16-cell honeycomb

16-cell honeycomb280pxPerspective projection: the first layer of adjacent 16-cell facetsTypeRegular 4-honeycombUniform 4-honeycombFamilyAlternated hypercube honeycombSchläfli symbol{3,3,4,3}Coxeter diagramsnode_1 3 node 3 node 4 node 3 nodenodes_10ru split2 node 3 node 4 node = node_h1 4 node 3 node 3 node 4 nodenodes_10ru split2 node split1 nodes = node_h1 4 node 3 node split1 nodeslabel2 branch_hh 4a4b nodes split2 node4-face type{3,3,4} 40pxCell type{3,3} 40pxFace type{3}Edge figurecubeVertex figure24-cell 60px|class=skin-invertCoxeter group = [3,3,4,3]Dual{3,4,3,3}Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every (triangular) face. Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B4, D4, or F4 lattice.http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html The Lattice F4http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html The Lattice D4 Alternate names Hexadecachoric tetracomb/honeycomb Demitesseractic tetracomb/honeycomb Coordinates Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even. D4 lattice The vertex arrangement of the 16-cell honeycomb is called the D4 lattice or F4 lattice. The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space;Conway Sloane 1998 9 1.4 n-dimensional packings its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003.Conway Sloane 1998 12 1.5 Sphere packing problem – summary of resultsO. R. Musin The problem of the twenty-five spheres 2003 Russ. Math. Surv. 58 4 794–795 10.1070/RM2003v058n04ABEH000651 2003RuMaS..58..794M The related D+ 4 lattice (also called D2 4) can be constructed by the union of two D4 lattices, and is identical to the C4 lattice:Conway Sloane 1998 119 7.3 The packing D3+ nodes_10ru split2 node split1 nodes ∪ nodes_01rd split2 node split1 nodes = node_1 4 node 3 node split1 nodes = node_1 4 node 3 node 3 node 4 node The kissing number for D+ 4 is 23 = 8, (2n − 1 for n < 8, 240 for n = 8, and 2n(n − 1) for n > 8).Conway Sloane 1998 119 The related D* 4 lattice (also called D4 4 and C2 4) can be constructed by the union of all four D4 lattices, but it is identical to the D4 lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.Conway Sloane 1998 120 7.4 The dual lattice D3* nodes_10ru split2 node split1 nodes ∪ nodes_01rd split2 node split1 nodes ∪ nodes split2 node split1 nodes_10lu ∪ nodes split2 node split1 nodes_01ld = nodes_10ru split2 node split1 nodes = nodes_10r 4a4b nodes split2 node ∪ nodes_01r 4a4b nodes split2 node. The kissing number of the D* 4 lattice (and D4 lattice) is 24Conway Sloane 1998 120 and its Voronoi tessellation is a 24-cell honeycomb, node_1 split1 nodes 4a4b nodes, containing all rectified 16-cells (24-cell) Voronoi cells, node 4 node 3 node_1 3 node or node_1 3 node 4 node 3 node.Conway Sloane 1998 466 Symmetry constructions There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16-cell facets. Coxeter groupSchläfli symbolCoxeter diagramVertex figureSymmetryFacets/verf = [3,3,4,3]{3,3,4,3}node_1 3 node 3 node 4 node 3 nodenode_1 3 node 4 node 3 node[3,4,3], order 115224: 16-cell = [31,1,3,4]= h{4,3,3,4}nodes_10ru split2 node 3 node 4 node = node_h1 4 node 3 node 3 node 4 nodenode 3 node_1 3 node 4 node[3,3,4], order 38416+8: 16-cell = [31,1,1,1]{3,31,1,1} = h{4,3,31,1}nodes_10ru split2 node split1 nodes = node_h1 4 node 3 node split1 nodesnode 3 node_1 split1 nodes[31,1,1], order 1928+8+8: 16-cell2×½ = [[(4,3,3,4,2+)]]ht0,4{4,3,3,4}label2 branch_hh 4a4b nodes split2 node8+4+4: 4-demicube8: 16-cell Related honeycombs It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb, {3,3,3,4,3}, with 5-orthoplex facets, the regular 4-polytope 24-cell, {3,4,3} with octahedral (3-orthoplex) cell, and cube {4,3}, with (2-orthoplex) square faces. It has a 2-dimensional analogue, {3,6}, and as an alternated form (the demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb. See also Regular and uniform honeycombs in 4-space: Tesseractic honeycomb 24-cell honeycomb Truncated 24-cell honeycomb Snub 24-cell honeycomb 5-cell honeycomb Truncated 5-cell honeycomb Omnitruncated 5-cell honeycomb Notes References Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover, 0-486-61480-8 pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4} = {4,4}; h{4,3,4} = {31,1,4}, h{4,3,3,4} = {3,3,4,3}, ... 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] George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) flat.htm 4D Euclidean tesselations (tetracombs) x3o3o4o3o - hext - O104 Conway J.H. John Horton Conway Sloane N.J.A. Neil Sloane 1998 Sphere Packings, Lattices and Groups Springer 3rd 0-387-98585-9 registration https://archive.org/details/spherepackingsla0000conw_b8u0 Conway Sloane 1998 Category:Honeycombs (geometry) Category:5-polytopes Category:Regular tessellations