uniform, polytope, graphs, three, regular, related, uniform, polytopes, simplex, rectified, simplex, truncated, simplex, cantellated, simplex, runcinated, simplex, stericated, simplex, pentellated, simplex, hexicated, simplex, heptellated, simplex, orthoplex, . Graphs of three regular and related uniform polytopes 8 simplex Rectified 8 simplex Truncated 8 simplex Cantellated 8 simplex Runcinated 8 simplex Stericated 8 simplex Pentellated 8 simplex Hexicated 8 simplex Heptellated 8 simplex 8 orthoplex Rectified 8 orthoplex Truncated 8 orthoplex Cantellated 8 orthoplex Runcinated 8 orthoplex Hexicated 8 orthoplex Cantellated 8 cube Runcinated 8 cube Stericated 8 cube Pentellated 8 cube Hexicated 8 cube Heptellated 8 cube 8 cube Rectified 8 cube Truncated 8 cube 8 demicube Truncated 8 demicube Cantellated 8 demicube Runcinated 8 demicube Stericated 8 demicube Pentellated 8 demicube Hexicated 8 demicube 421 142 241 In eight dimensional geometry an eight dimensional polytope or 8 polytope is a polytope contained by 7 polytope facets Each 6 polytope ridge being shared by exactly two 7 polytope facets A uniform 8 polytope is one which is vertex transitive and constructed from uniform 7 polytope facets Contents 1 Regular 8 polytopes 2 Characteristics 3 Uniform 8 polytopes by fundamental Coxeter groups 3 1 Uniform prismatic forms 3 2 The A8 family 3 3 The B8 family 3 4 The D8 family 3 5 The E8 family 4 Regular and uniform honeycombs 4 1 Regular and uniform hyperbolic honeycombs 5 References 6 External links Regular 8 polytopes edit Regular 8 polytopes can be represented by the Schlafli symbol p q r s t u v with v p q r s t u 7 polytope facets around each peak There are exactly three such convex regular 8 polytopes 3 3 3 3 3 3 3 8 simplex 4 3 3 3 3 3 3 8 cube 3 3 3 3 3 3 4 8 orthoplex There are no nonconvex regular 8 polytopes Characteristics edit The topology of any given 8 polytope is defined by its Betti numbers and torsion coefficients 91 1 93 The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions and is zero for all 8 polytopes whatever their underlying topology This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers 91 1 93 Similarly the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes and this led to the use of torsion coefficients 91 1 93 Uniform 8 polytopes by fundamental Coxeter groups edit Uniform 8 polytopes with reflective symmetry can be generated by these four Coxeter groups represented by permutations of rings of the Coxeter Dynkin diagrams Coxeter group Forms 1 A8 37 135 2 BC8 4 36 255 3 D8 35 1 1 191 64 unique 4 E8 34 2 1 255 Selected regular and uniform 8 polytopes from each family include Simplex family A8 37 135 uniform 8 polytopes as permutations of rings in the group diagram including one regular 37 8 simplex or ennea 9 tope or enneazetton Hypercube orthoplex family B8 4 36 255 uniform 8 polytopes as permutations of rings in the group diagram including two regular ones 4 36 8 cube or octeract 36 4 8 orthoplex or octacross Demihypercube D8 family 35 1 1 191 uniform 8 polytopes as permutations of rings in the group diagram including 3 35 1 8 demicube or demiocteract 151 also as h 4 36 3 3 3 3 3 31 1 8 orthoplex 511 E polytope family E8 family 34 1 1 255 uniform 8 polytopes as permutations of rings in the group diagram including 3 3 3 3 32 1 Thorold Gosset s semiregular 421 3 34 2 the uniform 142 3 3 34 1 the uniform 241 Uniform prismatic forms edit There are many uniform prismatic families including Uniform 8 polytope prism families Coxeter group Coxeter Dynkin diagram 7 1 1 A7A1 3 3 3 3 3 3 160 2 B7A1 4 3 3 3 3 3 160 3 D7A1 34 1 1 160 4 E7A1 33 2 1 160 6 2 1 A6I2 p 3 3 3 3 3 p 2 B6I2 p 4 3 3 3 3 p 3 D6I2 p 33 1 1 p 4 E6I2 p 3 3 3 3 3 p 6 1 1 1 A6A1A1 3 3 3 3 3 160 x 160 2 B6A1A1 4 3 3 3 3 160 x 160 3 D6A1A1 33 1 1 160 x 160 4 E6A1A1 3 3 3 3 3 160 x 160 5 3 1 A5A3 34 3 3 2 B5A3 4 33 3 3 3 D5A3 32 1 1 3 3 4 A5B3 34 4 3 5 B5B3 4 33 4 3 6 D5B3 32 1 1 4 3 7 A5H3 34 5 3 8 B5H3 4 33 5 3 9 D5H3 32 1 1 5 3 5 2 1 1 A5I2 p A1 3 3 3 p 160 2 B5I2 p A1 4 3 3 p 160 3 D5I2 p A1 32 1 1 p 160 5 1 1 1 1 A5A1A1A1 3 3 3 160 160 160 2 B5A1A1A1 4 3 3 160 160 160 3 D5A1A1A1 32 1 1 160 160 160 4 4 1 A4A4 3 3 3 3 3 3 2 B4A4 4 3 3 3 3 3 3 D4A4 31 1 1 3 3 3 4 F4A4 3 4 3 3 3 3 5 H4A4 5 3 3 3 3 3 6 B4B4 4 3 3 4 3 3 7 D4B4 31 1 1 4 3 3 8 F4B4 3 4 3 4 3 3 9 H4B4 5 3 3 4 3 3 10 D4D4 31 1 1 31 1 1 11 F4D4 3 4 3 31 1 1 12 H4D4 5 3 3 31 1 1 13 F4 F4 3 4 3 3 4 3 14 H4 F4 5 3 3 3 4 3 15 H4H4 5 3 3 5 3 3 4 3 1 1 A4A3A1 3 3 3 3 3 160 2 A4B3A1 3 3 3 4 3 160 3 A4H3A1 3 3 3 5 3 160 4 B4A3A1 4 3 3 3 3 160 5 B4B3A1 4 3 3 4 3 160 6 B4H3A1 4 3 3 5 3 160 7 H4A3A1 5 3 3 3 3 160 8 H4B3A1 5 3 3 4 3 160 9 H4H3A1 5 3 3 5 3 160 10 F4A3A1 3 4 3 3 3 160 11 F4B3A1 3 4 3 4 3 160 12 F4H3A1 3 4 3 5 3 160 13 D4A3A1 31 1 1 3 3 160 14 D4B3A1 31 1 1 4 3 160 15 D4H3A1 31 1 1 5 3 160 4 2 2 4 2 1 1 4 1 1 1 1 3 3 2 1 A3A3I2 p 3 3 3 3 p 2 B3A3I2 p 4 3 3 3 p 3 H3A3I2 p 5 3 3 3 p 4 B3B3I2 p 4 3 4 3 p 5 H3B3I2 p 5 3 4 3 p 6 H3H3I2 p 5 3 5 3 p 3 3 1 1 1 A32A12 3 3 3 3 160 160 2 B3A3A12 4 3 3 3 160 160 3 H3A3A12 5 3 3 3 160 160 4 B3B3A12 4 3 4 3 160 160 5 H3B3A12 5 3 4 3 160 160 6 H3H3A12 5 3 5 3 160 160 3 2 2 1 1 A3I2 p I2 q A1 3 3 p q 160 2 B3I2 p I2 q A1 4 3 p q 160 3 H3I2 p I2 q A1 5 3 p q 160 3 2 1 1 1 1 A3I2 p A13 3 3 p 160 x 160 160 2 B3I2 p A13 4 3 p 160 x 160 160 3 H3I2 p A13 5 3 p 160 x 160 160 3 1 1 1 1 1 1 A3A15 3 3 160 x 160 160 x 160 160 2 B3A15 4 3 160 x 160 160 x 160 160 3 H3A15 5 3 160 x 160 160 x 160 160 2 2 2 2 1 I2 p I2 q I2 r I2 s p q r s 2 2 2 1 1 1 I2 p I2 q I2 r A12 p q r 160 160 2 2 1 1 1 1 2 I2 p I2 q A14 p q 160 160 160 160 2 1 1 1 1 1 1 1 I2 p A16 p 160 160 160 160 160 160 1 1 1 1 1 1 1 1 1 A18 160 160 160 160 160 160 160 160 The A8 family edit The A8 family has symmetry of order 362880 9 factorial There are 135 forms based on all permutations of the Coxeter Dynkin diagrams with one or more rings 128 8 1 cases These are all enumerated below Bowers style acronym names are given in parentheses for cross referencing See also a list of 8 simplex polytopes for symmetric Coxeter plane graphs of these polytopes A8 uniform polytopes Coxeter Dynkin diagram Truncationindices Johnson name Basepoint Element counts 7 6 5 4 3 2 1 0 1 t0 8 simplex ene 0 0 0 0 0 0 0 0 1 9 36 84 126 126 84 36 9 2 t1 Rectified 8 simplex rene 0 0 0 0 0 0 0 1 1 18 108 336 630 576 588 252 36 3 t2 Birectified 8 simplex bene 0 0 0 0 0 0 1 1 1 18 144 588 1386 2016 1764 756 84 4 t3 Trirectified 8 simplex trene 0 0 0 0 0 1 1 1 1 1260 126 5 t0 1 Truncated 8 simplex tene 0 0 0 0 0 0 0 1 2 288 72 6 t0 2 Cantellated 8 simplex 0 0 0 0 0 0 1 1 2 1764 252 7 t1 2 Bitruncated 8 simplex 0 0 0 0 0 0 1 2 2 1008 252 8 t0 3 Runcinated 8 simplex 0 0 0 0 0 1 1 1 2 4536 504 9 t1 3 Bicantellated 8 simplex 0 0 0 0 0 1 1 2 2 5292 756 10 t2 3 Tritruncated 8 simplex 0 0 0 0 0 1 2 2 2 2016 504 11 t0 4 Stericated 8 simplex 0 0 0 0 1 1 1 1 2 6300 630 12 t1 4 Biruncinated 8 simplex 0 0 0 0 1 1 1 2 2 11340 1260 13 t2 4 Tricantellated 8 simplex 0 0 0 0 1 1 2 2 2 8820 1260 14 t3 4 Quadritruncated 8 simplex 0 0 0 0 1 2 2 2 2 2520 630 15 t0 5 Pentellated 8 simplex 0 0 0 1 1 1 1 1 2 5040 504 16 t1 5 Bistericated 8 simplex 0 0 0 1 1 1 1 2 2 12600 1260 17 t2 5 Triruncinated 8 simplex 0 0 0 1 1 1 2 2 2 15120 1680 18 t0 6 Hexicated 8 simplex 0 0 1 1 1 1 1 1 2 2268 252 19 t1 6 Bipentellated 8 simplex 0 0 1 1 1 1 1 2 2 7560 756 20 t0 7 Heptellated 8 simplex 0 1 1 1 1 1 1 1 2 504 72 21 t0 1 2 Cantitruncated 8 simplex 0 0 0 0 0 0 1 2 3 2016 504 22 t0 1 3 Runcitruncated 8 simplex 0 0 0 0 0 1 1 2 3 9828 1512 23 t0 2 3 Runcicantellated 8 simplex 0 0 0 0 0 1 2 2 3 6804 1512 24 t1 2 3 Bicantitruncated 8 simplex 0 0 0 0 0 1 2 3 3 6048 1512 25 t0 1 4 Steritruncated 8 simplex 0 0 0 0 1 1 1 2 3 20160 2520 26 t0 2 4 Stericantellated 8 simplex 0 0 0 0 1 1 2 2 3 26460 3780 27 t1 2 4 Biruncitruncated 8 simplex 0 0 0 0 1 1 2 3 3 22680 3780 28 t0 3 4 Steriruncinated 8 simplex 0 0 0 0 1 2 2 2 3 12600 2520 29 t1 3 4 Biruncicantellated 8 simplex 0 0 0 0 1 2 2 3 3 18900 3780 30 t2 3 4 Tricantitruncated 8 simplex 0 0 0 0 1 2 3 3 3 10080 2520 31 t0 1 5 Pentitruncated 8 simplex 0 0 0 1 1 1 1 2 3 21420 2520 32 t0 2 5 Penticantellated 8 simplex 0 0 0 1 1 1 2 2 3 42840 5040 33 t1 2 5 Bisteritruncated 8 simplex 0 0 0 1 1 1 2 3 3 35280 5040 34 t0 3 5 Pentiruncinated 8 simplex 0 0 0 1 1 2 2 2 3 37800 5040 35 t1 3 5 Bistericantellated 8 simplex 0 0 0 1 1 2 2 3 3 52920 7560 36 t2 3 5 Triruncitruncated 8 simplex 0 0 0 1 1 2 3 3 3 27720 5040 37 t0 4 5 Pentistericated 8 simplex 0 0 0 1 2 2 2 2 3 13860 2520 38 t1 4 5 Bisteriruncinated 8 simplex 0 0 0 1 2 2 2 3 3 30240 5040 39 t0 1 6 Hexitruncated 8 simplex 0 0 1 1 1 1 1 2 3 12096 1512 40 t0 2 6 Hexicantellated 8 simplex 0 0 1 1 1 1 2 2 3 34020 3780 41 img, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.