1.
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

2.
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

3.
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

4.
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

5.
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

6.
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

7.
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

8.
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

9.
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