph.svg 25px bgcolor e7dcc3 Faces 672 square geometry 4 File 2 cube.svg 25px bgcolor e7dcc3 Edges 448 bgcolor e7dcc3 Vertices 128 bgcolor e7dcc3 Vertex figure 6 simplex File 6 simplex graph.svg 25px bgcolor e7dcc3 Petrie polygon tetradecagon bgcolor e7dcc3 Coxeter group C sub 7 sub , 3 sup 5 sup ,4 bgcolor e7dcc3 Dual 7 orthoplex bgcolor e7dcc3 Properties Convex polytope convex In geometry , a 7 cube is a seven dimensional space seven dimensional hypercube with 128 Vertex geometry vertices , 448 Edge geometry edge s, 672 square Face geometry faces , 560 cubic Cell mathematics cells , 280 tesseract hypercell 4 faces , 84 penteract 5 faces, and 14 hexeract 6 faces. It can be named by its Schl fli symbol 4,3 sup 5 sup , being composed of 3 6 cube s around each 5 face. It can be called a hepteract , a portmanteau of tesseract the 4 cube and hepta for seven dimensions in Greek language Greek . It can also be called a regular tetradeca 7 tope or tetradecaexon , being a 7 polytope 7 dimensional polytope constructed from 14 regular Facet geometry facet s. Related polytopes It is a part of an infinite family of polytopes, called hypercube s. The Dual polytope dual of a Hepteract can be called a 7 orthoplex , and is a part of the infinite family of cros ... more details
class wikitable align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 PAGENAME bgcolor ffffff align center colspan 2 no image bgcolor e7dcc3 Type List of regular polytopes Tessellations of Euclidean Space Regular 7 space honeycomb bgcolor e7dcc3 Family Hypercube honeycomb bgcolor e7dcc3 Schl fli symbol 4,3,3,3,3,3,4 BR 4,3,3,3,3,3 sup 1,1 sup BR &infin sup 7 sup bgcolor e7dcc3 Coxeter Dynkin diagram s CDD node 1 4 node 3 node 3 node 3 node 3 node 3 node 4 node BR CDD node 1 3 node 3 node 3 node 3 node 3 node split1 nodes BR CDD node 1 4 node 3 node 3 node 3 node 3 node 3 node 4 node 1 bgcolor e7dcc3 7 face type hepteract 4,3,3,3,3,3 bgcolor e7dcc3 6 face type hexeract 4,3,3,3,3 bgcolor e7dcc3 5 face type penteract 4,3,3,3 bgcolor e7dcc3 4 face type tesseract 4,3,3 bgcolor e7dcc3 Cell type cube 4,3 bgcolor e7dcc3 Face type square geometry 4 bgcolor e7dcc3 Face figure cube 4,3 BR octahedron bgcolor e7dcc3 Edge figure 8 tesseract 4,3,3 BR 16 cell bgcolor e7dcc3 Vertex figure 128 hepteract 4,3,3,3,3,3 BR heptacross bgcolor e7dcc3 Coxeter group 4,3,3,3,3,3,4 bgcolor e7dcc3 Dual Self dual polytope self dual bgcolor e7dcc3 Properties vertex transitive , edge transitive , face transitive , cell transitive The 7 cubic honeycomb or hepteractic honeycomb is the only regular space filling tessellation or honeycomb geometry honeycomb in Euclidean 7 space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3 space. There are many different Wythoff construction s of this honeycomb. The most symmetric form is Regular polytope regular , with Schl fli symbol 4,3 sup 5 sup ,4 . Another form has two alternating 7 cube facets like a checkerboard with Schl fli symbol 4,3 sup 4 sup ,3 sup 1,1 sup . The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schl fli symbol &infin sup 7 sup . See also Tesseractic honeycomb Penteractic honeycomb Hexeractic honeycomb List of regular polytopes Referen ... more details
class wikitable align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 8 cube BR Octeract bgcolor ffffff align center colspan 2 File 8 cube.svg 285px BR Orthogonal projection BR inside Petrie polygon bgcolor e7dcc3 Type Regular 8 polytope bgcolor e7dcc3 Family hypercube bgcolor e7dcc3 Schl fli symbol 4,3 sup 6 sup bgcolor e7dcc3 Coxeter Dynkin diagram CDD node 1 4 node 3 node 3 node 3 node 3 node 3 node 3 node bgcolor e7dcc3 7 faces 16 hepteract 4,3 sup 5 sup Image 7 cube graph.svg 35px bgcolor e7dcc3 6 faces 112 hexeract 4,3 sup 4 sup Image 6 cube graph.svg 25px bgcolor e7dcc3 5 faces 448 penteract 4,3 sup 3 sup Image 5 cube graph.svg 25px bgcolor e7dcc3 4 faces 1120 tesseract 4,3 sup 2 sup Image 4 cube graph.svg 25px bgcolor e7dcc3 Cells 1792 cube 4,3 Image 3 cube.png 25px bgcolor e7dcc3 Faces 1792 square geometry 4 File 2 cube.svg 25px bgcolor e7dcc3 Edges 1024 bgcolor e7dcc3 Vertices 256 bgcolor e7dcc3 Vertex figure 7 simplex File 7 simplex graph.svg 25px bgcolor e7dcc3 Petrie polygon hexadecagon bgcolor e7dcc3 Coxeter group C sub 8 sub , 3 sup 6 sup ,4 bgcolor e7dcc3 Dual 8 orthoplex File 8 orthoplex.svg 25px bgcolor e7dcc3 Properties Convex polytope convex In geometry , an 8 cube is an eight dimension al hypercube 8 cube . It has 256 Vertex geometry vertices , 1024 Edge geometry edge s, 1792 square Face geometry faces , 1792 cubic Cell mathematics cells , 1120 tesseract hypercell 4 faces , 448 5 cube 5 faces, 112 6 cube 6 faces, and 16 7 cube 7 faces. It is represented by Schl fli symbol 4,3 sup 6 sup , being composed of 3 7 cube s around each 6 face. It is called an octeract , a portmanteau of tesseract the 4 cube and oct for eight dimensions in Greek language Greek . It can also be called a regular hexdeca 8 tope or hexadecazetton , being a 8 polytope 8 dimensional polytope constructed from 16 regular Facet geometry facet s. Related polytopes It is a part of an infinite family of polytopes, called hypercube s. The Dual polytope dual of an 8 cu ... more details
align right class wikitable width 320 Graphs of three List of regular polytopes Five dimensional regular polytopes and higher regular and three Uniform polypeton uniform polytopes align center valign top Image 6 simplex t0.svg 160px BR 6 simplex Heptapeton File 6 cube t5.svg 160px BR 6 orthoplex , 3 sub 11 sub align center valign top Image 6 cube t0.svg 160px BR 6 cube Hexeract Image 6 demicube t0 D6.svg 160px BR 6 demicube 1 sub 31 sub BR Demihexeract align center valign top Image E6 graph.svg 150px BR Gosset 2 21 polytope 2 sub 21 sub Image Gosset 1 22 polytope.svg 150px BR Gosset 1 22 polytope 1 sub 22 sub In Six dimensional space six dimensional geometry , a 6 polytope is a polytope , bounded by 5 polytope Facet mathematics facets . In the context of uniform polytope s, a 5 polytope A note on generality of terms for n polytopes and elements proposed name polypeton plural polypeta has been advocated, from the Greek language Greek root poly meaning many , a shortened Numerical prefix penta meaning five , and suffix on . Five refers to the dimension of the 5 polytope Facet mathematics facets . Definition A 6 polytope, or polypeton, is a closed six dimensional figure with Vertex geometry vertices , Edge geometry edges , face geometry faces , cell mathematics cells 3 faces , 4 faces, and 5 faces. A vertex is a Point geometry point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron . A 4 face is a polychoron , and a 5 face is a polyteron . Furthermore, the following requirements must be met Each 4 face must join exactly two 5 faces facets . Adjacent facets are not in the same five dimensional hyperplane . The figure is not a compound of other figures which meet the requirements. Regular 6 polytopes Regular 6 polytopes can be generated from Coxeter group s represented by the Schl fli symbol p,q,r,s,t with t p,q,r,s polyteron Facet mathematics facets around ... more details
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