1 32 polytope

Uniform polytope {{DISPLAYTITLE:1 32 polytope}} 120px321nodea_1 3a nodea 3a nodea 3a branch 3a nodea 3a nodea120px231nodea 3a nodea 3a nodea 3a branch 3a nodea 3a nodea_1120px132nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea150pxRectified 321nodea 3a nodea_1 3a nodea 3a branch 3a nodea 3a nodea150pxBirectified 321nodea 3a nodea 3a nodea_1 3a branch 3a nodea 3a nodea150pxRectified 231nodea 3a nodea 3a nodea 3a branch 3a nodea_1 3a nodea150pxRectified 132nodea 3a nodea 3a nodea 3a branch_10 3a nodea 3a nodeaOrthogonal projections in E7 Coxeter plane In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences. The rectified 132 is constructed by points at the mid-edges of the 132. These polytopes are part of a family of 127 (27−1) convex uniform polytopes in 7 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 3a nodea. 132 polytope 132TypeUniform 7-polytopeFamily1k2 polytopeSchläfli symbol {3,33,2}Coxeter symbol 132Coxeter diagramnodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea6-faces182:56 122 30px|class=skin-invert126 131 30px|class=skin-invert5-faces4284:756 121 25px|class=skin-invert1512 121 25px|class=skin-invert2016 {34} 25px|class=skin-invert4-faces23688:4032 {33} 25px|class=skin-invert7560 111 25px|class=skin-invert12096 {33} 25px|class=skin-invertCells50400:20160 {32} 25px|class=skin-invert30240 {32} 25px|class=skin-invertFaces40320 {3}25px|class=skin-invertEdges10080Vertices576Vertex figuret2{35} 25px|class=skin-invertPetrie polygonOctadecagonCoxeter groupE7, [33,2,1], order 2903040Propertiesconvex This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea. It is the Voronoi cell of the dual E7* 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 Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes.Elte, 1912 Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch. Pentacontahexa-hecatonicosihexa-exon (acronym: lin) - 56-126 facetted polyexon (Jonathan Bowers)Klitzing (o3o3o3x *c3o3o3o - lin) Images + Coxeter plane projectionsE7E6 / F4B7 / A6120px[18]120px[12]120px[7×2]A5D7 / B6D6 / B5120px[6]120px[12/2]120px[10]D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3120px[8]120px[6]120px[4] Construction 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, nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea Removing the node on the end of the 2-length branch leaves the 6-demicube, 131, nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea Removing the node on the end of the 3-length branch leaves the 122, nodea 3a 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 6-simplex, 032, nodea 3a nodea 3a nodea_1 3a nodea 3a nodea 3a nodea 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, p. 202–203 E7nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 3a nodeak-face fk f0 f1f2f3f4f5f6 k-figures NotesA6 nodea 3a nodea 3a nodea 3a nodes_0x 3a nodea 3a nodea( ) f0 5763521014021035105105214221772r{3,3,3,3,3} E7/A6 = 72·8!/7! = 576A3A2A1 nodea 3a nodea 3a nodea 2 nodes_x1 2 nodea 3a nodea{ } f1 21008012121841212612343{3,3}x{3} E7/A3A2A1 = 72·8!/4!/3!/2 = 10080A2A2A1 nodea 3a nodea 2 nodea_x 2 branch_01 2 nodea_x 2 nodea{3}f2 33403202316336132{ }∨{3} E7/A2A2A1 = 72·8!/3!/3!/2 = 40320A3A2 nodea 3a nodea 2 nodea_x 2 branch_01r 3a nodea 2 nodea_x{3,3}f3 46420160*13033031{3}∨( ) E7/A3A2 = 72·8!/4!/3! = 20160A3A1A1 nodea 2 nodea_x 2 nodea 3a branch_01l 2 nodea_x 2 nodea464*3024002214122Phyllic disphenoid E7/A3A1A1 = 72·8!/4!/2/2 = 30240A4A2 nodea 3a nodea 2 nodea_x 2 branch_01r 3a nodea 3a nodea{3,3,3}f4 51010504032**30030{3} E7/A4A2 = 72·8!/5!/3! = 4032D4A1 nodea 2 nodea_x 2 nodea 3a branch_01lr 3a nodea 2 nodea_x{3,3,4}8243288*7560*12021{ }∨( ) E7/D4A1 = 72·8!/8/4!/2 = 7560A4A1 nodea_x 2 nodea 3a nodea 3a branch_01l 2 nodea_x 2 nodea{3,3,3}5101005**1209602112 E7/A4A1 = 72·8!/5!/2 = 12096D5A1 nodea 2 nodea_x 2 nodea 3a branch_01lr 3a nodea 3a nodeah{4,3,3,3}f5 1680160804016100756**20{ } E7/D5A1 = 72·8!/16/5!/2 = 756D5 nodea_x 2 nodea 3a nodea 3a branch_01lr 3a nodea 2 nodea_x1680160408001016*1512*11 E7/D5 = 72·8!/16/5! = 1512A5A1 nodea 3a nodea 3a nodea 3a branch_01l 2 nodea_x 2 nodea{3,3,3,3,3} 61520015006**201602 E7/A5A1 = 72·8!/6!/2 = 2016E6 nodea_x 2 nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea{3,32,2} f6 727202160108010802162702162727056*( ) E7/E6 = 72·8!/72/6! = 56D6 nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 2 nodea_xh{4,3,3,3,3} 3224064016048006019201232*126 E7/D6 = 72·8!/32/6! = 126 Related polytopes and honeycombs The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134. Rectified 132 polytope Rectified 132TypeUniform 7-polytopeSchläfli symbol t1{3,33,2}Coxeter symbol 0321Coxeter-Dynkin diagramnodea 3a nodea 3a branch_10 3a nodea 3a nodea 3a nodea6-faces7585-faces123484-faces72072Cells191520Faces241920Edges120960Vertices10080Vertex figure{3,3}×{3}×{}Coxeter groupE7, [33,2,1], order 2903040Propertiesconvex The rectified 132 (also called 0321) is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}. Alternate names Rectified pentacontahexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym: lanq) (Jonathan Bowers)Klitzing (o3o3x3o *c3o3o3o - lanq) Construction It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, nodea 3a nodea 3a branch_10 3a nodea 3a nodea 3a nodea, and the ring represents the position of the active mirror(s). Removing the node on the end of the 3-length branch leaves the rectified 122 polytope, nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea Removing the node on the end of the 2-length branch leaves the demihexeract, 131, nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea Removing the node on the end of the 1-length branch leaves the birectified 6-simplex, nodea 3a nodea 3a nodea_1 3a nodea 3a nodea 3a nodea The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{}, nodea 3a nodea_1 2 nodea_1 2 nodea_1 3a nodea 3a nodea Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders. E7nodea 3a nodea 3a nodea 3a branch_10 3a nodea 3a nodeak-face fk f0 f1f2f3f4f5f6 k-figures NotesA3A2A1 nodea 3a nodea 3a nodea 2 nodes_x0 2 nodea 3a nodea ( )f010080242412368123618244121824126681263423{3,3}x{3}x{ } E7/A3A2A1 = 72·8!/4!/3!/2 = 10080A2A1A1 nodea 3a nodea 2 nodea_x 2 nodes_1x 2 nodea_x 2 nodea { }f121209602131263313663133621312( )v{3}v{ } E7/A2A1A1 = 72·8!/3!/2/2 = 120960A2A2 nodea 3a nodea 2 nodea_x 2 nodes_1x 3a nodea 2 nodea_x01f23380640**1130013330033310311{3}v( )v( ) E7/A2A2 = 72·8!/3!/3! = 80640A2A2A1 nodea 3a nodea 2 nodea_x 2 branch_10 2 nodea_x 2 nodea33*40320*0203010603030601302{3}v{ } E7/A2A2A1 = 72·8!/3!/3!/2 = 40320A2A1A1 nodea 2 nodea_x 2 nodea 3a nodes_1x 2 nodea_x 2 nodea33**1209600021201242112421212{ }v{ }v( ) E7/A2A1A1 = 72·8!/3!/2/2 = 120960A3A2 nodea 3a nodea 2 nodea_x 2 nodes_1x 3a nodea 3a nodea 02f3 4640020160****13000033000310{3}v( ) E7/A3A2 = 72·8!/4!/3! = 20160nodea 3a nodea 2 nodea_x 2 branch_10 3a nodea 2 nodea_x 011612440*20160***10300030300301 A3A1 nodea 2 nodea_x 2 nodea 3a nodes_1x 3a nodea 2 nodea_x612404**60480**01120012210211Sphenoid E7/A3A1 = 72·8!/4!/2 = 60480A3A1A1 nodea 2 nodea_x 2 nodea 3a branch_10 2 nodea_x 2 nodea612044***30240*00202010401202{ }v{ } E7/A3A1A1 = 72·8!/4!/2/2 = 30240A3A1 nodea_x 2 nodea 3a nodea 3a nodes_1x 2 nodea_x 2 nodea 0246004****6048000021101221112Sphenoid E7/A3A1 = 72·8!/4!/2 = 60480A4A2 nodea 3a nodea 2 nodea_x 2 branch_10 3a nodea 3a nodea021f4 103020100550004032*****30000300{3} E7/A4A2 = 72·8!/5!/3! = 4032A4A1 nodea 2 nodea_x 2 nodea 3a nodes_1x 3a nodea 3a nodea10302001050500*12096****12000210{ }v() E7/A4A1 = 72·8!/5!/2 = 12096D4A1 nodea 2 nodea_x 2 nodea 3a branch_10 3a nodea 2 nodea_x 0111249632323208880**7560***10200201 E7/D4A1 = 72·8!/8/4!/2 = 7560A4 nodea_x 2 nodea 3a nodea 3a nodes_1x 3a nodea 2 nodea_x02110301002000505***24192**01110111( )v( )v( ) E7/A4 = 72·8!/5! = 34192A4A1 nodea_x 2 nodea 3a nodea 3a branch_10 2 nodea_x 2 nodea10300102000055****12096*00201102{ }v() E7/A4A1 = 72·8!/5!/2 = 12096nodea 3a nodea 3a nodea 3a nodes_1x 2 nodea_x 2 nodea 03510001000005*****1209600021012D5A1 nodea 2 nodea_x 2 nodea 3a branch_10 3a nodea 3a nodea 0211 f580480320160160808080400161610000756****200{ } E7/D5A1 = 72·8!/16/5!/2 = 756A5 nodea_x 2 nodea 3a nodea 3a nodes_1x 3a nodea 3a nodea 02220906006015030015060600*4032***110 E7/A5 = 72·8!/6! = 4032D5 nodea_x 2 nodea 3a nodea 3a branch_10 3a nodea 2 nodea_x 021180480160160320040808080001016160**1512**101 E7/D5 = 72·8!/16/5! = 1512A5 nodea 3a nodea 3a nodea 3a nodes_1x 3a nodea 2 nodea_x0311560200600015030000606***4032*011 E7/A5 = 72·8!/6! = 4032A5A1 nodea 3a nodea 3a nodea 3a branch_10 2 nodea_x 2 nodea1560020600001530000066****2016002 E7/A5A1 = 72·8!/6!/2 = 2016E6 nodea_x 2 nodea 3a nodea 3a branch_10 3a nodea 3a nodea 0221f672064804320216043201080108021601080108021643227043221602772270056**( ) E7/E6 = 72·8!/72/6! = 56A6 nodea 3a nodea 3a nodea 3a nodes_1x 3a nodea 3a nodea 032352101400210350105010502104202107070*576* E7/A6 = 72·8!/7! = 576D6 nodea 3a nodea 3a nodea 3a branch_10 3a nodea 2 nodea_x 0311240192064064019200160480480960006019219219200123232**126 E7/D6 = 72·8!/32/6! = 126 Images + Coxeter plane projectionsE7E6 / F4B7 / A6120px[18]120px[12]120px[14]A5D7 / B6D6 / B5120px[6]120px[12/2]120px[10]D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3120px[8]120px[6]120px[4] See also List of E7 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] polyexa.htm 7D uniform polytopes (polyexa) with acronyms o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - lanq CITEREFKlitzing Category:7-polytopes