Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform tessellation which fills threedimensional Euclidean space with nonoverlapping convex uniform polyhedral cells.
Twentyeight such honeycombs exist:
 the familiar cubic honeycomb and 7 truncations thereof;
 the alternated cubic honeycomb and 4 truncations thereof;
 10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb);
 5 modifications of some of the above by elongation and/or gyration.
They can be considered the threedimensional analogue to the uniform tilings of the plane.
Contents
 1 History
 2 Compact Euclidean uniform tessellations (by their infinite Coxeter group families)
 3 Noncompact forms
 4 Hyperbolic forms
 5 References
 6 External links
History
 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and SemiRegular Figures in Space of n Dimensions, including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra.
 1905: Alfredo Andreini enumerated 25 of these tessellations.
 1991: Norman Johnson's manuscript Uniform Polytopes identified the complete list of 28.
 1994: Branko Grünbaum, in his paper Uniform tilings of 3space, also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991. He also mentions that I. Alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time.
 2006: George Olshevsky, in his manuscript Uniform Panoploid Tetracombs, along with repeating the derived list of 11 convex uniform tilings, and 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs (Honeycombs of uniform polychorons in 4space).
Only 14 of the convex uniform polyhedra appear in these patterns:
 three of the five Platonic solids,
 six of the thirteen Archimedean solids, and
 five of the infinite family of prisms.
Names
This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (nonregular) polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations.
The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in Uniform polychoron#Geometric derivations.)
For crossreferencing, they are given with list indices from [A]ndreini (122), [W]illiams(12,919), [J]ohnson (1119, 2125, 3134, 4149, 5152, 6165), and [G]runbaum(128).
Compact Euclidean uniform tessellations (by their infinite Coxeter group families)
The fundamental infinite Coxeter groups for 3space are:
 The , [4,3,4], cubic, (8 unique forms plus one alternation)
 The , [4,3^{1,1}], alternated cubic, (11 forms, 3 new)
 The cyclic group, [(3,3,3,3)], (5 forms, one new)
In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations.
The total unique honeycombs above are 18.
The prismatic stacks from infinite Coxeter groups for 3space are:
 The x, [4,4]x[∞] prismatic group, (2 new forms)
 The x, [6,3]x[∞] prismatic group, (7 unique forms)
 The x, (3 3 3)x[∞] prismatic group, (No new forms)
 The xx, [∞]x[∞]x[∞] prismatic group, (These all become a cubic honeycomb)
In addition there is one special elongated form of the triangular prismatic honeycomb.
The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10.
Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs.
The C^{~}_{3}, [4,3,4] group (cubic)
The regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, offers seven unique derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the cubic honeycomb.)
Reference
IndicesHoneycomb name
CoxeterDynkin
and Schläfli
symbolsCell counts/vertex
and positions in cubic honeycomb(0)
(1)
(2)
(3)
Solids
(Partial)Frames
(Perspective)Vertex figure J_{11,15}
A_{1}
W_{1}
G_{22}cubic
t_{0}{4,3,4}(8)
(4.4.4)
octahedronJ_{12,32}
A_{15}
W_{14}
G_{7}rectified cubic
t_{1}{4,3,4}(2)
(3.3.3.3)(4)
(3.4.3.4)
cuboidJ_{13}
A_{14}
W_{15}
G_{8}truncated cubic
t_{0,1}{4,3,4}(1)
(3.3.3.3)(4)
(3.8.8)
square pyramidJ_{14}
A_{17}
W_{12}
G_{9}cantellated cubic
t_{0,2}{4,3,4}(1)
(3.4.3.4)(2)
(4.4.4)(2)
(3.4.4.4)
obilique triangular prismJ_{11,15} runcinated cubic
(same as regular cubic)
t_{0,3}{4,3,4}(1)
(4.4.4)(3)
(4.4.4)(3)
(4.4.4)(1)
(4.4.4)
octahedronJ_{16}
A_{3}
W_{2}
G_{28}bitruncated cubic
t_{1,2}{4,3,4}(2)
(4.6.6)(2)
(4.6.6)
(disphenoid tetrahedron)J_{17}
A_{18}
W_{13}
G_{25}cantitruncated cubic
t_{0,1,2}{4,3,4}(1)
(4.6.6)(1)
(4.4.4)(2)
(4.6.8)
irregular tetrahedronJ_{18}
A_{19}
W_{19}
G_{20}runcitruncated cubic
t_{0,1,3}{4,3,4}(1)
(3.4.4.4)(1)
(4.4.4)(2)
(4.4.8)(1)
(3.8.8)
oblique trapezoidal pyramidJ_{19}
A_{22}
W_{18}
G_{27}omnitruncated cubic
t_{0,1,2,3}{4,3,4}(1)
(4.6.8)(1)
(4.4.8)(1)
(4.4.8)(1)
(4.6.8)
irregular tetrahedronJ_{21,31,51}
A_{2}
W_{9}
G_{1}alternated cubic
h_{0}{4,3,4}(6)
(3.3.3.3)(8)
(3.3.3)
cuboctahedronB^{~}_{4}, [4,3^{1,1}] group
The group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.
The honeycombs from this group are called alternated cubic because the first form can be seen as a cubic honeycomb with alternate vertices removed, reducing cubic cells to tetrahedra and creating octahedron cells in the gaps.
Nodes are indexed left to right as 0,1,0',3 with 0' being below and interchangeable with 0. The alternate cubic names given are based on this ordering.
Referenced
indicesHoneycomb name
CoxeterDynkin
diagramCells by location
(and count around each vertex)Solids
(Partial)Frames
(Perspective)vertex figure (0)
(1)
(0')
(3)
J_{21,31,51}
A_{2}
W_{9}
G_{1}alternated cubic
(6)
(3.3.3.3)(8)
(3.3.3)
cuboctahedronJ_{22,34}
A_{21}
W_{17}
G_{10}truncated alternated cubic
(1)
(3.4.3.4)(2)
(4.6.6)(2)
(3.6.6)
rectangular pyramidJ_{12,32}
A_{15}
W_{14}
G_{7}rectified cubic
(rectified alternate cubic)
(2)
(3.4.3.4)(2)
(3.4.3.4)(2)
(3.3.3.3)
cuboidJ_{12,32}
A_{15}
W_{14}
G_{7}rectified cubic
(cantellated alternate cubic)
(1)
(3.3.3.3)(1)
(3.3.3.3)(4)
(3.4.3.4)
cuboidJ_{16}
A_{3}
W_{2}
G_{28}bitruncated cubic
(cantitruncated alternate cubic)
(1)
(4.6.6)(1)
(4.6.6)(2)
(4.6.6)
isosceles tetrahedronJ_{13}
A_{14}
W_{15}
G_{8}truncated cubic
(bicantellated alternate cubic)
(2)
(3.8.8)(2)
(3.8.8)(1)
(3.3.3.3)
square pyramidJ_{11,15}
A_{1}
W_{1}
G_{22}cubic
(trirectified alternate cubic)
(4)
(4.4.4)(4)
(4.4.4)
octahedronJ_{23}
A_{16}
W_{11}
G_{5}runcinated alternated cubic
(1)
cube(3)
(3.4.4.4)(1)
(3.3.3)
tapered triangular prismJ_{14}
A_{17}
W_{12}
G_{9}cantellated cubic
(runcicantellated alternate cubic)
(1)
(3.4.4.4)(2)
(4.4.4)(1)
(3.4.4.4)(1)
(3.4.3.4)
obilique triangular prismJ_{24}
A_{20}
W_{16}
G_{21}cantitruncated alternated cubic
(or runcitruncated alternate cubic)
(1)
(3.8.8)(2)
(4.6.8)(1)
(3.6.6)
Irregular tetrahedronJ_{17}
A_{18}
W_{13}
G_{25}cantitruncated cubic
(omnitruncated alternated cubic)
(1)
(4.6.8)(1)
(4.4.4)(1)
(4.6.8)(1)
(4.6.6)
irregular tetrahedronA^{~}_{3}, [(3,3,3,3)] group
There are 5 forms^{[1]} constructed from the group, of which only the quarter cubic honeycomb is unique.
Referenced
indicesHoneycomb name
CoxeterDynkin
diagramCells by location
(and count around each vertex)Solids
(Partial)Frames
(Perspective)vertex figure (0)
(1)
(2)
(3)
J_{21,31,51}
A_{2}
W_{9}
G_{1}alternated cubic
(4)
(3.3.3)(6)
(3.3.3.3)(4)
(3.3.3)
cuboctahedronJ_{12,32}
A_{15}
W_{14}
G_{7}rectified cubic
(2)
(3.4.3.4)(1)
(3.3.3.3)(2)
(3.4.3.4)(1)
(3.3.3.3)
cuboidJ_{25,33}
A_{13}
W_{10}
G_{6}quarter cubic
(1)
(3.3.3)(1)
(3.3.3)(3)
(3.6.6)(3)
(3.6.6)
triangular antiprismJ_{22,34}
A_{21}
W_{17}
G_{10}truncated alternated cubic
(1)
(3.6.6)(1)
(3.4.3.4)(1)
(3.6.6)(2)
(4.6.6)
Rectangular pyramidJ_{16}
A_{3}
W_{2}
G_{28}bitruncated cubic
(1)
(4.6.6)(1)
(4.6.6)(1)
(4.6.6)(1)
(4.6.6)
isosceles tetrahedronNonwythoffian forms (gyrated and elongated)
Three more uniform honeycombs are generated by breaking one or another of the above honeycombs where its faces form a continuous plane, then rotating alternate layers by 60 or 90 degrees (gyration) and/or inserting a layer of prisms (elongation).
The elongated and gyroelongated alternated cubic tilings have the same vertex figure, but are not alike. In the elongated form, each prism meets a tetrahedron at one triangular end and an octahedron at the other. In the gyroelongated form, prisms that meet tetrahedra at both ends alternate with prisms that meet octahedra at both ends.
The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings; the two may be derived from the gyrated and plain triangular prismatic tilings, respectively, by inserting layers of cubes.
Referenced
indicessymbol Honeycomb name cell types (# at each vertex) Solids
(Partial)Frames
(Perspective)vertex figure J_{52}
A_{2'}
G_{2}h{4,3,4}:g gyrated alternated cubic tetrahedron (8)
octahedron (6)
triangular orthobicupolaJ_{61}
A_{?}
G_{3}h{4,3,4}:ge gyroelongated alternated cubic triangular prism (6)
tetrahedron (4)
octahedron (3) J_{62}
A_{?}
G_{4}h{4,3,4}:e elongated alternated cubic triangular prism (6)
tetrahedron (4)
octahedron (3)J_{63}
A_{?}
G_{12}{3,6}:g x {∞} gyrated triangular prismatic triangular prism (12) J_{64}
A_{?}
G_{15}{3,6}:ge x {∞} gyroelongated triangular prismatic triangular prism (6)
cube (4)Prismatic stacks
Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The vertex figure of each is an irregular bipyramid whose faces are isosceles triangles.
The C^{~}_{2}xI^{~}_{1}(∞), [4,4] x [∞], prismatic group
There's only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form.
Indices CoxeterDynkin
and Schläfli
symbolsHoneycomb name Plane
tilingSolids
(Partial)Tiling J_{11,15}
A_{1}
G_{22}
{4,4} x {∞}Cubic
(Square prismatic)(4.4.4.4) J_{45}
A_{6}
G_{24}
t_{0,1}{4,4} x {∞}Truncated/Bitruncated square prismatic (4.8.8) J_{11,15}
A_{1}
G_{22}
t_{1}{4,4} x {∞}Cubic
(Rectified square prismatic)(4.4.4.4) J_{11,15}
A_{1}
G_{22}
t_{0,2}{4,4} x {∞}Cubic
(Cantellated square prismatic)(4.4.4.4) J_{45}
A_{6}
G_{24}
t_{0,1,2}{4,4} x {∞}Truncated square prismatic
(Omnitruncated square prismatic)(4.8.8) J_{44}
A_{11}
G_{14}
s{4,4} x {∞}Snub square prismatic (3.3.4.3.4) The G^{~}_{2}xI^{~}_{1}(∞), [6,3] x [∞] prismatic group
Indices CoxeterDynkin
and Schläfli
symbolsHoneycomb name Plane
tilingSolids
(Partial)Tiling J_{42}
A_{5}
G_{26}
t_{0}{6,3} x {∞}Hexagonal prismatic (6^{3}) J_{46}
A_{7}
G_{19}
t_{0,1}{6,3} x {∞}Truncated hexagonal prismatic (3.12.12) J_{43}
A_{8}
G_{18}
t_{1}{6,3} x {∞}Trihexagonal prismatic (3.6.3.6) J_{42}
A_{5}
G_{26}
t_{1,2}{6,3} x {∞}Truncated triangular prismatic
Hexagonal prismatic(6.6.6) J_{41}
A_{4}
G_{11}
t_{2}{6,3} x {∞}Triangular prismatic (3^{6}) J_{47}
A_{9}
G_{16}
t_{0,2}{6,3} x {∞}Rhombitrihexagonal prismatic (3.4.6.4) J_{49}
A_{10}
G_{23}
t_{0,1,2}{6,3} x {∞}Omnitruncated trihexagonal prismatic (4.6.12) J_{48}
A_{12}
G_{17}
s{6,3} x {∞}Snub trihexagonal prismatic (3.3.3.3.6) J_{65}
A_{11'}
G_{13}{3,6}:e x {∞} elongated triangular prismatic (3.3.3.4.4) Examples
All 28 of these tessellations are found in crystal arrangements.^{[citation needed]}
The alternated cubic honeycomb is of special importance since its vertices form a cubic closepacking of spheres. The spacefilling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently rediscovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s). [2] [3] [4] [5]. Octet trusses are now among the most common types of truss used in construction.
Noncompact forms
Examples (partially drawn) Cubic slab honeycomb
Alternated hexagonal slab honeycomb
If cells are allowed to be uniform tilings, more uniform honeycombs can be defined:
Families:
 xA_{1}: [4,4]x[ ] Cubic prismatic slab honeycomb (3 forms)
 xA_{1}: [6,3]x[ ] Trihexagonal prismatic slab honeycomb (8 forms)
 xA_{1}: (3 3 3)x[ ] Triangular prismatic slab (No new forms)
 xA_{1}xA_{1}: [∞]x[ ]x[ ] = Cubic column honeycomb (1 form)
 I_{2}(p)x: [p]x[∞] Prismatic column honeycomb
 xxA_{1}: [∞]x[∞]x[ ] = [4,4]x[ ]  = (Same as cubic slab honeycomb family)
Hyperbolic forms
Main article: Convex uniform honeycombs in hyperbolic spaceThere are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3space, generated as Wythoff constructions, and represented by ring permutations of the CoxeterDynkin diagrams for each family.
From these 9 families, there are a total of 76 unique honeycombs generated:
 [3,5,3] :  9 forms
 [5,3,4] :  15 forms
 [5,3,5] :  9 forms
 [5,3^{1,1}] :  11 forms (7 overlap with [5,3,4] family, 4 are unique)
 (4 3 3 3) :  9 forms
 (4 3 4 3) :  6 forms
 (5 3 3 3) :  9 forms
 (5 3 4 3) :  9 forms
 (5 3 5 3) :  6 forms
The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of nonWythoffian exist. One known example is in the {3,5,3} family.
There are also 23 noncompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity:
Hyperbolic noncompact groups 7 , , , , , , 7 , , ,, , , 6 , , , , , 3 , , References
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292298, includes all the nonprismatic forms)
 George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
 Branko Grünbaum, (1994) Uniform tilings of 3space. Geombinatorics 4, 49  56.
 Norman Johnson (1991) Uniform Polytopes, Manuscript
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Chapter 5: Polyhedra packing and space filling)
 Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0500340331.
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [6]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (1.9 Uniform spacefillings)
 A. Andreini, (1905) 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 75–129.
 D. M. Y. Sommerville, (1930) An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, . 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
External links
 Weisstein, Eric W., "Honeycomb" from MathWorld.
 Uniform Honeycombs in 3Space VRML models
 Elementary Honeycombs Vertex transitive space filling honeycombs with nonuniform cells.
 Uniform partitions of 3space, their relatives and embedding, 1999
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
 octet truss animation
 Review: A. F. Wells, Threedimensional nets and polyhedra, H. S. M. Coxeter (Source: Bull. Amer. Math. Soc. Volume 84, Number 3 (1978), 466470.)
 Richard Klitzing, 3D, Euclidean tesselations
Categories: Honeycombs (geometry)
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