1.
6-simplex
–
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices,21 edges,35 triangle faces,35 tetrahedral cells,21 5-cell 4-faces and its dihedral angle is cos−1, or approximately 80. 41°. It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions, the name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop, the regular 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 Coxeter plane orthographic projections. 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. 6D uniform polytopes x3o3o3o3o - hix, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary

2.
10-demicube
–
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
–
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
–
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 42 polytope
–
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

6.
1 52 honeycomb
–
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

7.
2 31 polytope
–
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

8.
2 41 polytope
–
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

9.
2 51 honeycomb
–
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

10.
3 21 polytope
–
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

11.
3 31 honeycomb
–
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*

12.
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

13.
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

14.
6-polytope
–
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. A 6-polytope is a closed figure with vertices, edges, faces, cells, 4-faces. A vertex is a point where six or more edges meet, an edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A 4-face is a polychoron, and a 5-face is a 5-polytope, furthermore, the following requirements must be met, Each 4-face must join exactly two 5-faces. Adjacent facets are not in the same five-dimensional hyperplane, the figure is not a compound of other figures which meet the requirements. The topology of any given 6-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-polytopes, 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, 6-polytopes may be classified by properties like convexity and symmetry. Self-intersecting 6-polytope are also known as star 6-polytopes, from analogy with the shapes of the non-convex Kepler-Poinsot polyhedra. A regular 6-polytope has all identical regular 5-polytope facets, a semi-regular 6-polytope contains two or more types of regular 4-polytope facets. There is only one figure, called 221. A uniform 6-polytope has a group under which all vertices are equivalent. The faces of a uniform polytope must be regular, a prismatic 6-polytope is constructed by the Cartesian product of two lower-dimensional polytopes. A prismatic 6-polytope is uniform if its factors are uniform, the 6-cube is prismatic, but is considered separately because it has symmetries other than those inherited from its factors. A 5-space tessellation is the division of five-dimensional Euclidean space into a grid of 5-polytope facets. Strictly speaking, tessellations are not 6-polytopes as they do not bound a 6D volume, a uniform 5-space tessellation is one whose vertices are related by a space group and whose facets are uniform 5-polytopes. Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol with t 5-polytope facets around each cell, There are only three such convex regular 6-polytopes, - 6-simplex - 6-cube - 6-orthoplex There are no nonconvex regular polytopes of 5 or more dimensions. For the 3 convex regular 6-polytopes, their elements are, Here are six simple uniform convex 6-polytopes, the expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb

15.
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

16.
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

17.
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

18.
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

19.
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

20.
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

21.
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

22.
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

23.
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

24.
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

25.
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

26.
A6 polytope
–
In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices, each can be visualized as symmetric orthographic projections in Coxeter planes of the A6 Coxeter group, and other subgroups. Symmetric orthographic projections of these 35 polytopes can be made in the A6, A5, A4, A3, for even k and symmetric ringed diagrams, symmetry doubles to. These 63 polytopes are each shown in these 5 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

27.
Cantellated 6-simplexes
–
In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex. There are unique 4 degrees of cantellation for the 6-simplex, including truncations, small rhombated heptapeton The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the cantellated 7-orthoplex, small prismated heptapeton The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the bicantellated 7-orthoplex, great rhombated heptapeton The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the cantitruncated 7-orthoplex, great birhombated heptapeton The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the bicantitruncated 7-orthoplex, the truncated 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 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, x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary

28.
Coxeter group
–
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

29.
E9 honeycomb
–
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

30.
Heptagon
–
In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is also referred to as the septagon, using sept- together with the Greek suffix -agon meaning angle. A regular heptagon, in all sides and all angles are equal, has internal angles of 5π/7 radians. The area of a regular heptagon of side length a is given by, the apothem is half the cotangent of π /7, and the area of each of the 14 small triangles is one-fourth of the apothem. This expression cannot be rewritten without complex components, since the indicated cubic function is casus irreducibilis. As 7 is a Pierpont prime but not a Fermat prime and this type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector, the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π7 ≈1.247 is a zero of the irreducible cubic x3 + x2 − 2x −1. Consequently, this polynomial is the polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0. 2% is shown in the drawing and it is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle, then B D =12 B C gives an approximation for the edge of the heptagon. Example to illustrate the error, At a circumscribed circle radius r =1 m, since 7 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z7, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptagon, john Conway labels these by a letter and group order. Full symmetry of the form is r14 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g7 subgroup has no degrees of freedom but can seen as directed edges. However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis. A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon and angles π /7,2 π /7, thus its sides coincide with one side and two particular diagonals of the regular heptagon. Two kinds of star heptagons can be constructed from regular heptagons, labeled by Schläfli symbols, blue, and green star heptagons inside a red heptagon

31.
Heptagram
–
A heptagram, septagram, or septogram is a seven-point star drawn with seven straight strokes. The name heptagram combines a numeral prefix, hepta-, with the Greek suffix -gram, the -gram suffix derives from γραμμῆς meaning a line. In general, a heptagram is any self-intersecting heptagon, there are two regular heptagrams, labeled as and, with the second number representing the vertex interval step from a regular heptagon. This is the smallest star polygon that can be drawn in two forms, as irreducible fractions, the two heptagrams are sometimes called the heptagram and the great heptagram. The previous one, the hexagram, is a compound of two triangles. The smallest star polygon is the pentagram, the next one is the octagram, followed by the regular enneagram, which also has two forms, and, as well as one compound of three triangles. The heptagram was used in Christianity to symbolize the seven days of creation, the heptagram is a symbol of perfection in many Christian sects. The heptagram is used in the symbol for Babalon in Thelema, the heptagram is known among neopagans as the Elven Star or Fairy Star. It is treated as a symbol in various modern pagan. Blue Star Wicca also uses the symbol, where it is referred to as a septegram, the second heptagram is a symbol of magical power in some pagan spiritualities. The heptagram is used by members of the otherkin subculture as an identifier. In alchemy, a star can refer to the seven planets which were known to ancient alchemists. The seven-pointed star is incorporated into the flags of the bands of the Cherokee Nation. The Bennington flag, a historical American Flag, has thirteen seven-pointed stars along with the numerals 76 in the canton, the Flag of Jordan contains a seven-pointed star. The Flag of Australia employs five heptagrams and one pentagram to depict the Southern Cross constellation, some old versions of the coat of arms of Georgia including the Georgian Soviet Socialist Republic used the heptagram as an element. A seven-pointed star is used as the badge in many sheriffs departments, the seven-pointed star is used as the logo for the international Danish shipping company A. P. Moller–Maersk Group, sometimes known simply as Maersk. In George R. R. Martins novel series A Song of Ice and Fire, Star polygon Stellated polygons Two-dimensional regular polytopes Bibliography Grünbaum, B. and G. C. Shephard, Tilings and Patterns, New York, W. H. Freeman & Co, polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes

32.
Hexicated 7-simplexes
–
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-simplex. There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed, in seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication of the regular 7-simplex, or alternately can be seen as an expansion operation. Its 56 vertices represent the vectors of the simple Lie group A7. Expanded 7-simplex Small petated hexadecaexon The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexicated 8-orthoplex. This construction is based on facets of the hexitruncated 8-orthoplex, petirhombated octaexon The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexicantellated 8-orthoplex, petiprismated hexadecaexon The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexiruncinated 8-orthoplex, petigreatorhombated octaexon The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexicantitruncated 8-orthoplex, petiprismatotruncated octaexon The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexiruncitruncated 8-orthoplex, in seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope. Petiprismatorhombated octaexon The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexiruncicantellated 8-orthoplex. Peticellitruncated octaexon The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexisteritruncated 8-orthoplex. Peticellirhombihexadecaexon The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexistericantellated 8-orthoplex. Petiteritruncated hexadecaexon The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexipentitruncated 8-orthoplex. Petigreatoprismated octaexon The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexiruncicantitruncated 8-orthoplex. Peticelligreatorhombated octaexon The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexistericantitruncated 8-orthoplex. Peticelliprismatotruncated octaexon The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the hexisteriruncitruncated 8-orthoplex

33.
Pentellated 6-simplexes
–
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

34.
Petrie polygon
–
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

35.
Rectified 6-simplexes
–
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex. There are three degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex, vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as. Rectified heptapeton The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the rectified 7-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as. Birectified heptapeton The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the birectified 7-orthoplex. The rectified 6-simplex polytope is the figure of the 7-demicube. These polytopes are a part of 35 uniform 6-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. O3x3o3o3o3o - ril, o3x3o3o3o3o - bril Olshevsky, George, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary

36.
Runcinated 6-simplexes
–
In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex. There are 8 unique runcinations of the 6-simplex with permutations of truncations, small prismated heptapeton The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the runcinated 7-orthoplex, small biprismated tetradecapeton The vertices of the biruncinted 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the biruncinated 7-orthoplex, note, Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram. Prismatotruncated heptapeton The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the runcitruncated 7-orthoplex. Biprismatorhombated heptapeton The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the biruncitruncated 7-orthoplex. Prismatorhombated heptapeton The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the runcicantellated 7-orthoplex. Runcicantitruncated heptapeton Great prismated heptapeton The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the runcicantitruncated 7-orthoplex. Biruncicantitruncated heptapeton Great biprismated tetradecapeton The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the biruncicantitruncated 7-orthoplex. Note, Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram, the truncated 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 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, x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary

37.
Simplex
–
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 ∨

38.
Six-dimensional space
–
Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are a number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes, six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature. Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space, as such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined, and can be used to calculate the metric,6 ×6 matrices can be used to describe transformations such as rotations that keep the origin fixed. More generally, any space that can be described locally with six coordinates, one example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidean space ℝ7 that are equidistant from the origin and this constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, a polytope in six dimensions is called a 6-polytope. The most studied are the regular polytopes, of which there are three in six dimensions, the 6-simplex, 6-cube, and 6-orthoplex. A wider family are the uniform 6-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 6-demicube is a unique polytope from the D6 family, and 221 and 122 polytopes from the E6 family. The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point and it has symbol S5, and the equation for the 5-sphere, radius r, centre the origin is S5 =. The volume of space bounded by this 5-sphere is V6 = π3 r 66 which is 5.16771 × r6. The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point and it has symbol S6, and the equation for the 6-sphere, radius r, centre the origin is S6 =. The volume of the bounded by this 6-sphere is V7 =16 π3 r 7105 which is 4.72477 × r7. In three dimensional space a transformation has six degrees of freedom, three translations along the three coordinate axes and three from the rotation group SO. Often these transformations are handled separately as they have different geometrical structures. In screw theory angular and linear velocity are combined into one six-dimensional object, a similar object called a wrench combines forces and torques in six dimensions

39.
Stericated 6-simplexes
–
In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations of the regular 6-simplex. There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, small cellated heptapeton The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the stericated 7-orthoplex, cellirhombated heptapeton The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the steritruncated 7-orthoplex, cellirhombated heptapeton The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the stericantellated 7-orthoplex, celligreatorhombated heptapeton The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the stericantitruncated 7-orthoplex, celliprismated heptapeton The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the steriruncinated 7-orthoplex, celliprismatotruncated heptapeton The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the steriruncitruncated 7-orthoplex, bistericantitruncated 6-simplex as t1,2,3,5 Celliprismatorhombated heptapeton The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the steriruncicantellated 7-orthoplex, great cellated heptapeton The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the steriruncicantitruncated 7-orthoplex, the truncated 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 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, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary

40.
Truncated 6-simplexes
–
In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation, vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the cells of the 6-simplex. Truncated heptapeton The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the truncated 7-orthoplex. Bitruncated heptapeton The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the bitruncated 7-orthoplex. The tritruncated 6-simplex is a uniform polytope, with 14 identical bitruncated 5-simplex facets. The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration, and, tetradecapeton The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the bitruncated 7-orthoplex, alternately it can be centered on the origin as permutations of. Note, Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram, the truncated 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 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, o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary

41.
Uniform 1 k2 polytope
–
In geometry, 1k2 polytope is a uniform polytope in n-dimensions constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram and it can be named by an extended Schläfli symbol. The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube in 5-dimensions, each polytope is constructed from 1k-1,2 and -demicube facets. Each has a figure of a polytope is a birectified n-simplex. The sequence ends with k=6, as a tessellation of 9-dimensional hyperbolic space. Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings, alicia Boole Stott, Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, Vol.11, No. 1, pp. 1–24 plus 3 plates,1910, Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings. Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam, H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin,1940 N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Coxeter, Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin,1988 PolyGloss v0.05, Gosset figures

42.
Uniform 2 k1 polytope
–
In geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by their Coxeter symbol as 2k1 by its bifurcating Coxeter-Dynkin diagram and it can be named by an extended Schläfli symbol. The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex in 5-dimensions, each polytope is constructed from -simplex and 2k-1,1 -polytope facets, each has a vertex figure as an -demicube. The sequence ends with k=6, as an infinite tessellation of 9-space. Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings, alicia Boole Stott, Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, Vol.11, No. 1, pp. 1–24 plus 3 plates,1910, Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings. Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam, H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin,1940 N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Coxeter, Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin,1988 PolyGloss v0.05, Gosset figures

43.
Uniform 6-polytope
–
In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes, the complete set of convex uniform polypeta has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams, each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes, the 6-simplex, the 6-cube, Regular polytopes,1852, Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex uniform polytopes,1940, The search was expanded systematically by H. S. M, Coxeter in his publication Regular and Semi-Regular Polytopes. Nonregular uniform star polytopes, Ongoing, Thousands of nonconvex uniform polypeta are known, participating researchers include Jonathan Bowers, Richard Klitzing and Norman Johnson. Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, There are four fundamental reflective symmety groups which generate 153 unique uniform 6-polytopes. Uniform prism There are 6 categorical uniform prisms based on the uniform 5-polytopes, Uniform duoprism There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope, in addition, there are 105 uniform 6-polytope constructions based on prisms of the uniform 5-polytopes. In addition, there are many uniform 6-polytope based on. There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram and they are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex. Bowers-style acronym names are given in parentheses for cross-referencing, the A6 family has symmetry of order 5040. The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, see also list of A6 polytopes for graphs of these polytopes. There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, the B6 family has symmetry of order 46080. They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube, Bowers names and acronym names are given for cross-referencing. See also list of B6 polytopes for graphs of these polytopes, the D6 family has symmetry of order 23040. This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D6 Coxeter-Dynkin diagram, of these,31 are repeated from the B6 family and 16 are unique to this family. The 16 unique forms are enumerated below, bowers-style acronym names are given for cross-referencing

44.
Uniform k 21 polytope
–
In geometry, a uniform k21 polytope is a polytope in k +4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gossets semiregular figures. Gosset named them by their dimension from 5 to 9, for example the 5-ic semiregular figure, the sequence as identified by Gosset ends as an infinite tessellation in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice,621 and it is a tessellation of hyperbolic 9-space constructed of The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness, the demipenteract also exists in the demihypercube family. They are also named by their symmetry group, like E6 polytope. The orthoplex faces are constructed from the Coxeter group Dn−1 and have a Schläfli symbol of rather than the regular and this construction is an implication of two facet types. Half the facets around each orthoplex ridge are attached to another orthoplex, in contrast, every simplex ridge is attached to an orthoplex. Each has a figure as the previous form. For example, the rectified 5-cell has a figure as a triangular prism. Uniform 2k1 polytope family Uniform 1k2 polytope family T. B, Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings. Alicia Boole Stott, Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, Vol.11,1, pp. 1–24 plus 3 plates,1910. Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings, analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam, vol 11.5,1913. H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 H. S. M, Coxeter, Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin,1985 H. S. M