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 28.1
- 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).23
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 C ~ 3 {\displaystyle {\tilde {C}}_{3}} , [4,3,4], cubic, (8 unique forms plus one alternation)
- The B ~ 3 {\displaystyle {\tilde {B}}_{3}} , [4,31,1], alternated cubic, (11 forms, 3 new)
- The A ~ 3 {\displaystyle {\tilde {A}}_{3}} cyclic group, [(3,3,3,3)] or [3[4]], (5 forms, one new)
There is a correspondence between all three families. Removing one mirror from C ~ 3 {\displaystyle {\tilde {C}}_{3}} produces B ~ 3 {\displaystyle {\tilde {B}}_{3}} , and removing one mirror from B ~ 3 {\displaystyle {\tilde {B}}_{3}} produces A ~ 3 {\displaystyle {\tilde {A}}_{3}} . 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 C ~ 2 {\displaystyle {\tilde {C}}_{2}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} , [4,4,2,∞] prismatic group, (2 new forms)
- The G ~ 2 {\displaystyle {\tilde {G}}_{2}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} , [6,3,2,∞] prismatic group, (7 unique forms)
- The A ~ 2 {\displaystyle {\tilde {A}}_{2}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} , [(3,3,3),2,∞] prismatic group, (No new forms)
- The I ~ 1 {\displaystyle {\tilde {I}}_{1}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} , [∞,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.
| C3 honeycombs | |||||
|---|---|---|---|---|---|
| Spacegroup | Fibrifold | Extendedsymmetry | Extendeddiagram | Order | Honeycombs |
| Pm3m(221) | 4−:2 | [4,3,4] | ×1 | 1, 2, 3, 4, 5, 6 | |
| Fm3m(225) | 2−:2 | [1+,4,3,4]↔ [4,31,1] | ↔ | Half | 7, 11, 12, 13 |
| I43m(217) | 4o:2 | [[(4,3,4,2+)]] | Half × 2 | (7), | |
| Fd3m(227) | 2+:2 | [[1+,4,3,4,1+]]↔ [[3[4]]] | ↔ | Quarter × 2 | 10, |
| Im3m(229) | 8o:2 | [[4,3,4]] | ×2 | ||
| ReferenceIndices | Honeycomb nameCoxeter diagramand Schläfli symbol | Cell counts/vertexand positions in cubic honeycomb | Frames(Perspective) | Vertex figure | Dual cell | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| (0) | (1) | (2) | (3) | Alt | Solids(Partial) | |||||
| J11,15A1W1G22δ4 | cubic (chon) t0{4,3,4}{4,3,4} | (8)(4.4.4) | octahedron | Cube, | ||||||
| J12,32A15W14G7O1 | rectified cubic (rich) t1{4,3,4}r{4,3,4} | (2)(3.3.3.3) | (4)(3.4.3.4) | cuboid | Square bipyramid | |||||
| J13A14W15G8t1δ4O15 | truncated cubic (tich) t0,1{4,3,4}t{4,3,4} | (1)(3.3.3.3) | (4)(3.8.8) | square pyramid | Isosceles square pyramid | |||||
| J14A17W12G9t0,2δ4O14 | cantellated 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 prism | Triangular bipyramid | ||||
| J17A18W13G25t0,1,2δ4O17 | cantitruncated 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 tetrahedron | Triangular pyramidille | ||||
| J18A19W19G20t0,1,3δ4O19 | runcitruncated 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 | |||
| J21,31,51A2W9G1hδ4O21 | alternated cubic (octet)h{4,3,4} | (8)(3.3.3) | (6)(3.3.3.3) | cuboctahedron | Dodecahedrille | |||||
| J22,34A21W17G10h2δ4O25 | Cantic cubic (tatoh) ↔ | (1)(3.4.3.4) | (2)(3.6.6) | (2)(4.6.6) | rectangular pyramid | Half oblate octahedrille | ||||
| J23A16W11G5h3δ4O26 | Runcic cubic (sratoh) ↔ | (1)(4.4.4) | (1)(3.3.3) | (3)(3.4.4.4) | tapered triangular prism | Quarter cubille | ||||
| J24A20W16G21h2,3δ4O28 | Runcicantic cubic (gratoh) ↔ | (1)(3.8.8) | (1)(3.6.6) | (2)(4.6.8) | Irregular tetrahedron | Half pyramidille | ||||
| Nonuniformb | snub 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 icosahedron | ||||
| Nonuniform | Cantic snub cubic (casch)2s0{4,3,4} | (1)(3.3.3.3.3) | (2)(3.4.4.4) | (3)(3.4.4) | ||||||
| Nonuniform | Runcicantic snub cubic (rusch) | (1)(3.4.3.4) | (2)(4.4.4) | (1)(3.3.3) | (1)(3.6.6) | (3)Tricup | ||||
| Nonuniform | Runcic 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) | ||||
| ReferenceIndices | Honeycomb nameCoxeter diagramand Schläfli symbol | Cell counts/vertexand positions in cubic honeycomb | Solids(Partial) | Frames(Perspective) | Vertex figure | Dual cell | ||
|---|---|---|---|---|---|---|---|---|
| (0,3) | (1,2) | Alt | ||||||
| J11,15A1W1G22δ4O1 | runcinated cubic(same as regular cubic) (chon)t0,3{4,3,4} | (2)(4.4.4) | (6)(4.4.4) | octahedron | Cube | |||
| J16A3W2G28t1,2δ4O16 | bitruncated cubic (batch) t1,2{4,3,4}2t{4,3,4} | (4)(4.6.6) | (disphenoid) | Oblate tetrahedrille | ||||
| J19A22W18G27t0,1,2,3δ4O20 | omnitruncated cubic (gippich)t0,1,2,3{4,3,4} | (2)(4.6.8) | (2)(4.4.8) | irregular tetrahedron | Eighth pyramidille | |||
| J21,31,51A2W9G1hδ4O27 | Quarter cubic honeycomb (cytatoh)ht0ht3{4,3,4} | (2)(3.3.3) | (6)(3.6.6) | elongated triangular antiprism | Oblate cubille | |||
| J21,31,51A2W9G1hδ4O21 | Alternated 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) | cuboctahedron | |||
| Nonuniform | Biorthosnub cubic honeycomb (gabreth)2s0,3{(4,2,4,3)} | (2)(4.6.6) | (2)(4.4.4) | (2)(4.4.6) | ||||
| Nonuniforma | Alternated bitruncated cubic (bisch)h2t{4,3,4} | (4)(3.3.3.3.3) | (4)(3.3.3) | |||||
| Nonuniform | Cantic bisnub cubic (cabisch)2s0,3{4,3,4} | (2)(3.4.4.4) | (2)(4.4.4) | (2)(4.4.4) | ||||
| Nonuniformc | Alternated 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 B ~ 3 {\displaystyle {\tilde {B}}_{3}} , [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.
| B3 honeycombs | |||||
|---|---|---|---|---|---|
| Spacegroup | Fibrifold | Extendedsymmetry | Extendeddiagram | Order | Honeycombs |
| Fm3m(225) | 2−:2 | [4,31,1]↔ [4,3,4,1+] | ↔ | ×1 | 1, 2, 3, 4 |
| Fm3m(225) | 2−:2 | <[1+,4,31,1]>↔ <[3[4]]> | ↔ | ×2 | (1), (3) |
| Pm3m(221) | 4−:2 | <[4,31,1]> | ×2 | ||
| Referencedindices | Honeycomb nameCoxeter diagrams | Cells by location(and count around each vertex) | Solids(Partial) | Frames(Perspective) | vertex figure | |||
|---|---|---|---|---|---|---|---|---|
| (0) | (1) | (0') | (3) | |||||
| J21,31,51A2W9G1hδ4O21 | Alternated cubic (octet) ↔ | (6)(3.3.3.3) | (8)(3.3.3) | cuboctahedron | ||||
| J22,34A21W17G10h2δ4O25 | Cantic cubic (tatoh) ↔ | (1)(3.4.3.4) | (2)(4.6.6) | (2)(3.6.6) | rectangular pyramid | |||
| J23A16W11G5h3δ4O26 | Runcic cubic (sratoh) ↔ | (1)cube | (3)(3.4.4.4) | (1)(3.3.3) | tapered triangular prism | |||
| J24A20W16G21h2,3δ4O28 | Runcicantic cubic (gratoh) ↔ | (1)(3.8.8) | (2)(4.6.8) | (1)(3.6.6) | Irregular tetrahedron | |||
| Referencedindices | Honeycomb nameCoxeter diagrams ↔ | Cells by location(and count around each vertex) | Solids(Partial) | Frames(Perspective) | vertex figure | |||
|---|---|---|---|---|---|---|---|---|
| (0,0') | (1) | (3) | Alt | |||||
| J11,15A1W1G22δ4O1 | Cubic (chon) ↔ | (8)(4.4.4) | octahedron | |||||
| J12,32A15W14G7t1δ4O15 | Rectified cubic (rich) ↔ | (4)(3.4.3.4) | (2)(3.3.3.3) | cuboid | ||||
| Rectified cubic (rich) ↔ | (2)(3.3.3.3) | (4)(3.4.3.4) | cuboid | |||||
| J13A14W15G8t0,1δ4O14 | Truncated cubic (tich) ↔ | (4)(3.8.8) | (1)(3.3.3.3) | square pyramid | ||||
| J14A17W12G9t0,2δ4O17 | Cantellated cubic (srich) ↔ | (2)(3.4.4.4) | (2)(4.4.4) | (1)(3.4.3.4) | obilique triangular prism | |||
| J16A3W2G28t0,2δ4O16 | Bitruncated cubic (batch) ↔ | (2)(4.6.6) | (2)(4.6.6) | isosceles tetrahedron | ||||
| J17A18W13G25t0,1,2δ4O18 | Cantitruncated cubic (grich) ↔ | (2)(4.6.8) | (1)(4.4.4) | (1)(4.6.6) | irregular tetrahedron | |||
| J21,31,51A2W9G1hδ4O21 | Alternated cubic (octet) ↔ | (8)(3.3.3) | (6)(3.3.3.3) | cuboctahedron | ||||
| J22,34A21W17G10h2δ4O25 | Cantic cubic (tatoh) ↔ | (2)(3.6.6) | (1)(3.4.3.4) | (2)(4.6.6) | rectangular pyramid | |||
| Nonuniforma | Alternated bitruncated cubic (bisch) ↔ | (2)(3.3.3.3.3) | (2)(3.3.3.3.3) | (4)(3.3.3) | ||||
| Nonuniformb | Alternated 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[4]] group
There are 5 forms4 constructed from the A ~ 3 {\displaystyle {\tilde {A}}_{3}} , [3[4]] Coxeter group, of which only the quarter cubic honeycomb is unique. There is one index 2 subgroup [3[4]]+ which generates the snub form, which is not uniform, but included for completeness.
| A3 honeycombs | ||||||
|---|---|---|---|---|---|---|
| Spacegroup | Fibrifold | Square symmetry | Extendedsymmetry | Extendeddiagram | Extendedgroup | Honeycomb diagrams |
| F43m(216) | 1o:2 | a1 | [3[4]] | A ~ 3 {\displaystyle {\tilde {A}}_{3}} | (None) | |
| Fm3m(225) | 2−:2 | d2 | <[3[4]]>↔ [4,31,1] | ↔ | A ~ 3 {\displaystyle {\tilde {A}}_{3}} ×21↔ B ~ 3 {\displaystyle {\tilde {B}}_{3}} | 1, 2 |
| Fd3m(227) | 2+:2 | g2 | [[3[4]]] or [2+[3[4]]] | ↔ | A ~ 3 {\displaystyle {\tilde {A}}_{3}} ×22 | 3 |
| Pm3m(221) | 4−:2 | d4 | <2[3[4]]>↔ [4,3,4] | ↔ | A ~ 3 {\displaystyle {\tilde {A}}_{3}} ×41↔ C ~ 3 {\displaystyle {\tilde {C}}_{3}} | 4 |
| I3(204) | 8−o | r8 | [4[3[4]]]+↔ [[4,3+,4]] | ↔ | ½ A ~ 3 {\displaystyle {\tilde {A}}_{3}} ×8↔ ½ C ~ 3 {\displaystyle {\tilde {C}}_{3}} ×2 | (*) |
| Im3m(229) | 8o:2 | [4[3[4]]]↔ [[4,3,4]] | A ~ 3 {\displaystyle {\tilde {A}}_{3}} ×8↔ C ~ 3 {\displaystyle {\tilde {C}}_{3}} ×2 | 5 | ||
| Referencedindices | Honeycomb nameCoxeter diagrams | Cells by location(and count around each vertex) | Solids(Partial) | Frames(Perspective) | vertex figure | |
|---|---|---|---|---|---|---|
| (0,1) | (2,3) | |||||
| J25,33A13W10G6qδ4O27 | quarter cubic (cytatoh) ↔ q{4,3,4} | (2)(3.3.3) | (6)(3.6.6) | triangular antiprism | ||
| Referencedindices | Honeycomb nameCoxeter diagrams ↔ | Cells by location(and count around each vertex) | Solids(Partial) | Frames(Perspective) | vertex figure | ||
|---|---|---|---|---|---|---|---|
| 0 | (1,3) | 2 | |||||
| J21,31,51A2W9G1hδ4O21 | alternated cubic (octet) ↔ ↔ h{4,3,4} | (8)(3.3.3) | (6)(3.3.3.3) | cuboctahedron | |||
| J22,34A21W17G10h2δ4O25 | cantic cubic (tatoh) ↔ ↔ h2{4,3,4} | (2)(3.6.6) | (1)(3.4.3.4) | (2)(4.6.6) | Rectangular pyramid | ||
| Referencedindices | Honeycomb nameCoxeter diagrams ↔ | Cells by location(and count around each vertex) | Solids(Partial) | Frames(Perspective) | vertex figure | |
|---|---|---|---|---|---|---|
| (0,2) | (1,3) | |||||
| J12,32A15W14G7t1δ4O1 | rectified cubic (rich) ↔ ↔ ↔ r{4,3,4} | (2)(3.4.3.4) | (1)(3.3.3.3) | cuboid | ||
| Referencedindices | Honeycomb nameCoxeter diagrams ↔ ↔ | Cells by location(and count around each vertex) | Solids(Partial) | Frames(Perspective) | vertex figure | |
|---|---|---|---|---|---|---|
| (0,1,2,3) | Alt | |||||
| J16A3W2G28t1,2δ4O16 | bitruncated cubic (batch) ↔ ↔ 2t{4,3,4} | (4)(4.6.6) | isosceles tetrahedron | |||
| Nonuniforma | Alternated cantitruncated cubic (bisch) ↔ ↔ h2t{4,3,4} | (4)(3.3.3.3.3) | (4)(3.3.3) | |||
Nonwythoffian 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.
| Referencedindices | symbol | Honeycomb name | cell types (# at each vertex) | Solids(Partial) | Frames(Perspective) | vertex figure |
|---|---|---|---|---|---|---|
| J52A2'G2O22 | h{4,3,4}:g | gyrated alternated cubic (gytoh) | tetrahedron (8)octahedron (6) | triangular orthobicupola | ||
| J61A?G3O24 | h{4,3,4}:ge | gyroelongated alternated cubic (gyetoh) | triangular prism (6)tetrahedron (4)octahedron (3) | |||
| J62A?G4O23 | h{4,3,4}:e | elongated alternated cubic (etoh) | triangular prism (6)tetrahedron (4)octahedron (3) | |||
| J63A?G12O12 | {3,6}:g × {∞} | gyrated triangular prismatic (gytoph) | triangular prism (12) | |||
| J64A?G15O13 | {3,6}:ge × {∞} | gyroelongated triangular prismatic (gyetaph) | 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 C̃2×Ĩ1(∞), [4,4,2,∞], prismatic group
There are 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 | Coxeter-Dynkinand Schläflisymbols | Honeycomb name | Planetiling | Solids(Partial) | Tiling |
|---|---|---|---|---|---|
| J11,15A1G22 | {4,4}×{∞} | Cubic(Square prismatic) (chon) | (4.4.4.4) | ||
| r{4,4}×{∞} | |||||
| rr{4,4}×{∞} | |||||
| J45A6G24 | t{4,4}×{∞} | Truncated/Bitruncated square prismatic (tassiph) | (4.8.8) | ||
| tr{4,4}×{∞} | |||||
| J44A11G14 | sr{4,4}×{∞} | Snub square prismatic (sassiph) | (3.3.4.3.4) | ||
| Nonuniform | ht0,1,2,3{4,4,2,∞} |
The G̃2xĨ1(∞), [6,3,2,∞] prismatic group
| Indices | Coxeter-Dynkinand Schläflisymbols | Honeycomb name | Planetiling | Solids(Partial) | Tiling |
|---|---|---|---|---|---|
| J41A4G11 | {3,6} × {∞} | Triangular prismatic (tiph) | (36) | ||
| J42A5G26 | {6,3} × {∞} | Hexagonal prismatic (hiph) | (63) | ||
| t{3,6} × {∞} | |||||
| J43A8G18 | r{6,3} × {∞} | Trihexagonal prismatic (thiph) | (3.6.3.6) | ||
| J46A7G19 | t{6,3} × {∞} | Truncated hexagonal prismatic (thaph) | (3.12.12) | ||
| J47A9G16 | rr{6,3} × {∞} | Rhombi-trihexagonal prismatic (srothaph) | (3.4.6.4) | ||
| J48A12G17 | sr{6,3} × {∞} | Snub hexagonal prismatic (snathaph) | (3.3.3.3.6) | ||
| J49A10G23 | tr{6,3} × {∞} | truncated trihexagonal prismatic (grothaph) | (4.6.12) | ||
| J65A11'G13 | {3,6}:e × {∞} | elongated triangular prismatic (etoph) | (3.3.3.4.4) | ||
| J52A2'G2 | h3t{3,6,2,∞} | gyrated tetrahedral-octahedral (gytoh) | (36) | ||
| s2r{3,6,2,∞} | |||||
| Nonuniform | ht0,1,2,3{3,6,2,∞} |
Enumeration of Wythoff forms
All nonprismatic Wythoff constructions by Coxeter groups are given below, along with their alternations. Uniform solutions are indexed with Branko Grünbaum's listing. Green backgrounds are shown on repeated honeycombs, with the relations are expressed in the extended symmetry diagrams.
| Coxeter group | Extendedsymmetry | Honeycombs | Chiralextendedsymmetry | Alternation honeycombs | ||
|---|---|---|---|---|---|---|
| [4,3,4] | [4,3,4] | 6 | 22 | 7 | 89 | 25 | 20 | [1+,4,3+,4,1+] | (2) | 1 | b |
| [2+[4,3,4]] = | (1) | 22 | [2+[(4,3+,4,2+)]] | (1) | 1 | 6 | |
| [2+[4,3,4]] | 1 | 28 | [2+[(4,3+,4,2+)]] | (1) | a | |
| [2+[4,3,4]] | 2 | 27 | [2+[4,3,4]]+ | (1) | c | |
| [4,31,1] | [4,31,1] | 4 | 1 | 7 | 10 | 28 | |||
| [1[4,31,1]]=[4,3,4] = | (7) | 22 | 7 | 22 | 7 | 9 | 28 | 25 | [1[1+,4,31,1]]+ | (2) | 1 | 6 | a | |
| [1[4,31,1]]+=[4,3,4]+ | (1) | b | ||||
| [3[4]] | [3[4]] | (none) | ||||
| [2+[3[4]]] | 1 | 6 | ||||
| [1[3[4]]]=[4,31,1] = | (2) | 1 | 10 | ||||
| [2[3[4]]]=[4,3,4] = | (1) | 7 | ||||
| [(2+,4)[3[4]]]=[2+[4,3,4]] = | (1) | 28 | [(2+,4)[3[4]]]+= [2+[4,3,4]]+ | (1) | a | |
Examples
The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s). [3] [4] [5] [6]. Octet trusses are now among the most common types of truss used in construction.
Frieze forms
If cells are allowed to be uniform tilings, more uniform honeycombs can be defined:
Families:
- C ~ 2 {\displaystyle {\tilde {C}}_{2}} × A 1 {\displaystyle A_{1}} : [4,4,2] Cubic slab honeycombs (3 forms)
- G ~ 2 {\displaystyle {\tilde {G}}_{2}} × A 1 {\displaystyle A_{1}} : [6,3,2] Tri-hexagonal slab honeycombs (8 forms)
- A ~ 2 {\displaystyle {\tilde {A}}_{2}} × A 1 {\displaystyle A_{1}} : [(3,3,3),2] Triangular slab honeycombs (No new forms)
- I ~ 1 {\displaystyle {\tilde {I}}_{1}} × A 1 {\displaystyle A_{1}} × A 1 {\displaystyle A_{1}} : [∞,2,2] = Cubic column honeycombs (1 form)
- I 2 ( p ) {\displaystyle I_{2}(p)} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} : [p,2,∞] Polygonal column honeycombs (analogous to duoprisms: these look like a single infinite tower of p-gonal prisms, with the remaining space filled with apeirogonal prisms)
- I ~ 1 {\displaystyle {\tilde {I}}_{1}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} × A 1 {\displaystyle A_{1}} : [∞,2,∞,2] = [4,4,2] - = (Same as cubic slab honeycomb family)
| Cubic slab honeycomb | Alternated hexagonal slab honeycomb | Trihexagonal slab honeycomb |
|---|---|---|
| (4) 43: cube(1) 44: square tiling | (4) 33: tetrahedron(3) 34: octahedron(1) 36: triangular tiling | (2) 3.4.4: triangular prism(2) 4.4.6: hexagonal prism(1) (3.6)2: trihexagonal tiling |
The first two forms shown above are semiregular (uniform with only regular facets), and were listed by Thorold Gosset in 1900 respectively as the 3-ic semi-check and tetroctahedric semi-check.5
Scaliform honeycomb
A scaliform honeycomb is vertex-transitive, like a uniform honeycomb, with regular polygon faces while cells and higher elements are only required to be orbiforms, equilateral, with their vertices lying on hyperspheres. For 3D honeycombs, this allows a subset of Johnson solids along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, pyramid and cupola gaps.6
Euclidean honeycomb scaliforms| Frieze slabs | Prismatic stacks | ||
|---|---|---|---|
| s3{2,6,3}, | s3{2,4,4}, | s{2,4,4}, | 3s4{4,4,2,∞}, |
| (1) 3.4.3.4: triangular cupola(2) 3.4.6: triangular cupola(1) 3.3.3.3: octahedron(1) 3.6.3.6: trihexagonal tiling | (1) 3.4.4.4: square cupola(2) 3.4.8: square cupola(1) 3.3.3: tetrahedron(1) 4.8.8: truncated square tiling | (1) 3.3.3.3: square pyramid(4) 3.3.4: square pyramid(4) 3.3.3: tetrahedron(1) 4.4.4.4: square tiling | (1) 3.3.3.3: square pyramid(4) 3.3.4: square pyramid(4) 3.3.3: tetrahedron(4) 4.4.4: cube |
Hyperbolic forms
Main article: Uniform honeycombs in hyperbolic space
There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin 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
- [5,31,1] : - 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
Several non-Wythoffian forms outside the list of 76 are known; it is not known how many there are.
Paracompact hyperbolic forms
Main article: Paracompact uniform honeycombs
There are also 23 paracompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity:
Simplectic hyperbolic paracompact group summary| Type | Coxeter groups | Unique honeycomb count |
|---|---|---|
| Linear graphs | | | | | | | | 4×15+6+8+8 = 82 |
| Tridental graphs | | | | 4+4+0 = 8 |
| Cyclic graphs | | | | | | | | | | 4×9+5+1+4+1+0 = 47 |
| Loop-n-tail graphs | | | | | 4+4+4+2 = 14 |
- John 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)
- Branko Grünbaum, (1994) Uniform tilings of 3-space. 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 0-486-23729-X. (Chapter 5: Polyhedra packing and space filling)
- Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1.
- 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 [7]
- (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, (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. PDF [8]
- 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
- Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 5. Joining polyhedra
- Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi (2009), p. 54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry
External links
Wikimedia Commons has media related to Uniform tilings of Euclidean 3-space.- Weisstein, Eric W. "Honeycomb". MathWorld.
- Uniform Honeycombs in 3-Space VRML models
- Elementary Honeycombs Vertex transitive space filling honeycombs with non-uniform cells.
- Uniform partitions of 3-space, their relatives and embedding, 1999
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- octet truss animation
- Review: A. F. Wells, Three-dimensional nets and polyhedra, H. S. M. Coxeter (Source: Bull. Amer. Math. Soc. Volume 84, Number 3 (1978), 466-470.)
- Klitzing, Richard. "3D Euclidean tesselations".
- (sequence A242941 in the OEIS)
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} | C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} | B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} | D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} | G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}} |
| E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
| En−1 | Uniform (n−1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |
References
Sloane, N. J. A. (ed.). "Sequence A242941 (Convex uniform tessellations in dimension n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩
George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) [1] http://bendwavy.org/4HONEYS.pdf ↩
Sloane, N. J. A. (ed.). "Sequence A242941 (Convex uniform tessellations in dimension n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩
[2], A000029 6-1 cases, skipping one with zero marks http://mathworld.wolfram.com/Necklace.html ↩
Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48. /wiki/Thorold_Gosset ↩
"Polytope-tree". http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform ↩