1 42 polytope

Uniform 8 dimensional polytope {{DISPLAYTITLE:142 polytope}} 150px421nodea_1 3a nodea 3a nodea 3a nodea 3a branch 3a nodea 3a nodea150px142nodea 3a nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea150px241nodea 3a nodea 3a nodea 3a nodea 3a branch 3a nodea 3a nodea_1150pxRectified 421nodea 3a nodea_1 3a nodea 3a nodea 3a branch 3a nodea 3a nodea150pxRectified 142nodea 3a nodea 3a nodea 3a nodea 3a branch_10 3a nodea 3a nodea150pxRectified 241nodea 3a nodea 3a nodea 3a nodea 3a branch 3a nodea_1 3a nodea150pxBirectified 421nodea 3a nodea 3a nodea_1 3a nodea 3a branch 3a nodea 3a nodea150pxTrirectified 421nodea 3a nodea 3a nodea 3a nodea_1 3a branch 3a nodea 3a nodeaOrthogonal projections in E6 Coxeter plane In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421. These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8 dimensions, made of uniform polytope facets and vertex figures, defined by all non-empty combinations of rings in this Coxeter-Dynkin diagram: nodea 3a nodea 3a branch 3a nodea 3a nodea 3a nodea 3a nodea. 142 polytope 142TypeUniform 8-polytopeFamily1k2 polytopeSchläfli symbol {3,34,2}Coxeter symbol 142Coxeter diagramsnodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea 3a nodeanode_1 3 node split1 nodes 3ab nodes 3a nodea 3a nodea7-faces2400:240 132 35px|class=skin-invert2160 141 25px|class=skin-invert6-faces106080:6720 122 25px|class=skin-invert30240 131 25px|class=skin-invert69120 {35} 25px|class=skin-invert5-faces725760:60480 112 25px|class=skin-invert181440 121 25px|class=skin-invert483840 {34} 25px|class=skin-invert4-faces2298240:241920 102 25px|class=skin-invert604800 111 25px|class=skin-invert1451520 {33} 25px|class=skin-invertCells3628800:1209600 101 25px|class=skin-invert2419200 {32} 25px|class=skin-invertFaces2419200 {3} 25px|class=skin-invertEdges483840Vertices17280Vertex figuret2{36} 25px|class=skin-invertPetrie polygon30-gonCoxeter groupE8, [34,2,1]Propertiesconvex The 142 is composed of 2400 facets: 240 132 polytopes, and 2160 7-demicubes (141). Its vertex figure is a birectified 7-simplex. This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 152, and Coxeter-Dynkin diagram: nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea 3a nodea 3a nodea. Alternate names E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V17280 for its 17280 vertices. Elte E. L. The Semiregular Polytopes of the Hyperspaces University of Groningen Groningen 1912 Coxeter named it 142 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch. Diacositetraconta-dischiliahectohexaconta-zetton (acronym: bif) – 240-2160 facetted polyzetton (Jonathan Bowers)Klitzing (o3o3o3x *c3o3o3o3o - bif) Coordinates The 17280 vertices can be defined as sign and location permutations of: All sign combinations (32): (280 × 32 = 8960 vertices) (4, 2, 2, 2, 2, 0, 0, 0) Half of the sign combinations (128): ((1 + 8 + 56) × 128 = 8320 vertices) (2, 2, 2, 2, 2, 2, 2, 2) (5, 1, 1, 1, 1, 1, 1, 1) (3, 3, 3, 1, 1, 1, 1, 1) The edge length is 22 in this coordinate set, and the polytope radius is 42. Construction It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-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 3a nodea. Removing the node on the end of the 2-length branch leaves the 7-demicube, 141, nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea 3a nodea. Removing the node on the end of the 4-length branch leaves the 132, nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 3a nodea. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 042, nodea 3a nodea 3a nodea_1 3a nodea 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 Configuration matrixE8nodea 3a nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 3a nodeak-face fk f0 f1f2f3f4f5f6f7 k-figureNotesA7 nodea 3a nodea 3a nodea 3a nodea 3a nodes_0x 3a nodea 3a nodea ( )f017280564202805607028042056168168285628882r{36} E8/A7 = 192·10!/8! = 17280A4A2A1 nodea 3a nodea 3a nodea 3a nodea 2 nodes_x1 2 nodea 3a nodea { }f12483840151530530301030151015353{3}x{3,3,3} E8/A4A2A1 = 192·10!/5!/2/2 = 483840A3A2A1 nodea 3a nodea 3a nodea 2 nodea_x 2 branch_01 2 nodea_x 2 nodea {3}f233241920024186412468142{3.3}v{ } E8/A3A2A1 = 192·10!/4!/3!/2 = 2419200A3A3 nodea 3a nodea 3a nodea 2 nodea_x 2 branch_01r 3a nodea 2 nodea_x110f34641209600*14046064041{3,3}v( ) E8/A3A3 = 192·10!/4!/4! = 1209600A3A2A1 nodea 3a nodea 2 nodea_x 2 nodea 3a branch_01l 2 nodea_x 2 nodea464*241920002316336132{3}v{ } E8/A3A2A1 = 192·10!/4!/3!/2 = 2419200A4A3 nodea 3a nodea 3a nodea 2 nodea_x 2 branch_01r 3a nodea 3a nodea 120f45101050241920**40060040{3,3} E8/A4A3 = 192·10!/4!/4! = 241920D4A2 nodea 3a nodea 2 nodea_x 2 nodea 3a branch_01lr 3a nodea 2 nodea_x 1118243288*604800*13033031{3}v( ) E8/D4A2 = 192·10!/8/4!/3! = 604800A4A1A1 nodea 2 nodea_x 2 nodea 3a nodea 3a branch_01l 2 nodea_x 2 nodea 120 5101005**145152002214122{ }v{ } E8/A4A1A1 = 192·10!/5!/2/2 = 1451520D5A2 nodea 3a nodea 2 nodea_x 2 nodea 3a branch_01lr 3a nodea 3a nodea 121f5168016080401610060480**30030{3} E8/D5A2 = 192·10!/16/5!/3! = 40480D5A1 nodea 2 nodea_x 2 nodea 3a nodea 3a branch_01lr 3a nodea 2 nodea_x1680160408001016*181440*12021{ }v( ) E8/D5A1 192·10!/16/5!/2 = 181440A5A1 nodea_x 2 nodea 3a nodea 3a nodea 3a branch_01l 2 nodea_x 2 nodea 13061520015006**48384002112 E8/A5A1 = 192·10!/6!/2 = 483840E6A1 nodea 2 nodea_x 2 nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 122f672720216010801080216270216272706720**20{ } E8/E6A1 = 192·10!/72/6!/2 = 6720D6 nodea_x 2 nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 2 nodea_x 1313224064016048006019201232*30240*11 E8/D6 = 192·10!/32/6! = 30240A6A1 nodea 3a nodea 3a nodea 3a nodea 3a branch_01l 2 nodea_x 2 nodea 140721350350021007**6912002 E8/A6A1 = 192·10!/7!/2 = 69120E7 nodea_x 2 nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 132f757610080403202016030240403275601209675615122016561260240*( ) E8/E7 = 192·10!/72/8! = 240D7 nodea 3a nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 2 nodea_x 14164672224056022400280134408444801464*2160 E8/D7 = 192·10!/64/7! = 2160 Projections E8[30]E7[18]E6[12]168px(1)168px(1,3,6)168px(8,16,24,32,48,64,96)[20][24][6]168px168px168px(1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20)Vertices on the image have nine colors only: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet, whereas the multiplicities listed in the image caption are 16 in number, which is impossible. The image or its caption (or both) is incorrect. May 2026 230px|thumb|The projection of 142 to the E8 Coxeter plane (also known as the Petrie projection) with polytope radius is shown below with 483,840 edges of length culled 53% on the interior to only 226,440. Orthographic projections are shown for the sub-symmetries of E8: E7, E6, B8, B7, B6, B5, B4, B3, B2, A7, and A5 Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their number of overlap in each projective plane in increasing order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet. D3 / B2 / A3[4]D4 / B3 / A2[6]D5 / B4[8]190px(32,160,192,240,480,512,832,960)190px(72,216,432,720,864,1080)190px(8,16,24,32,48,64,96)D6 / B5 / A4[10]D7 / B6[12]D8 / B7 / A6[14]190px190px190pxB8[16/2]A5[6]A7[8]190px190px190px thumb|230px|Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry: u (1, φ, 0, −1, φ, 0, 0, 0) v (φ, 0, 1, φ, 0, −1, 0, 0) w (0, 1, φ, 0, −1, φ, 0, 0) The 17280 projected 142 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. Notice the last two outer hulls are a combination of two overlapped dodecahedrons (40) and a nonuniform rhombicosidodecahedron (60). Related polytopes and honeycombs Rectified 142 polytope Rectified 142TypeUniform 8-polytopeSchläfli symbol t1{3,34,2}Coxeter symbol 0421Coxeter diagramsnodea 3a nodea 3a branch_10 3a nodea 3a nodea 3a nodea 3a nodeanode 3 node_1 split1 nodes 3ab nodes 3a nodea 3a nodea7-faces196806-faces3825605-faces26611204-faces9072000Cells16934400Faces16934400Edges7257600Vertices483840Vertex figure{3,3,3}×{3}×{}Coxeter groupE8, [34,2,1]Propertiesconvex The rectified 142 is named from being a rectification of the 142 polytope, with vertices positioned at the mid-edges of the 142. It can also be called a 0421 polytope with the ring at the center of 3 branches of length 4, 2, and 1. Alternate names 0421 polytope Birectified 241 polytope Quadrirectified 421 polytope Rectified diacositetraconta-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym: buffy) (Jonathan Bowers)Klitzing (o3o3x3o *c3o3o3o3o - buffy) Construction It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram: nodea 3a nodea 3a branch_10 3a nodea 3a nodea 3a nodea 3a nodea. Removing the node on the end of the 1-length branch leaves the birectified 7-simplex, nodea 3a nodea 3a nodea_1 3a nodea 3a nodea 3a nodea 3a nodea Removing the node on the end of the 2-length branch leaves the birectified 7-cube, nodea 3a branch_10 3a nodea 3a nodea 3a nodea 3a nodea. Removing the node on the end of the 3-length branch leaves the rectified 132, nodea 3a nodea 3a branch_10 3a nodea 3a nodea 3a nodea. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell-triangle duoprism prism, nodea 3a nodea_1 2 nodea_1 2 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. Configuration matrixE8nodea 3a nodea 3a nodea 3a nodea 3a branch_10 3a nodea 3a nodeak-face fk f0 f1f2f3f4f5f6f7 k-figureA4A2A1 nodea 3a nodea 3a nodea 3a nodea 2 nodes_x0 2 nodea 3a nodea ( )f0483840303015601015603060520306030301020303015610101563523{3,3,3}x{3,3}x{} A3A1A1 nodea 3a nodea 3a nodea 2 nodea_x 2 nodes_1x 2 nodea_x 2 nodea { }f127257600214128461481264461284164821412 A3A2 nodea 3a nodea 3a nodea 2 nodea_x 2 nodes_1x 3a nodea 2 nodea_x {3}f2334838400**1140014460046640064410411 A3A2A1 nodea 3a nodea 3a nodea 2 nodea_x 2 branch_10 2 nodea_x 2 nodea33*2419200*02040108060401204060801402A2A2A1 nodea 3a nodea 2 node_x 2 nodea 3a nodes_1x 2 nodea_x 2 nodea33**96768000021301263313663133621312 A3A3 nodea 3a nodea 3a nodea 2 nodea_x 2 nodes_1x 3a nodea 3a nodea 0200f3464001209600****14000046000064000410 nodea 3a nodea 3a nodea 2 nodea_x 2 branch_10 3a nodea 2 nodea_x0110612440*1209600***10400040600060400401A3A2 nodea 3a nodea 2 nodea_x 2 nodea 3a nodes_1x 3a nodea 2 nodea_x612404**4838400**01130013330033310311A3A2A1 nodea 3a nodea 2 nodea_x 2 nodea 3a branch_10 2 nodea_x 2 nodea612044***2419200*00203010603030601302A3A1A1 nodea 2 nodea_x 2 nodea 3a nodea 3a nodes_1x 2 nodea_x 2 nodea 020046004****725760000021201242112421212A4A3 nodea 3a nodea 3a nodea 2 nodea_x 2 branch_10 3a nodea 3a nodea0210f410302010055000241920*****40000060000400 A4A2 nodea 3a nodea 2 nodea_x 2 nodea 3a nodes_1x 3a nodea 3a nodea10302001050500*967680****13000033000310 D4A2 nodea 3a nodea 2 nodea_x 2 nodea 3a branch_10 3a nodea 2 nodea_x 0111249632323208880**604800***10300030300301A4A1 nodea 2 nodea_x 2 nodea 3a nodea 3a nodes_1x 3a nodea 2 nodea_x021010301002000505***2903040**01120012210211 A4A1A1 nodea 2 nodea_x 2 nodea 3a nodea 3a branch_10 2 nodea_x 2 nodea10300102000055****1451520*00202010401202 A4A1 nodea_x 2 nodea 3a nodea 3a nodea 3a nodes_1x 2 nodea_x 2 nodea0300 510001000005*****290304000021101221112 D5A2 nodea 3a nodea 2 nodea_x 2 nodea 3a branch_10 3a nodea 3a nodea 0211f58048032016016080808040016161000060480*****30000300 {3}A5A1 nodea 2 nodea_x 2 nodea 3a nodea 3a nodes_1x 3a nodea 3a nodea 022020906006015030015060600*483840****12000210{ }v()D5A1 nodea 2 nodea_x 2 nodea 3a nodea 3a branch_10 3a nodea 2 nodea_x 021180480160160320040808080001016160**181440***10200201A5 nodea_x 2 nodea 3a nodea 3a nodea 3a nodes_1x 3a nodea 2 nodea_x03101560200600015030000606***967680**01110111 ( )v( )v()A5A1 nodea_x 2 nodea 3a nodea 3a nodea 3a branch_10 2 nodea_x 2 nodea1560020600001530000066****483840*00201102{ }v() nodea 3a nodea 3a nodea 3a nodea 3a nodes_1x 2 nodea_x 2 nodea 04006150020000015000006*****48384000021012E6A1 nodea 2 nodea_x 2 nodea 3a nodea 3a branch_10 3a nodea 3a nodea 0221f672064804320216043201080108021601080108021643227043221602772270006720****200 { }A6 nodea_x 2 nodea 3a nodea 3a nodea 3a nodes_1x 3a nodea 3a nodea 03203521014002103501050105021042021070700*138240***110D6 nodea_x 2 nodea 3a nodea 3a nodea 3a branch_10 3a nodea 2 nodea_x 03112401920640640192001604804809600060192192192001232320**30240**101A6 nodea 3a nodea 3a nodea 3a nodea 3a nodes_1x 3a nodea 2 nodea_x0410211053501400035010500021042000707***138240*011A6A1 nodea 3a nodea 3a nodea 3a nodea 3a branch_10 2 nodea_x 2 nodea211050351400003510500002142000077****69120002E7 nodea_x 2 nodea 3a nodea 3a nodea 3a branch_10 3a nodea 3a nodea 0321f710080120960806404032012096020160201606048030240604804032120967560241921209612096756403215124032201605657612600240**( )A7 nodea 3a nodea 3a nodea 3a nodea 3a nodes_1x 3a nodea 3a nodea 042056420280056070028004200560168016802805602808080*17280*D7 nodea 3a nodea 3a nodea 3a nodea 3a branch_10 3a nodea 2 nodea_x 04116726720224022408960056022402240672000280134413442688008444844844800146464**2160 Projections Orthographic projections are shown for the sub-symmetries of B6, B5, B4, B3, B2, A7, and A5 Coxeter planes. Vertices are shown as circles, colored by their number of overlap in each projective plane in increasing order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet. (Planes for E8: E7, E6, B8, B7, [24] are not shown for being too large to display, but projections in E7 and E6 Coxeter planes can be seen at E8 polytope Graphs and E6 in the main table above.) D3 / B2 / A3[4]D4 / B3 / A2[6]D5 / B4[8]200px200px200pxD6 / B5 / A4[10]D7 / B6[12][6]200px200px200pxA5[6]A7[8] [20]200px200px200px See also List of E8 polytopes Notes References 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] polyzetta.htm 8D Uniform polytopes (polyzetta) with acronyms o3o3o3x *c3o3o3o3o - bif, o3o3x3o *c3o3o3o3o - buffy CITEREFKlitzing Category:8-polytopes Category:E8 (mathematics)