1 22 polytope

Uniform 6-polytope {{DISPLAYTITLE:1 22 polytope}} 150px|class=skin-invert122nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea150px|class=skin-invertRectified 122nodea 3a nodea 3a branch_10 3a nodea 3a nodea150px|class=skin-invertBirectified 122nodea 3a nodea_1 3a branch 3a nodea_1 3a nodeaImage needed June 2025Trirectified 122nodea_1 3a nodea 3a branch 3a nodea 3a nodea_1150px|class=skin-invertTruncated 122nodea 3a nodea 3a branch_11 3a nodea 3a nodea150px|class=skin-invert221nodea_1 3a nodea 3a branch 3a nodea 3a nodea150px|class=skin-invertRectified 221nodea 3a nodea_1 3a branch 3a nodea 3a nodeaOrthogonal projections in E6 Coxeter plane In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).Elte, 1912 Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, constructed by positions points on the elements of 122. The rectified 122 is constructed by points at the mid-edges of the 122. The birectified 122 is constructed by points at the triangle face centers of the 122. These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: nodea 3a nodea 3a branch 3a nodea 3a nodea. 122 polytope 122 polytopeTypeUniform 6-polytopeFamily1k2 polytopeSchläfli symbol {3,32,2}Coxeter symbol 122Coxeter-Dynkin diagramnodea 3a nodea 3a branch_01lr 3a nodea 3a nodea or node_1 3 node split1 nodes 3ab nodes5-faces54:27 121 30px|class=skin-invert27 121 30px|class=skin-invert4-faces702:270 111 25px|class=skin-invert432 120 25px|class=skin-invertCells2160:1080 110 25px|class=skin-invert1080 {3,3} 25px|class=skin-invertFaces2160 {3} 25px|class=skin-invertEdges720Vertices72Vertex figureBirectified 5-simplex:022 50px|class=skin-invertPetrie polygonDodecagonCoxeter groupE6, [[3,32,2]], order 103680Propertiesconvex, isotopic The 122 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E6. Alternate names Pentacontatetrapeton (Acronym: mo) - 54-facetted polypeton (Jonathan Bowers)Klitzing, (o3o3o3o3o *c3x - mo) Images Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green. The multiplicities of vertices by color are given in parentheses. + Coxeter plane orthographic projectionsE6[12]D5[8]D4 / A2[6]120px(1,2)120px(1,3)120px(1,9,12)B6[12/2]A5[6]A45 = [10]A3 / D3[4]120px(1,2)120px(2,3,6)120px(1,2)120px(1,6,8,12) Construction It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram, nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea. Removing the node on either of 2-length branches leaves the 5-demicube, 121, nodea 3a branch_01lr 3a nodea 3a nodea. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 022, node 3 node 3 node_1 3 node 3 node. Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203 E6nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea k-face fk f0 f1f2f3f4f5 k-figure NotesA5 nodea 3a nodea 3a nodes_0x 3a nodea 3a nodea ( ) f0722090606015153066r{3,3,3} E6/A5 = 72·6!/6! = 72A2A2A1 nodea 3a nodea 2 nodes_x1 2 nodea 3a nodea { } f1272099933933{3}×{3} E6/A2A2A1 = 72·6!/3!/3!/2 = 720A2A1A1 nodea 2 nodea_x 2 branch_01 2 nodea_x 2 nodea {3} f23321602211422s{2,4} E6/A2A1A1 = 72·6!/3!/2/2 = 2160A3A1 nodea_x 2 nodea 3a branch_01l 2 nodea_x 2 nodea{3,3}f34641080*10221{ }∨( ) E6/A3A1 = 72·6!/4!/2 = 1080nodea 2 nodea_x 2 branch_01r 3a nodea 2 nodea_x464*108001212A4A1 nodea 2 nodea_x 2 branch_01r 3a nodea 3a nodea {3,3,3}f45101050216**20{ } E6/A4A1 = 72·6!/5!/2 = 216nodea 3a nodea 3a branch_01l 2 nodea_x 2 nodea5101005*216*02D4 nodea_x 2 nodea 3a branch_01lr 3a nodea 2 nodea_x h{4,3,3}8243288**27011 E6/D4 = 72·6!/8/4! = 270D5 nodea 3a nodea 3a branch_01lr 3a nodea 2 nodea_xh{4,3,3,3}f5168016080401601027*( ) E6/D5 = 72·6!/16/5! = 27nodea_x 2 nodea 3a branch_01lr 3a nodea 3a nodea 1680160408001610*27 Related complex polyhedron thumb|Orthographic projection in Aut(E6) Coxeter plane with 18-gonal symmetry for complex polyhedron, 3{3}3{4}2. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. The regular complex polyhedron 3{3}3{4}2, 3node_1 3 3node 4 node, in has a real representation as the 122 polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is 3[3]3[4]2, order 1296. It has a half-symmetry quasiregular construction as 3node 3 3node_1 3 3node, as a rectification of the Hessian polyhedron, 3node_1 3 3node 3 3node. Coxeter, H. S. M., Regular Complex Polytopes, (1991) second edition, Cambridge University Press, p. 30 and p. 47 Related polytopes and honeycomb Along with the semiregular polytope, 221, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: nodea 3a nodea 3a branch 3a nodea 3a nodea. Geometric folding The 122 is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 122 in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 122. E6/F4 Coxeter planes160px122160px24-cellD4/B4 Coxeter planes160px122160px24-cell Tessellations This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222, node_1 3 node 3 node split1 nodes 3ab nodes. Rectified 122 polytope Rectified 122TypeUniform 6-polytopeSchläfli symbol 2r{3,3,32,1}r{3,32,2}Coxeter symbol 0221Coxeter-Dynkin diagramnodea 3a nodea 3a branch_10 3a nodea 3a nodeaor node 3 node_1 split1 nodes 3ab nodes5-faces1264-faces1566Cells6480Faces6480Edges6480Vertices720Vertex figure3-3 duoprism prismPetrie polygonDodecagonCoxeter groupE6, [[3,32,2]], order 103680Propertiesconvex The rectified 122 polytope (also called 0221) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).The Voronoi Cells of the E6* and E7* Lattices https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html 2016-01-30 , Edward Pervin Alternate names Birectified 221 polytope Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers)Klitzing, (o3o3x3o3o *c3o - ram) Images Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple. + Coxeter plane orthographic projectionsE6[12]D5[8]D4 / A2[6]B6[12/2]120px120px120px120pxA5[6]A4[5]A3 / D3[4]120px120px120px Construction Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: nodea 3a nodea 3a branch_10 3a nodea 3a nodea. Removing the ring on the short branch leaves the birectified 5-simplex, nodea 3a nodea 3a nodea_1 3a nodea 3a nodea. Removing the ring on either of 2-length branches leaves the birectified 5-orthoplex in its alternated form: t2(211), nodea 3a branch_10 3a nodea 3a nodea. The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, node 3 node_1 2 node_1 2 node_1 3 node. Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.ram E6nodea 3a nodea 3a branch_10 3a nodea 3a nodeak-face fk f0 f1f2f3f4f5 k-figure NotesA2A2A1 nodea 3a nodea 2 nodes_x0 2 nodea 3a nodea ( )f0720181818961896963693233{3}×{3}×{ } E6/A2A2A1 = 72·6!/3!/3!/2 = 720A1A1A1 nodea 2 nodea_x 2 nodes_1x 2 nodea_x 2 nodea { }f1264802211421221241122{ }∨{ }∨( ) E6/A1A1A1 = 72·6!/2/2/2 = 6480A2A1 nodea_x 2 nodea 3a nodes_1x 2 nodea_x 2 nodea{3}f2334320**1210021120121Sphenoid E6/A2A1 = 72·6!/3!/2 = 4320nodea 2 nodea_x 2 nodes_1x 3a nodea 2 nodea_x33*4320*0201110221112A2A1A1 nodea 2 nodea_x 2 branch_10 2 nodea_x 2 nodea33**21600020201041022{ }∨{ } E6/A2A1A1 = 72·6!/3!/2/2 = 2160A2A1 nodea 3a nodea 3a nodes_1x 2 nodea_x 2 nodea{3,3}f3464001080****21000120{ }∨( ) E6/A2A1 = 72·6!/3!/2 = 1080A3 nodea_x 2 nodea 3a nodes_1x 3a nodea 2 nodea_xr{3,3}612440*2160***10110111{3} E6/A3 = 72·6!/4! = 2160A3A1 nodea_x 2 nodea 3a branch_10 2 nodea_x 2 nodea612404**1080**01020021{ }∨( ) E6/A3A1 = 72·6!/4!/2 = 1080nodea 2 nodea_x 2 nodes_1x 3a nodea 3a nodea {3,3}46040***1080*00201102nodea 2 nodea_x 2 branch_10 3a nodea 2 nodea_x r{3,3}612044****108000021012A4 nodea 3a nodea 3a nodes_1x 3a nodea 2 nodea_x r{3,3,3}f410302010055000432****110{ } E6/A4 = 72·6!/5! = 432A4A1 nodea 3a nodea 3a branch_10 2 nodea_x 2 nodea10302001050500*216***020 E6/A4A1 = 72·6!/5!/2 = 216A4 nodea_x 2 nodea 3a nodes_1x 3a nodea 3a nodea10301020005050**432**101 E6/A4 = 72·6!/5! = 432D4 nodea_x 2 nodea 3a branch_10 3a nodea 2 nodea_x {3,4,3}249632323208808***270*011 E6/D4 = 72·6!/8/4! = 270A4A1 nodea 2 nodea_x 2 branch_10 3a nodea 3a nodea r{3,3,3}10300201000055****216002 E6/A4A1 = 72·6!/5!/2 = 216A5 nodea 3a nodea 3a nodes_1x 3a nodea 3a nodea 2r{3,3,3,3}f5209060600153001506060072**( ) E6/A5 = 72·6!/6! = 72D5 nodea 3a nodea 3a branch_10 3a nodea 2 nodea_x2r{4,3,3,3}8048032016016080808004016160100*27* E6/D5 = 72·6!/16/5! = 27nodea_x 2 nodea 3a branch_10 3a nodea 3a nodea8048016032016008040808000161016**27 Truncated 122 polytope Truncated 122TypeUniform 6-polytopeSchläfli symbol t{3,32,2}Coxeter symbol t(122)Coxeter-Dynkin diagramnodea 3a nodea 3a branch_11 3a nodea 3a nodeaor node_1 3 node_1 split1 nodes 3ab nodes5-faces72+27+274-faces32+216+432+270+216Cells1080+2160+1080+1080+1080Faces4320+4320+2160Edges6480+720Vertices1440Vertex figure( )v{3}x{3}Petrie polygonDodecagonCoxeter groupE6, [[3,32,2]], order 103680Propertiesconvex Alternate names Truncated 122 polytope (Acronym: tim)Klitzing, (o3o3x3o3o *c3x - tim) Construction Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: nodea 3a nodea 3a branch_11 3a nodea 3a nodea. Images Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue. + Coxeter plane orthographic projectionsE6[12]D5[8]D4 / A2[6]B6[12/2]120px120px120px120pxA5[6]A4[5]A3 / D3[4]120px120px120px Birectified 122 polytope Birectified 122 polytopeTypeUniform 6-polytopeSchläfli symbol 2r{3,32,2}Coxeter symbol 2r(122)Coxeter-Dynkin diagramnodea 3a nodea_1 3a branch 3a nodea_1 3a nodeaor node 3 node split1 nodes_11 3ab nodes5-faces1264-faces2286Cells10800Faces19440Edges12960Vertices2160Vertex figureCoxeter groupE6, [[3,32,2]], order 103680Propertiesconvex Alternate names Bicantellated 221 Birectified pentacontatetrapeton (barm) (Jonathan Bowers)Klitzing, (o3x3o3x3o *c3o - barm) Images Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet. + Coxeter plane orthographic projectionsE6[12]D5[8]D4 / A2[6]B6[12/2]120px120px120px120pxA5[6]A4[5]A3 / D3[4]120px120px120px Trirectified 122 polytope Trirectified 122 polytopeTypeUniform 6-polytopeSchläfli symbol 3r{3,32,2}Coxeter symbol 3r(122)Coxeter-Dynkin diagramnodea_1 3a nodea 3a branch 3a nodea 3a nodea_1or node 3 node split1 nodes 3ab nodes_115-faces5584-faces4608Cells8640Faces6480Edges2160Vertices270Vertex figureCoxeter groupE6, [[3,32,2]], order 103680Propertiesconvex Alternate names Tricantellated 221 Trirectified pentacontatetrapeton (Acronym: trim, old: cacam, tram, mak) (Jonathan Bowers)Klitzing, (x3o3o3o3x *c3o - trim) See also List of E6 polytopes Notes References Elte E. L. The Semiregular Polytopes of the Hyperspaces University of Groningen Groningen 1912 H. S. M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 978-0-471-01003-6 (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45], p. 334 (figure 3.6a) by Peter McMullen: (12-gonal node-edge graph of 122) polypeta.htm 6D uniform polytopes (polypeta) with acronyms o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3o3x3o3o *c3x - tim, o3x3o3x3o *c3o - barm, x3o3o3o3x *c3o - trim Category:6-polytopes