1.
5-simplex
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In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices,15 edges,20 triangle faces,15 tetrahedral cells and it has a dihedral angle of cos−1, or approximately 78. 46°. It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions, the name hexateron is derived from hexa- for having six facets and teron for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix, the hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell. These construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron, the 5-simplex, as 220 polytope is first in dimensional series 22k. The regular 5-simplex is one of 19 uniform polytera based on the Coxeter group, the 5-simplex can also be considered a 5-cell pyramid, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 5D uniform polytopes x3o3o3o3o - hix, archived from the original on 4 February 2007. Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary

2.
10-demicube
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In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 171 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or. Cartesian coordinates for the vertices of a demidekeract centered at the origin are alternate halves of the dekeract, with an odd number of plus signs. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 10D uniform polytopes x3o3o *b3o3o3o3o3o3o3o - hede, archived from the original on 4 February 2007

3.
10-orthoplex
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It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 711. It is one of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube, decacross is derived from combining the family name cross polytope with deca for ten in Greek Chilliaicositetraxennon as a 1024-facetted 10-polytope. Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are, Every vertex pair is connected by an edge, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 10D uniform polytopes x3o3o3o3o3o3o3o3o4o - ka, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary

4.
10-simplex
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In geometry, a 10-simplex is a self-dual regular 10-polytope. Its dihedral angle is cos−1, or approximately 84. 26° and it can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn, having 9-dimensional facets and this construction is based on facets of the 11-orthoplex. The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices,55 edges, but only 1/3 the faces. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 10D uniform polytopes x3o3o3o3o3o3o3o3o3o - ux, Polytopes of Various Dimensions Multi-dimensional Glossary

5.
1 32 polytope
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In 7-dimensional geometry,132 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, the rectified 132 is constructed by points at the mid-edges of the 132. This polytope can tessellate 7-dimensional space, with symbol 133, and it is the Voronoi cell of the dual E7* lattice. Emanuel Lodewijk Elte named it V576 in his 1912 listing of semiregular polytopes, Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch. Pentacontihexa-hecatonicosihexa-exon - 56-126 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space and this makes the birectified 6-simplex,032, The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb,134, the rectified 132 is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a prism, the product of a regular tetrahedra and triangle, doubled into a prism. Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space and these mirrors are represented by its Coxeter-Dynkin diagram, and the ring represents the position of the active mirror. This makes the tetrahedron-triangle duoprism prism, ××, List of E7 polytopes Elte, E. L. The Semiregular Polytopes of the Hyperspaces, Groningen, University of Groningen H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Klitzing, Richard. O3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin

6.
1 33 honeycomb
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In 7-dimensional geometry,133 is a uniform honeycomb, also given by Schläfli symbol, and is composed of 132 facets. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing a node on the end of one of the 3-length branch leaves the 132, the vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex,033, the edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, ×, each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others, the best known for 7 dimensions is 126. The E ~7 group is related to the F ~4 by a geometric folding, E ~7 contains A ~7 as a subgroup of index 144. Both E ~7 and A ~7 can be seen as affine extension from A7 from different nodes, The E7* lattice has double the symmetry, the Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb. The 133 is fourth in a series of uniform polytopes and honeycombs. The final is a noncompact hyperbolic honeycomb,134, the rectified 133 or 0331, Coxeter diagram has facets and, and vertex figure. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Coxeter The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 978-0-486-40919-1 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi-Regular Polytopes III

7.
1 42 polytope
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In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, the rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421. The 142 is composed of 2400 facets,240132 polytopes and its vertex figure is a birectified 7-simplex. This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 152, Coxeter named it 142 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 7-demicube,141, Removing the node on the end of the 4-length branch leaves the 132. The vertex figure is determined by removing the ringed node and ringing the neighboring node and this makes the birectified 7-simplex,042. Orthographic projections are shown for the sub-symmetries of E8, E7, E6, B8, B7, B6, B5, B4, B3, B2, A7, vertices are shown as circles, colored by their order of overlap in each projective plane. The rectified 142 is named from being a rectification of the 142 polytope, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 1-length branch leaves the birectified 7-simplex, Removing the node on the end of the 3-length branch leaves the 132. The vertex figure is determined by removing the ringed node and ringing the neighboring node and this makes the 5-cell-triangle duoprism prism. Orthographic projections are shown for the sub-symmetries of B6, B5, B4, B3, B2, A7, vertices are shown as circles, colored by their order of overlap in each projective plane. List of E8 polytopes H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Klitzing, Richard. O3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy

8.
1 52 honeycomb
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In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure and it is the final figure in the 1k2 polytope family. It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 8-demicube,151, removing the node on the end of the 5-length branch leaves the 142. The vertex figure is determined by removing the ringed node and ringing the neighboring node and this makes the birectified 8-simplex,052. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 GoogleBook H. S. M. Coxeter, Regular and Semi-Regular Polytopes III

9.
2 21 polytope
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In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 6-ic semi-regular figure. It is also called the Schläfli polytope and its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, the rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the face centers of the 221. The 221 has 27 vertices, and 99 facets,27 5-orthoplexes and 72 5-simplices and its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon and its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements can also be extracted and drawn on this projection, the Schläfli graph contains the 1-skeleton of this polytope. E. L. Elte named it V27 in his 1912 listing of semiregular polytopes, icosihepta-heptacontidi-peton - 27-72 facetted polypeton The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope, Its construction is based on the E6 group. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the 5-simplex. Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form, every simplex facet touches an 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node, vertices are colored by their multiplicity in this projection, in progressive order, red, orange, yellow. The number of vertices by color are given in parentheses, the 221 is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. This can be seen in the Coxeter plane projections, the 24 vertices of the 24-cell are projected in the same two rings as seen in the 221. This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram, the regular complex polygon 333, in C2 has a real representation as the 221 polytope, in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess and it has 27 vertices,72 3-edges, and 2733 faces. Its complex reflection group is 333, order 648, the 221 is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope, Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes

10.
2 22 honeycomb
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In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol and it is constructed from 221 facets and has a 122 vertex figure, with 54221 polytopes around every vertex. Its vertex arrangement is the E6 lattice, and the system of the E6 Lie group so it can also be called the E6 honeycomb. It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space, the facet information can be extracted from its Coxeter–Dynkin diagram. Removing a node on the end of one of the 2-node branches leaves the 221, its only facet type, The vertex figure is determined by removing the ringed node, the edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t2. The face figure is the figure of the edge figure, here being a triangular duoprism. Each vertex of this tessellation is the center of a 5-sphere in the densest known packing in 6 dimensions, with kissing number 72, the 222 honeycombs vertex arrangement is called the E6 lattice. The E62 lattice, with symmetry, can be constructed by the union of two E6 lattices, ∪ The E6* lattice with symmetry, the Voronoi cell of the E6* lattice is the rectified 122 polytope, and the Voronoi tessellation is a bitruncated 222 honeycomb. It is constructed by 3 copies of the E6 lattice vertices, the 222 honeycomb is one of 127 uniform honeycombs with E ~6 symmetry. 24 of them have doubled symmetry with 2 equally ringed branches and, the birectified 222 honeycomb, has within its symmetry construction 3 copies of facets. The 222 honeycomb, is fourth in a series of uniform polytopes. The final is a paracompact hyperbolic honeycomb,322, each progressive uniform polytope is constructed from the previous as its vertex figure. The 222 honeycomb is third in another dimensional series 22k. S. M, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 GoogleBook H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, R. T. Worley, The Voronoi Region of E6*. Conway, John H. Sloane, Neil J. A, p125-126,8.3 The 6-dimensional lattices, E6 and E6*

11.
2 31 polytope
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In 7-dimensional geometry,231 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, the rectified 231 is constructed by points at the mid-edges of the 231. The 231 is composed of 126 vertices,2016 edges,10080 faces,20160 cells,16128 4-faces,4788 5-faces,632 6-faces and its vertex figure is a 6-demicube. Its 126 vertices represent the vectors of the simple Lie group E7. This polytope is the figure for a uniform tessellation of 7-dimensional space,331. E. L. Elte named it V126 in his 1912 listing of semiregular polytopes and it was called 231 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. Pentacontihexa-pentacosiheptacontihexa-exon - 56-576 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the branch leaves the 6-simplex. There are 576 of these facets and these facets are centered on the locations of the vertices of the 321 polytope. Removing the node on the end of the 3-length branch leaves the 221, there are 56 of these facets. These facets are centered on the locations of the vertices of the 132 polytope, the vertex figure is determined by removing the ringed node and ringing the neighboring node. The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231, rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the rectified 6-simplex. Removing the node on the end of the 2-length branch leaves the, removing the node on the end of the 3-length branch leaves the rectified 221. The vertex figure is determined by removing the ringed node and ringing the neighboring node, list of E7 polytopes Elte, E. L. The Semiregular Polytopes of the Hyperspaces, Groningen, University of Groningen H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Klitzing, Richard. X3o3o3o *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq

12.
2 41 polytope
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In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 241, describing its bifurcating Coxeter-Dynkin diagram, the rectified 241 is constructed by points at the mid-edges of the 241. The birectified 241 is constructed by points at the face centers of the 241. The 241 is composed of 17,520 facets,144,960 6-faces,544,320 5-faces,1,209,600 4-faces,1,209,600 cells,483,840 faces,69,120 edges and its vertex figure is a 7-demicube. This polytope is a facet in the uniform tessellation,251 with Coxeter-Dynkin diagram and it is named 241 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. The facet information can be extracted from its Coxeter-Dynkin diagram, Removing the node on the short branch leaves the 7-simplex. There are 17280 of these facets Removing the node on the end of the 4-length branch leaves the 231, there are 240 of these facets. They are centered at the positions of the 240 vertices in the 421 polytope, the vertex figure is determined by removing the ringed node and ringing the neighboring node. Petrie polygon projections can be 12,18, or 30-sided based on the E6, E7, the 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown, the rectified 241 is a rectification of the 241 polytope, with vertices positioned at the mid-edges of the 241. The facet information can be extracted from its Coxeter-Dynkin diagram, Removing the node on the short branch leaves the rectified 7-simplex. Removing the node on the end of the 4-length branch leaves the rectified 231, Removing the node on the end of the 2-length branch leaves the 7-demicube,141. The vertex figure is determined by removing the ringed node and ringing the neighboring node and this makes the rectified 6-simplex prism. Petrie polygon projections can be 12,18, or 30-sided based on the E6, E7, the 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown, list of E8 polytopes Elte, E. L. The Semiregular Polytopes of the Hyperspaces, Groningen, University of Groningen H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Klitzing, Richard. X3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay

13.
2 51 honeycomb
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In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube vertex figure and it is the final figure in the 2k1 family. It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the branch leaves the 8-simplex. Removing the node on the end of the 5-length branch leaves the 241, the vertex figure is determined by removing the ringed node and ringing the neighboring node. The edge figure is the figure of the vertex figure. This makes the rectified 7-simplex,051. S. M, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi-Regular Polytopes III

14.
3 21 polytope
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In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 7-ic semi-regular figure. Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, the rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the face centers of the 321. The trirectified 321 is constructed by points at the centers of the 321. In 7-dimensional geometry, the 321 is a uniform polytope and it has 56 vertices, and 702 facets,126311 and 576 6-simplexes. For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon and its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements can also be extracted and drawn on this projection, the 1-skeleton of the 321 polytope is called a Gosset graph. This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and it is also called the Hess polytope for Edmund Hess who first discovered it. It was enumerated by Thorold Gosset in his 1900 paper and he called it an 7-ic semi-regular figure. E. L. Elte named it V56 in his 1912 listing of semiregular polytopes. Coxeter called it 321 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3,2, and 1, Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the 6-simplex. Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its form,311. Every simplex facet touches an 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex, the vertex figure is determined by removing the ringed node and ringing the neighboring node. The 321 is fifth in a series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope, Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes. It is in a series of uniform polytopes and honeycombs

15.
3 31 honeycomb
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In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. The edge figure is determined by removing the ringed node and ringing the neighboring node, the face figure is determined by removing the ringed node and ringing the neighboring node. The cell figure is determined by removing the ringed node of the face figure, each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions, its kissing number is 126, represented by the vertices of its vertex figure 231. The 331 honeycombs vertex arrangement is called the E7 lattice, E ~7 contains A ~7 as a subgroup of index 144. The Voronoi cell of the E7* lattice is the 132 polytope and it is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 GoogleBook H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, R. T. Worley, The Voronoi Region of E7*. Conway, John H. Sloane, Neil J. A, p124-125,8.2 The 7-dimensinoal lattices, E7 and E7*

16.
4 21 polytope
–
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 8-ic semi-regular figure. Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, the rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the face centers of the 421. The trirectified 421 is constructed by points at the centers of the 421. The 421 is composed of 17,280 7-simplex and 2,160 7-orthoplex facets and its vertex figure is the 321 polytope. For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon and its 6720 edges are drawn between the 240 vertices. Specific higher elements can also be extracted and drawn on this projection, as its 240 vertices represent the root vectors of the simple Lie group E8, the polytope is sometimes referred to as the E8 polytope. The vertices of this polytope can be obtained by taking the 240 integral octonions of norm 1, because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop. This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure and it is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets. E. L. Elte named it V240 in his 1912 listing of semiregular polytopes, Coxeter called it 421 because its Coxeter-Dynkin diagram has three branches of length 4,2, and 1, with a single node on the terminal node of the 4 branch. Dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton - 2160-17280 facetted polyzetton It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space and these 56 points are the vertices of a 321 polytope in 7 dimensions. These 126 points are the vertices of a 231 polytope in 7 dimensions. Each vertex also has 56 third nearest neighbors, which are the negatives of its nearest neighbors, there are 17,280 simplex facets and 2160 orthoplex facets. Since every 7-simplex has 7 6-simplex facets, each incident to no other 6-simplex, since every 7-orthoplex has 128 6-simplex facets, half of which are not incident to 7-simplexes, the 421 polytope has 138,240 6-simplex faces that are not facets of 7-simplexes. The 421 polytope thus has two kinds of 6-simplex faces, not interchanged by symmetries of this polytope, the total number of 6-simplex faces is 259200. The vertex figure of a polytope is obtained by removing the ringed node. These graphs represent orthographic projections in the E8, E7, E6, the vertex colors are by overlapping multiplicity in the projection, colored by increasing order of multiplicities as red, orange, yellow, green

17.
5-polytope
–
In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two 4-polytope facets, a 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five 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, and a 4-face is a 4-polytope, furthermore, the following requirements must be met, Each cell must join exactly two 4-faces. Adjacent 4-faces are not in the same four-dimensional hyperplane, the figure is not a compound of other figures which meet the requirements. The topology of any given 5-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, 5-polytopes may be classified based on properties like convexity and symmetry. Self-intersecting 5-polytopes are also known as star polytopes, from analogy with the shapes of the non-convex Kepler-Poinsot polyhedra. A uniform 5-polytope has a group under which all vertices are equivalent. The faces of a uniform polytope must be regular, a semi-regular 5-polytope contains two or more types of regular 4-polytope facets. There is only one figure, called a demipenteract. A regular 5-polytope has all identical regular 4-polytope facets, a prismatic 5-polytope is constructed by a Cartesian product of two lower-dimensional polytopes. A prismatic 5-polytope is uniform if its factors are uniform, the hypercube is prismatic, but is considered separately because it has symmetries other than those inherited from its factors. A 4-space tessellation is the division of four-dimensional Euclidean space into a grid of polychoral facets. Strictly speaking, tessellations are not polytopes as they do not bound a 5D volume, a uniform 4-space tessellation is one whose vertices are related by a space group and whose facets are uniform 4-polytopes. Regular 5-polytopes can be represented by the Schläfli symbol, with s polychoral facets around each face, the 5-demicube honeycomb, vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes, or 5-pyramids, can be generated by a 4-polytope base in a 4-space hyperplane connected to a point off the hyperplane, the 5-simplex is the simplest example with a 4-simplex base

18.
5 21 honeycomb
–
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram and this honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure. Each vertex of the 521 honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplices, the vertex figure of Gossets honeycomb is the semiregular 421 polytope. It is the figure in the k21 family. This honeycomb is highly regular in the sense that its symmetry group acts transitively on the k-faces for k ≤6, all of the k-faces for k ≤7 are simplices. It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 8-orthoplex,611, removing the node on the end of the 1-length branch leaves the 8-simplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node, the edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. The face figure is determined from the figure by removing the ringed node. The cell figure is determined from the figure by removing the ringed node. Each vertex of this tessellation is the center of a 7-sphere in the densest known packing in 8 dimensions, its number is 240. E ~8 contains A ~8 as a subgroup of index 5760, both E ~8 and A ~8 can be seen as affine extensions of A8 from different nodes, E ~8 contains D ~8 as a subgroup of index 270. Both E ~8 and D ~8 can be seen as extensions of D8 from different nodes. Its elements are in proportion as 1 vertex,80 3-edges,27033 faces. The 521 is seventh in a series of semiregular polytopes. Each member of the sequence has the previous member as its vertex figure, all facets of these polytopes are regular polytopes, namely simplexes and orthoplexes. E8 lattice 152 honeycomb 251 honeycomb Coxeter The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 978-0-486-40919-1 Coxeter, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, N. W

19.
6-demicube
–
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches. It can named similarly by a 3-dimensional exponential Schläfli symbol or, cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract, with an odd number of plus signs. The fifth figure is a Euclidean honeycomb,331, and the final is a noncompact hyperbolic honeycomb,431, each progressive uniform polytope is constructed from the previous as its vertex figure. It is also the second in a series of uniform polytopes and honeycombs. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb,134. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 6D uniform polytopes x3o3o *b3o3o3o – hax, archived from the original on 4 February 2007

20.
6-orthoplex
–
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices,60 edges,160 triangle faces,240 tetrahedron cells,192 5-cell 4-faces, and 64 5-faces. It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets and it is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract, hexacross, derived from combining the family name cross polytope with hex for six in Greek. A lowest symmetry construction is based on a dual of a 6-orthotope, cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are, Every vertex pair is connected by an edge, except opposites. This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D.1966 Klitzing, Richard. 6D uniform polytopes x3o3o3o3o4o - gee, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary

21.
7-demicube
–
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or. Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract, there are 95 uniform polytopes with D6 symmetry,63 are shared by the B6 symmetry, and 32 are unique, H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 7D uniform polytopes x3o3o *b3o3o3o3o - hesa, archived from the original on 4 February 2007

22.
7-orthoplex
–
In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices,84 edges,280 triangle faces,560 tetrahedron cells,672 5-cells 4-faces,448 5-faces, and 128 6-faces. It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets and it is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract, heptacross, derived from combining the family name cross polytope with hept for seven in Greek. A lowest symmetry construction is based on a dual of a 7-orthotope, cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are, Every vertex pair is connected by an edge, except opposites. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, 7D uniform polytopes x3o3o3o3o3o4o - zee. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary

23.
7-simplex
–
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices,28 edges,56 triangle faces,70 tetrahedral cells,56 5-cell 5-faces,28 5-simplex 6-faces and its dihedral angle is cos−1, or approximately 81. 79°. It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions, the name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca, the Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are, More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of. This construction is based on facets of the 8-orthoplex and this polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram, This polytope is one of 71 uniform 7-polytopes with A7 symmetry. Polytopes of Various Dimensions Multi-dimensional Glossary

24.
7-simplex honeycomb
–
In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation. The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex and these facet types occur in proportions of 2,2,2,1 respectively in the whole honeycomb. This vertex arrangement is called the A7 lattice or 7-simplex lattice, the 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the A ~7 Coxeter group. It is the 7-dimensional case of a simplectic honeycomb, E ~7 contains A ~7 as a subgroup of index 144. Both E ~7 and A ~7 can be seen as extensions from A7 from different nodes. The A47 lattice is the union of four A7 lattices, S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley–Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III

25.
8-demicube
–
In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 151 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or. Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube and this polytope is the vertex figure for the uniform tessellation,251 with Coxeter-Dynkin diagram, H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Olshevsky, George. Archived from the original on 4 February 2007

26.
8-orthoplex
–
It has two constructive forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 511. It is a part of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract. A lowest symmetry construction is based on a dual of an 8-orthotope, cartesian coordinates for the vertices of an 8-cube, centered at the origin are, Every vertex pair is connected by an edge, except opposites. It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 8D uniform polytopes x3o3o3o3o3o3o4o - ek, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary

27.
8-simplex
–
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices,36 edges,84 triangle faces,126 tetrahedral cells,126 5-cell 4-faces,84 5-simplex 5-faces,36 6-simplex 6-faces and its dihedral angle is cos−1, or approximately 82. 82°. It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions, the name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on. This construction is based on facets of the 9-orthoplex and this polytope is a facet in the uniform tessellations,251, and 521 with respective Coxeter-Dynkin diagrams, This polytope is one of 135 uniform 8-polytopes with A8 symmetry. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 8D uniform polytopes x3o3o3o3o3o3o3o - ene, Polytopes of Various Dimensions Multi-dimensional Glossary

28.
8-simplex honeycomb
–
In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation. The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex and these facet types occur in proportions of 1,1,1,1 respectively in the whole honeycomb. This vertex arrangement is called the A8 lattice or 8-simplex lattice, the 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the A ~8 Coxeter group. It is the 8-dimensional case of a simplectic honeycomb, E ~8 contains A ~8 as a subgroup of index 5760. Both E ~8 and A ~8 can be seen as extensions of A8 from different nodes. This honeycomb is one of 45 unique uniform honeycombs constructed by the A ~8 Coxeter group, S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III

29.
9-demicube
–
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 161 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or. Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract, with an odd number of plus signs. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 9D uniform polytopes x3o3o *b3o3o3o3o3o3o - henne, archived from the original on 4 February 2007

30.
9-orthoplex
–
It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 611. It is one of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract, cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are, Every vertex pair is connected by an edge, except opposites. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, 9D uniform polytopes x3o3o3o3o3o3o3o4o - vee. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary

31.
9-simplex
–
In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices,45 edges,120 triangle faces,210 tetrahedral cells,252 5-cell 4-faces,210 5-simplex 5-faces,120 6-simplex 6-faces,45 7-simplex 7-faces and its dihedral angle is cos−1, or approximately 83. 62°. It can also be called a decayotton, or deca-9-tope, as a 10-facetted polytope in 9-dimensions, the name decayotton is derived from deca for ten facets in Greek and -yott, having 8-dimensional facets, and -on. This construction is based on facets of the 10-orthoplex, Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 9D uniform polytopes x3o3o3o3o3o3o3o3o - day, Polytopes of Various Dimensions Multi-dimensional Glossary

32.
A5 polytope
–
In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices, each can be visualized as symmetric orthographic projections in Coxeter planes of the A5 Coxeter group, and other subgroups. Symmetric orthographic projections of these 19 polytopes can be made in the A5, A4, A3, for even k and symmetrically nodea_1ed-diagrams, symmetry doubles to. These 19 polytopes are each shown in these 4 symmetry planes, with vertices and edges drawn, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, N. W, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Klitzing, Richard

33.
Cantellated 5-simplexes
–
In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex. There are unique 4 degrees of cantellation for the 5-simplex, including truncations, the cantellated 5-simplex has 60 vertices,240 edges,290 faces,135 cells, and 27 4-faces. Cantellated hexateron Small rhombated hexateron The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of or of and these represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively. Cantitruncated hexateron Great rhombated hexateron The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of or of and these construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively. The cantellated 5-simplex is one of 19 uniform 5-polytopes based on the Coxeter group, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. X3o3x3o3o - sarx, o3x3o3x3o - sibrid, x3x3x3o3o - garx, o3x3x3x3o - gibrid Glossary for hyperspace, Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary

34.
Coxeter element
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. Note that this assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple classes of Coxeter elements. There are many different ways to define the Coxeter number h of a root system. A Coxeter element is a product of all simple reflections, the product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the number of roots divided by the rank, the number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element, if the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi The dimension of the corresponding Lie algebra is n, where n is the rank and h is the Coxeter number. The Coxeter number is the highest degree of an invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant then so is h +2 − m, the eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m =2, these include the primitive hth root of unity, ζh = e2πi/h, an example, has h=30, so 64*30/g =12 -3 -6 -5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 =14400. Coxeter elements of A n −1 ≅ S n, considered as the group on n elements, are n-cycles, for simple reflections the adjacent transpositions, …. The dihedral group Dihm is generated by two reflections that form an angle of 2 π /2 m, and thus their product is a rotation by 2 π / m. For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h and this is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi/h. This plane was first systematically studied in, and subsequently used in to provide uniform proofs about properties of Coxeter elements, for polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids, in three dimensions, the symmetry of a regular polyhedron, with one directed petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, order h. Adding a mirror, the symmetry can be doubled to symmetry, Dhd. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, in four dimension, the symmetry of a regular polychoron, with one directed petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h, order h. In five dimension, the symmetry of a regular polyteron, with one directed petrie polygon marked, is represented by the composite of 5 reflections

35.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators

36.
Cyclotruncated 5-simplex honeycomb
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In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation. It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1,1,1. Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids from the cell of one side to a point on the other, the vertex figure has 8 vertices and 12 5-cells. It can be constructed as six sets of parallel hyperplanes that divide space, the hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane. This honeycomb is one of 12 unique uniform honeycombs constructed by the A ~5 Coxeter group, the extended symmetry of the hexagonal diagram of the A ~5 Coxeter group allows for automorphisms that map diagram nodes on to each other. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III

37.
E9 honeycomb
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In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. T ¯9, also is a hyperbolic group, so either facets or vertex figures will not be bounded. E10 is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1, there are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph, the 621 honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E10 Coxeter group. This honeycomb is highly regular in the sense that its symmetry group acts transitively on the k-faces for k ≤7, all of the k-faces for k ≤8 are simplices. It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 9-orthoplex,711, removing the node on the end of the 1-length branch leaves the 9-simplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node, the edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. The face figure is determined from the figure by removing the ringed node. The cell figure is determined from the figure by removing the ringed node. The 621 is last in a series of semiregular polytopes and honeycombs. Each member of the sequence has the previous member as its vertex figure, all facets of these polytopes are regular polytopes, namely simplexes and orthoplexes. The 261 honeycomb is composed of 251 9-honeycomb and 9-simplex facets and it is the final figure in the 2k1 family. It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the branch leaves the 9-simplex. Removing the node on the end of the 6-length branch leaves the 251 honeycomb and this is an infinite facet because E10 is a paracompact hyperbolic group. The vertex figure is determined by removing the ringed node and ringing the neighboring node, the edge figure is the vertex figure of the edge figure. This makes the rectified 8-simplex,051, the face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node

38.
Five-dimensional space
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A five-dimensional space is a space with five dimensions. If interpreted physically, that is one more than the three spatial dimensions and the fourth dimension of time used in relativitistic physics. It is an abstraction which occurs frequently in mathematics, where it is a legitimate construct, in physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space. Whether or not the universe is five-dimensional is a topic of debate, although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century. To explain why this dimension would not be observable, Klein suggested that the fifth dimension would be rolled up into a tiny. While not detectable, it would imply a connection between seemingly unrelated forces. Superstring theory then evolved into a generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings, the other 10 dimensions are compacted, or rolled up, to a size below the subatomic level. Kaluza–Klein theory today is seen as essentially a gauge theory, with the gauge being the circle group, the fifth dimension is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence. Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct and these theories make reference to Hilbert space, a concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. They suggested that electromagnetism resulted from a field that is “polarized” in the fifth dimension. The main novelty of Einstein and Bergmann was to consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor. But they then reneged, modifying the theory to break its five-dimensional symmetry, minkowski space and Maxwells equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal particle. T Hooft has speculated that the dimension is really the spacetime fabric. According to Klein’s definition, a geometry is the study of the invariant properties of a spacetime, therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. In five or more dimensions, only three regular polytopes exist, in five dimensions, they are, The 5-simplex of the simplex family, with 6 vertices,15 edges,20 faces,15 cells, and 6 hypercells. The 5-cube of the family, with 32 vertices,80 edges,80 faces,40 cells

39.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations

40.
Hypersimplex
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In polyhedral combinatorics, a hypersimplex, Δd, k, is a convex polytope that generalizes the simplex. It is determined by two parameters d and k, and is defined as the hull of the d-dimensional vectors whose coefficients consist of k ones. It forms a polytope, because all of these vectors lie in a single -dimensional hyperplane. The number of vertices in Δd, k is, the graph formed by the vertices and edges of a hypersimplex Δd, k is the Johnson graph J. An alternative construction is to take the hull of all -dimensional -vectors that have either or k nonzero coordinates. This has the advantage of operating in a space that is the dimension as the resulting polytope. A hypersimplex Δd, k is also the matroid polytope for a matroid with d elements. The hypersimplex with parameters is a -simplex, with d vertices, the hypersimplex with parameters is an octahedron, and the hypersimplex with parameters is a rectified 5-cell. Generally, every -hypersimplex, Δd, k, corresponds to a polytope, being the -rectified -simplex. The hypersimplices were first studied and named in the computation of characteristic classes, by Gabrièlov, hibi, Takayuki, Solus, Liam, Facets of the r-stable n, k-hypersimplex, arXiv,1408.5932

41.
Pentellated 6-simplexes
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In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, the simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the pentisteriruncicantitruncated 6-simplex, is called an omnitruncated 6-simplex with all of the nodes ringed, expanded 6-simplex Small terated tetradecapeton The vertices of the pentellated 6-simplex can be positioned in 7-space as permutations of. This construction is based on facets of the pentellated 7-orthoplex, a second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations of, Its 42 vertices represent the root vectors of the simple Lie group A6. It is the figure of the 6-simplex honeycomb. Note, Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram, teracellated heptapeton The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the runcitruncated 7-orthoplex, teriprismated heptapeton The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the penticantellated 7-orthoplex, terigreatorhombated heptapeton The vertices of the penticantitruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the penticantitruncated 7-orthoplex, tericellirhombated heptapeton The vertices of the pentiruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the pentiruncitruncated 7-orthoplex, teriprismatorhombated tetradecapeton The vertices of the pentiruncicantellated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the pentiruncicantellated 7-orthoplex, note, Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram. Terigreatoprismated heptapeton The vertices of the pentiruncicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the pentiruncicantitruncated 7-orthoplex. Tericellitruncated tetradecapeton The vertices of the pentisteritruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the pentisteritruncated 7-orthoplex. Note, Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram, Great teracellirhombated heptapeton The vertices of the pentistericantittruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the pentistericantitruncated 7-orthoplex, the omnitruncated 6-simplex has 5040 vertices,15120 edges,16800 faces,8400 cells,1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-simplex, pentisteriruncicantitruncated 6-simplex Omnitruncated heptapeton Great terated tetradecapeton The omnitruncated 6-simplex is the permutohedron of order 7. The omnitruncated 6-simplex is a zonotope, the Minkowski sum of seven line segments parallel to the seven lines through the origin, like all uniform omnitruncated n-simplices, the omnitruncated 6-simplex can tessellate space by itself, in this case 6-dimensional space with three facets around each hypercell. The vertices of the omnitruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the pentisteriruncicantitruncated 7-orthoplex, t0,1,2,3,4,5

42.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21

43.
Rectified 5-simplexes
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In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex, vertices of the birectified 5-simplex are located in the triangular face centers of the 5-simplex. In five dimensional geometry, a rectified 5-simplex, is a uniform 5-polytope with 15 vertices,60 edges,80 triangular faces,45 cells and it is also called 03,1 for its branching Coxeter-Dynkin diagram, shown as. E. L. Elte identified it in 1912 as a semiregular polytope, the rectified 5-simplex,031, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 13k series. The fifth figure is a Euclidean honeycomb,331, and the final is a noncompact hyperbolic honeycomb,431, each progressive uniform polytope is constructed from the previous as its vertex figure. Rectified hexateron The vertices of the rectified 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of or and these construction can be seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively. The birectified 5-simplex is isotopic, with all 12 of its facets as rectified 5-cells and it has 20 vertices,90 edges,120 triangular faces,60 cells. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 02,2 for its branching Coxeter-Dynkin diagram, shown as. It is seen in the figure of the 6-dimensional 122. Birectified hexateron dodecateron The A5 projection has an appearance to Metatrons Cube. The birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration, the vertices of a birectification exist at the center of the faces of the original polytope. It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cubes long diagonal orthogonally, in this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell. This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space, the vertices of the birectified 5-simplex can also be positioned on a hyperplane in 6-space as permutations of. This construction can be seen as facets of the birectified 6-orthoplex, the birectified 5-simplex,022, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The birectified 5-simplex is the figure for the third, the 122. The fourth figure is a Euclidean honeycomb,222, and the final is a noncompact hyperbolic honeycomb,322, each progressive uniform polytope is constructed from the previous as its vertex figure. This polytope is the figure of the 6-demicube, and the edge figure of the uniform 231 polytope

44.
Rectified 7-simplexes
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In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four degrees of rectifications, including the zeroth. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex, vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the cell centers of the 7-simplex. The rectified 7-simplex is the figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S17. Rectified octaexon The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the rectified 8-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as. Birectified octaexon The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the birectified 8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S37. This polytope is the figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, hexadecaexon The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the trirectified 8-orthoplex, the trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space and these polytopes are three of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he Olshevsky, George

45.
Runcinated 5-simplexes
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In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations of the regular 5-simplex. There are 4 unique runcinations of the 5-simplex with permutations of truncations and these polytopes are in a set of 19 uniform 5-polytopes based on the Coxeter group, all shown here in A5 Coxeter plane orthographic projections. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3o3o3x3o - spidtix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix Glossary for hyperspace, George Olshevsky. Polytopes of Various Dimensions, Jonathan Bowers Runcinated uniform polytera, Jonathan Bowers Multi-dimensional Glossary

46.
Simplex
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In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a polytope which is the convex hull of its k +1 vertices. More formally, suppose the k +1 points u 0, …, u k ∈ R k are affinely independent, then, the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices, a regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex, the associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. A 1-simplex is a line segment, the convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. In particular, the hull of a subset of size m+1 is an m-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, in general, the number of m-faces is equal to the binomial coefficient. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascals triangle, a simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex, see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn, an -simplex can be constructed as a join of an n-simplex and a point. An -simplex can be constructed as a join of an m-simplex, the two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points, ∨ =2, a general 2-simplex is the join of 3 points, ∨∨. An isosceles triangle is the join of a 1-simplex and a point, a general 3-simplex is the join of 4 points, ∨∨∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points, a 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point,3. ∨ or ∨