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*