It can be realized as the Voronoi tessellation of the body-centred cubic lattice. Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) was the optimal soap bubble foam. However, a number of less symmetrical structures have later been found to be more efficient foams of soap bubbles, among which the Weaire–Phelan structure appears to be the best.
The honeycomb represents the permutohedron tessellation for 3-space. The coordinates of the vertices for one octahedron represent a hyperplane of integers in 4-space, specifically permutations of (1,2,3,4). The tessellation is formed by translated copies within the hyperplane.
The tessellation is the highest tessellation of parallelohedrons in 3-space.
Projections
The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a nonuniform rhombitrihexagonal tiling. A square symmetry projection forms two overlapping truncated square tiling, which combine together as a chamfered square tiling.
The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.
This honeycomb can be alternated, creating pyritohedral icosahedra from the truncated octahedra with disphenoid tetrahedral cells created in the gaps. There are three constructions from three related Coxeter-Dynkin diagrams: , , and . These have symmetry [4,3+,4], [4,(31,1)+] and [3[4]]+ respectively. The first and last symmetry can be doubled as [[4,3+,4]] and [[3[4]]]+.
This honeycomb is represented in the boron atoms of the α-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.[2]
Related polytopes
Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with truncated octahedra and hexagonal prisms (as ditrigonal trapezoprisms). Its vertex figure is a C2v-symmetric triangular bipyramid.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN978-0-471-01003-6[2]
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
bitruncated, cubic, honeycomb, type, uniform, honeycombschläfli, symbol, coxeter, dynkin, diagramcell, type, face, types, square, hexagon, edge, figure, isosceles, triangle, vertex, figure, tetragonal, disphenoid, space, groupfibrifold, notationcoxeter, notati. Bitruncated cubic honeycomb Type Uniform honeycombSchlafli symbol 2t 4 3 4 t1 2 4 3 4 Coxeter Dynkin diagramCell type 4 6 6 Face types square 4 hexagon 6 Edge figure isosceles triangle 3 Vertex figure tetragonal disphenoid Space groupFibrifold notationCoxeter notation Im3 m 229 8o 2 4 3 4 Coxeter group C 3 displaystyle tilde C 3 4 3 4 Dual Oblate tetrahedrilleDisphenoid tetrahedral honeycombCell Properties isogonal isotoxal isochoricThe bitruncated cubic honeycomb is a space filling tessellation or honeycomb in Euclidean 3 space made up of truncated octahedra or equivalently bitruncated cubes It has 4 truncated octahedra around each vertex Being composed entirely of truncated octahedra it is cell transitive It is also edge transitive with 2 hexagons and one square on each edge and vertex transitive It is one of 28 uniform honeycombs The bitruncated cubic honeycomb shown here in relation to a cubic honeycomb John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic and catoptric tessellation list with its dual called an oblate tetrahedrille also called a disphenoid tetrahedral honeycomb Although a regular tetrahedron can not tessellate space alone this dual has identical disphenoid tetrahedron cells with isosceles triangle faces Contents 1 Geometry 1 1 Projections 2 Symmetry 3 Related polyhedra and honeycombs 3 1 Alternated form 3 2 Related polytopes 4 See also 5 Notes 6 References 7 External linksGeometry EditIt can be realized as the Voronoi tessellation of the body centred cubic lattice Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb with curved faces and edges but the same combinatorial structure was the optimal soap bubble foam However a number of less symmetrical structures have later been found to be more efficient foams of soap bubbles among which the Weaire Phelan structure appears to be the best The honeycomb represents the permutohedron tessellation for 3 space The coordinates of the vertices for one octahedron represent a hyperplane of integers in 4 space specifically permutations of 1 2 3 4 The tessellation is formed by translated copies within the hyperplane The tessellation is the highest tessellation of parallelohedrons in 3 space Projections Edit The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements The highest hexagonal symmetry form projects into a nonuniform rhombitrihexagonal tiling A square symmetry projection forms two overlapping truncated square tiling which combine together as a chamfered square tiling Orthogonal projections Symmetry p6m 632 p4m 442 pmm 2222 Solid Frame Symmetry EditThe vertex figure for this honeycomb is a disphenoid tetrahedron and it is also the Goursat tetrahedron fundamental domain for the A 3 displaystyle tilde A 3 Coxeter group This honeycomb has four uniform constructions with the truncated octahedral cells having different Coxeter groups and Wythoff constructions These uniform symmetries can be represented by coloring differently the cells in each construction Five uniform colorings by cell Space group Im3 m 229 Pm3 m 221 Fm3 m 225 F4 3m 216 Fd3 m 227 Fibrifold 8o 2 4 2 2 2 1o 2 2 2Coxeter group C 3 displaystyle tilde C 3 2 4 3 4 4 3 4 C 3 displaystyle tilde C 3 4 3 4 2 3 4 B 3 displaystyle tilde B 3 4 31 1 lt 3 4 gt A 3 displaystyle tilde A 3 3 4 A 3 displaystyle tilde A 3 2 3 4 3 4 Coxeter diagram truncated octahedra 1 1 1 2 1 1 1 1 1 1 1 1 Vertex figure Vertexfiguresymmetry 2 4 order 8 2 order 4 order 2 order 1 2 order 2 ImageColored bycell Related polyhedra and honeycombs Edit The regular skew apeirohedron 6 4 4 contains the hexagons of this honeycomb The 4 3 4 Coxeter group generates 15 permutations of uniform tessellations 9 with distinct geometry including the alternated cubic honeycomb The expanded cubic honeycomb also known as the runcinated tesseractic honeycomb is geometrically identical to the cubic honeycomb C3 honeycombsSpacegroup Fibrifold Extendedsymmetry Extendeddiagram Order HoneycombsPm3 m 221 4 2 4 3 4 1 1 2 3 4 5 6Fm3 m 225 2 2 1 4 3 4 4 31 1 Half 7 11 12 13I4 3m 217 4o 2 4 3 4 2 Half 2 7 Fd3 m 227 2 2 1 4 3 4 1 3 4 Quarter 2 10 Im3 m 229 8o 2 4 3 4 2 1 8 9The 4 31 1 Coxeter group generates 9 permutations of uniform tessellations 4 with distinct geometry including the alternated cubic honeycomb B3 honeycombsSpacegroup Fibrifold Extendedsymmetry Extendeddiagram Order HoneycombsFm3 m 225 2 2 4 31 1 4 3 4 1 1 1 2 3 4Fm3 m 225 2 2 lt 1 4 31 1 gt lt 3 4 gt 2 1 3 Pm3 m 221 4 2 lt 4 31 1 gt 2 5 6 7 6 9 10 11This honeycomb is one of five distinct uniform honeycombs 1 constructed by the A 3 displaystyle tilde A 3 Coxeter group The symmetry can be multiplied by the symmetry of rings in the Coxeter Dynkin diagrams A3 honeycombsSpacegroup Fibrifold Squaresymmetry Extendedsymmetry Extendeddiagram Extendedgroup Honeycomb diagramsF4 3m 216 1o 2 a1 3 4 A 3 displaystyle tilde A 3 None Fm3 m 225 2 2 d2 lt 3 4 gt 4 31 1 A 3 displaystyle tilde A 3 21 B 3 displaystyle tilde B 3 1 2Fd3 m 227 2 2 g2 3 4 or 2 3 4 A 3 displaystyle tilde A 3 22 3Pm3 m 221 4 2 d4 lt 2 3 4 gt 4 3 4 A 3 displaystyle tilde A 3 41 C 3 displaystyle tilde C 3 4I3 204 8 o r8 4 3 4 4 3 4 A 3 displaystyle tilde A 3 8 C 3 displaystyle tilde C 3 2 Im3 m 229 8o 2 4 3 4 4 3 4 A 3 displaystyle tilde A 3 8 C 3 displaystyle tilde C 3 2 5 Alternated form Edit Alternated bitruncated cubic honeycombType Convex honeycombSchlafli symbol 2s 4 3 4 2s 4 31 1 sr 3 4 Coxeter diagrams Cells tetrahedronicosahedronVertex figure Coxeter group 4 3 4 C 3 displaystyle tilde C 3 Dual Ten of diamonds honeycombCell Properties vertex transitiveThis honeycomb can be alternated creating pyritohedral icosahedra from the truncated octahedra with disphenoid tetrahedral cells created in the gaps There are three constructions from three related Coxeter Dynkin diagrams and These have symmetry 4 3 4 4 31 1 and 3 4 respectively The first and last symmetry can be doubled as 4 3 4 and 3 4 The dual honeycomb is made of cells called ten of diamonds decahedra Five uniform colorings Space group I3 204 Pm3 200 Fm3 202 Fd3 203 F23 196 Fibrifold 8 o 4 2 2o 1oCoxeter group 4 3 4 4 3 4 4 31 1 3 4 3 4 Coxeter diagram Order double full half quarterdouble quarterImagecolored by cells This honeycomb is represented in the boron atoms of the a rhombohedral crystal The centers of the icosahedra are located at the fcc positions of the lattice 2 Related polytopes Edit Nonuniform variants with 4 3 4 symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with truncated octahedra and hexagonal prisms as ditrigonal trapezoprisms Its vertex figure is a C2v symmetric triangular bipyramid This honeycomb can then be alternated to produce another nonuniform honeycomb with pyritohedral icosahedra octahedra as triangular antiprisms and tetrahedra as sphenoids Its vertex figure has C2v symmetry and consists of 2 pentagons 4 rectangles 4 isosceles triangles divided into two sets of 2 and 4 scalene triangles See also Edit Wikimedia Commons has media related to Bitruncated cubic honeycomb Architectonic and catoptric tessellation Cubic honeycomb Brillouin zoneNotes Edit 1 A000029 6 1 cases skipping one with zero marks Williams 1979 p 199 Figure 5 38 References EditJohn H Conway Heidi Burgiel Chaim Goodman Strauss 2008 The Symmetries of Things ISBN 978 1 56881 220 5 Chapter 21 Naming the Archimedean and Catalan polyhedra and tilings Architectonic and Catoptric tessellations p 292 298 includes all the nonprismatic forms George Olshevsky Uniform Panoploid Tetracombs Manuscript 2006 Complete list of 11 convex uniform tilings 28 convex uniform honeycombs and 143 convex uniform tetracombs Branko Grunbaum Uniform tilings of 3 space Geombinatorics 4 1994 49 56 Kaleidoscopes Selected Writings of H S M Coxeter edited by F Arthur Sherk Peter McMullen Anthony C Thompson Asia Ivic Weiss Wiley Interscience Publication 1995 ISBN 978 0 471 01003 6 2 Paper 22 H S M Coxeter Regular and Semi Regular Polytopes I Math Zeit 46 1940 380 407 MR 2 10 1 9 Uniform space fillings A Andreini Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative On the regular and semiregular nets of polyhedra and on the corresponding correlative nets Mem Societa Italiana della Scienze Ser 3 14 1905 75 129 Klitzing Richard 3D Euclidean Honeycombs o4x3x4o batch O16 Uniform Honeycombs in 3 Space 05 Batch Williams Robert 1979 The Geometrical Foundation of Natural Structure A Source Book of Design Dover Publications Inc ISBN 0 486 23729 X External links EditWeisstein Eric W Space filling polyhedron MathWorld Retrieved from https en wikipedia org w index php title Bitruncated cubic honeycomb amp oldid 1134836045, wikipedia, wiki, book, books, library,
