1.
Icosagon
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In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagons interior angles is 3240 degrees, the regular icosagon has Schläfli symbol, and can also be constructed as a truncated decagon, t, or a twice-truncated pentagon, tt. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°, the area of a regular icosagon with edge length t is A =5 t 2 ≃31.5687 t 2. In terms of the radius R of its circumcircle, the area is A =5 R22, since the area of the circle is π R2, the Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section. The Globe, the outdoor theater used by William Shakespeares acting company, was discovered to have built on an icosagonal foundation when a partial excavation was done in 1989. As a golygonal path, the swastika is considered to be an irregular icosagon, a regular square, pentagon, and icosagon can completely fill a plane vertex. E20 E1 ¯ E1 F ¯ = E20 F ¯ E20 E1 ¯ =1 +52 = φ ≈1.618 The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries, and, and 6 cyclic group symmetries and these 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order, full symmetry of the regular form is r40 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 g20 subgroup has no degrees of freedom but can seen as directed edges. These two forms are duals of each other and have half the order of the regular icosagon. An icosagram is a 20-sided star polygon, represented by symbol, there are three regular forms given by Schläfli symbols, and. There are also five regular star figures using the vertex arrangement,2,4,5,2,4. Deeper truncations of the regular decagon and decagram can produce isogonal intermediate icosagram forms with equally spaced vertices, a regular icosagram, can be seen as a quasitruncated decagon, t=. Similarly a decagram, has a quasitruncation t=, and finally a simple truncation of a decagram gives t=

2.
Uniform 10-polytope
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In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets, Regular 10-polytopes can be represented by the Schläfli symbol, with x 9-polytope facets around each peak. There are exactly three convex regular 10-polytopes, - 10-simplex - 10-cube - 10-orthoplex There are no nonconvex regular 10-polytopes. The topology of any given 10-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 10-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, There are 512+16-1=527 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 31 are shown below, all one and two ringed forms, and the final omnitruncated form, bowers-style acronym names are given in parentheses for cross-referencing. There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, twelve cases are shown below, ten single-ring forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing, the D10 family has symmetry of order 1,857,945,600. This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram, of these,511 are repeated from the B10 family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing, however, there are 3 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, 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, polytope names Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary Glossary for hyperspace, George Olshevsky

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