1.
4 21 polytope
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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

2.
Heptagon
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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

3.
7-simplex
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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

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

5.
Heptagram
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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

6.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21

7.
Uniform k 21 polytope
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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

8.
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

9.
Uniform 6-polytope
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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

10.
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

11.
E9 honeycomb
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In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. T ¯9, also is a hyperbolic group, so either facets or vertex figures will not be bounded. E10 is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1, there are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph, the 621 honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E10 Coxeter group. This honeycomb is highly regular in the sense that its symmetry group acts transitively on the k-faces for k ≤7, all of the k-faces for k ≤8 are simplices. It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 9-orthoplex,711, removing the node on the end of the 1-length branch leaves the 9-simplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node, the edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. The face figure is determined from the figure by removing the ringed node. The cell figure is determined from the figure by removing the ringed node. The 621 is last in a series of semiregular polytopes and honeycombs. Each member of the sequence has the previous member as its vertex figure, all facets of these polytopes are regular polytopes, namely simplexes and orthoplexes. The 261 honeycomb is composed of 251 9-honeycomb and 9-simplex facets and it is the final figure in the 2k1 family. It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the branch leaves the 9-simplex. Removing the node on the end of the 6-length branch leaves the 251 honeycomb and this is an infinite facet because E10 is a paracompact hyperbolic group. The vertex figure is determined by removing the ringed node and ringing the neighboring node, the edge figure is the vertex figure of the edge figure. This makes the rectified 8-simplex,051, the face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node