In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
Twenty-eight such honeycombs are known: 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 three-dimensional analogue to the uniform tilings of the plane.
The Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra.
History 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular 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 list of 28. 1994: Branko Grünbaum, in his paper Uniform tilings of 3-space, 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 4-polytopes in 4-space).George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Only 14 of the convex uniform polyhedra appear in these patterns: three of the five Platonic solids (the tetrahedron, cube, and octahedron), six of the thirteen Archimedean solids (the ones with reflective tetrahedral or octahedral symmetry), and five of the infinite family of prisms (the 3-, 4-, 6-, 8-, and 12-gonal ones; the 4-gonal prism duplicates the cube).
The icosahedron, snub cube, and square antiprism appear in some alternations, but those honeycombs cannot be realised with all edges unit length.
Names This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (non-regular) 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 4-polytope#Geometric derivations for 46 nonprismatic Wythoffian uniform 4-polytopes)
For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1–2,9-19), Johnson (11–19, 21–25, 31–34, 41–49, 51–52, 61–65), and Grünbaum(1-28). Coxeter uses δ4 for a cubic honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb, with subscripts for other forms based on the ring patterns of the Coxeter diagram.
Compact Euclidean uniform tessellations (by their infinite Coxeter group families)
The fundamental infinite Coxeter groups for 3-space are: The , [4,3,4], cubic, (8 unique forms plus one alternation) The , [4,31,1], alternated cubic, (11 forms, 3 new) The cyclic group, [(3,3,3,3)] or [3], (5 forms, one new)
There is a correspondence between all three families. Removing one mirror from produces , and removing one mirror from produces . This allows multiple constructions of the same honeycombs. If cells are colored based on unique positions within each Wythoff construction, these different symmetries can be shown.
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 3-space are: The ×, [4,4,2,∞] prismatic group, (2 new forms) The ×, [6,3,2,∞] prismatic group, (7 unique forms) The ×, [(3,3,3),2,∞] prismatic group, (No new forms) The ××, [∞,2,∞,2,∞] 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 C̃3, [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.) The reflectional symmetry is the affine Coxeter group [4,3,4]. There are four index 2 subgroups that generate alternations: [1+,4,3,4], [(4,3,4,2+)], [4,3+,4], and [4,3,4]+, with the first two generated repeated forms, and the last two are nonuniform.
+ [4,3,4], space group Pmm (221)ReferenceIndicesHoneycomb nameCoxeter diagramand Schläfli symbolCell counts/vertexand positions in cubic honeycombFrames(Perspective)Vertex figureDual cell(0) (1) (2) (3)AltSolids(Partial)J11,15A1W1G22δ4cubic (chon) t0{4,3,4}{4,3,4} (8)(4.4.4) octahedron Cube, J12,32A15W14G7O1rectified cubic (rich) t1{4,3,4}r{4,3,4}(2)(3.3.3.3) (4)(3.4.3.4) cuboidSquare bipyramidJ13A14W15G8t1δ4O15truncated cubic (tich) t0,1{4,3,4}t{4,3,4}(1)(3.3.3.3) (4)(3.8.8) square pyramidIsosceles square pyramidJ14A17W12G9t0,2δ4O14cantellated cubic (srich) t0,2{4,3,4}rr{4,3,4}(1)(3.4.3.4)(2)(4.4.4) (2)(3.4.4.4) oblique triangular prismTriangular bipyramidJ17A18W13G25t0,1,2δ4O17cantitruncated cubic (grich) t0,1,2{4,3,4}tr{4,3,4}(1)(4.6.6)(1)(4.4.4) (2)(4.6.8) irregular tetrahedronTriangular pyramidilleJ18A19W19G20t0,1,3δ4O19runcitruncated cubic (prich)t0,1,3{4,3,4}(1)(3.4.4.4)(1)(4.4.4)(2)(4.4.8)(1)(3.8.8) oblique trapezoidal pyramid Square quarter pyramidille J21,31,51A2W9G1hδ4O21alternated cubic (octet)h{4,3,4} (8)(3.3.3)(6)(3.3.3.3)cuboctahedronDodecahedrilleJ22,34A21W17G10h2δ4O25Cantic cubic (tatoh) ↔ (1)(3.4.3.4) (2)(3.6.6)(2)(4.6.6)rectangular pyramidHalf oblate octahedrilleJ23A16W11G5h3δ4O26Runcic cubic (sratoh) ↔ (1)(4.4.4) (1)(3.3.3)(3)(3.4.4.4)tapered triangular prismQuarter cubilleJ24A20W16G21h2,3δ4O28Runcicantic cubic (gratoh) ↔ (1)(3.8.8) (1)(3.6.6)(2)(4.6.8)Irregular tetrahedronHalf pyramidilleNonuniformbsnub rectified cubic (serch)sr{4,3,4}(1)(3.3.3.3.3)(1)(3.3.3) (2)(3.3.3.3.4)(4)(3.3.3)Irr. tridiminished icosahedronNonuniformCantic snub cubic (casch)2s0{4,3,4}(1)(3.3.3.3.3)(2)(3.4.4.4)(3)(3.4.4)NonuniformRuncicantic snub cubic (rusch)(1)(3.4.3.4)(2)(4.4.4)(1)(3.3.3)(1)(3.6.6)(3)TricupNonuniformRuncic cantitruncated cubic (esch) sr3{4,3,4}(1)(3.3.3.3.4)(1)(4.4.4)(1)(4.4.4)(1)(3.4.4.4)(3)(3.4.4)
+ honeycombs, space group Imm (229)ReferenceIndicesHoneycomb nameCoxeter diagramand Schläfli symbolCell counts/vertexand positions in cubic honeycombSolids(Partial)Frames(Perspective)Vertex figureDual cell(0,3)(1,2)AltJ11,15A1W1G22δ4O1runcinated cubic(same as regular cubic) (chon)t0,3{4,3,4}(2)(4.4.4)(6)(4.4.4) octahedron CubeJ16A3W2G28t1,2δ4O16bitruncated cubic (batch) t1,2{4,3,4}2t{4,3,4}(4)(4.6.6) (disphenoid)Oblate tetrahedrilleJ19A22W18G27t0,1,2,3δ4O20omnitruncated cubic (gippich)t0,1,2,3{4,3,4}(2)(4.6.8)(2)(4.4.8) irregular tetrahedronEighth pyramidilleJ21,31,51A2W9G1hδ4O27Quarter cubic honeycomb (cytatoh)ht0ht3{4,3,4}(2)(3.3.3)(6)(3.6.6)elongated triangular antiprismOblate cubilleJ21,31,51A2W9G1hδ4O21Alternated runcinated cubic (octet)(same as alternated cubic)ht0,3{4,3,4}(2)(3.3.3)(6)(3.3.3)(6)(3.3.3.3)cuboctahedronNonuniformBiorthosnub cubic honeycomb (gabreth)2s0,3{(4,2,4,3)}(2)(4.6.6)(2)(4.4.4)(2)(4.4.6)NonuniformaAlternated bitruncated cubic (bisch)h2t{4,3,4} (4)(3.3.3.3.3) (4)(3.3.3)NonuniformCantic bisnub cubic (cabisch)2s0,3{4,3,4}(2)(3.4.4.4)(2)(4.4.4)(2)(4.4.4)NonuniformcAlternated omnitruncated cubic (snich)ht0,1,2,3{4,3,4}(2)(3.3.3.3.4)(2)(3.3.3.4)(4)(3.3.3)
B̃3, [4,31,1] group The , [4,3] group offers 11 derived forms via truncation operations, four being unique uniform honeycombs. There are 3 index 2 subgroups that generate alternations: [1+,4,31,1], [4,(31,1)+], and [4,31,1]+. The first generates repeated honeycomb, and the last two are nonuniform but included for completeness.
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.
+ [4,31,1] uniform honeycombs, space group Fmm (225)ReferencedindicesHoneycomb nameCoxeter diagramsCells by location(and count around each vertex)Solids(Partial)Frames(Perspective)vertex figure(0) (1) (0') (3)J21,31,51A2W9G1hδ4O21Alternated cubic (octet) ↔ (6)(3.3.3.3)(8)(3.3.3)cuboctahedronJ22,34A21W17G10h2δ4O25Cantic cubic (tatoh) ↔ (1)(3.4.3.4) (2)(4.6.6) (2)(3.6.6)rectangular pyramidJ23A16W11G5h3δ4O26Runcic cubic (sratoh) ↔ (1)cube (3)(3.4.4.4) (1)(3.3.3)tapered triangular prismJ24A20W16G21h2,3δ4O28Runcicantic cubic (gratoh) ↔ (1)(3.8.8) (2)(4.6.8) (1)(3.6.6)Irregular tetrahedron
+ <[4,31,1]> uniform honeycombs, space group Pmm (221)ReferencedindicesHoneycomb nameCoxeter diagrams ↔ Cells by location(and count around each vertex)Solids(Partial)Frames(Perspective)vertex figure(0,0') (1) (3)AltJ11,15A1W1G22δ4O1Cubic (chon) ↔ (8)(4.4.4) octahedronJ12,32A15W14G7t1δ4O15Rectified cubic (rich) ↔ (4)(3.4.3.4) (2)(3.3.3.3) cuboidRectified cubic (rich) ↔ (2)(3.3.3.3) (4)(3.4.3.4) cuboidJ13A14W15G8t0,1δ4O14Truncated cubic (tich) ↔ (4)(3.8.8) (1)(3.3.3.3) square pyramidJ14A17W12G9t0,2δ4O17Cantellated cubic (srich) ↔ (2)(3.4.4.4) (2)(4.4.4) (1)(3.4.3.4) obilique triangular prismJ16A3W2G28t0,2δ4O16Bitruncated cubic (batch) ↔ (2)(4.6.6) (2)(4.6.6) isosceles tetrahedronJ17A18W13G25t0,1,2δ4O18Cantitruncated cubic (grich) ↔ (2)(4.6.8) (1)(4.4.4)(1)(4.6.6) irregular tetrahedronJ21,31,51A2W9G1hδ4O21Alternated cubic (octet) ↔ (8)(3.3.3) (6)(3.3.3.3)cuboctahedronJ22,34A21W17G10h2δ4O25Cantic cubic (tatoh) ↔ (2)(3.6.6) (1)(3.4.3.4) (2)(4.6.6)rectangular pyramidNonuniformaAlternated bitruncated cubic (bisch) ↔ (2)(3.3.3.3.3) (2)(3.3.3.3.3) (4)(3.3.3)NonuniformbAlternated cantitruncated cubic (serch) ↔ (2)(3.3.3.3.4) (1)(3.3.3) (1)(3.3.3.3.3) (4)(3.3.3)Irr. tridiminished icosahedron
Ã3, [3] group There are 5 forms, A000029 6-1 cases, skipping one with zero marks constructed from the , [3] Coxeter group, of which only the quarter cubic honeycomb is unique. There is one index 2 subgroup [3]+ which generates the snub form, which is not uniform, but included for completeness.
+ uniform honeycombs, space group Fdm (227)ReferencedindicesHoneycomb nameCoxeter diagramsCells by location(and count around each vertex)Solids(Partial)Frames(Perspective)vertex figure(0,1)(2,3)J25,33A13W10G6qδ4O27quarter cubic (cytatoh) ↔ q{4,3,4} (2)(3.3.3) (6)(3.6.6) triangular antiprism