Convex uniform honeycomb

{{Short description|Spatial tiling of convex uniform polyhedra}} {{More footnotes needed|date=August 2025}} [[File:Tetrahedral-octahedral honeycomb.svg|320px|thumb|The ''alternated cubic honeycomb'' is one of 28 space-filling uniform tessellations in Euclidean 3-space, composed of alternating yellow tetrahedra and red octahedra.]] 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.<ref name=OEIS/> * '''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).<ref>George Olshevsky, (2006, ''Uniform Panoploid Tetracombs'', Manuscript ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' [http://bendwavy.org/4HONEYS.pdf]</ref><ref name=OEIS>{{Cite OEIS|A242941|Convex uniform tessellations in dimension ''n''}}</ref>

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 '''A'''ndreini (1-22), '''W'''illiams(1–2,9-19), '''J'''ohnson (11–19, 21–25, 31–34, 41–49, 51–52, 61–65), and '''G'''rünbaum(1-28). Coxeter uses &delta;₄ for a cubic honeycomb, h&delta;₄ for an alternated cubic honeycomb, q&delta;₄ 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) == 380px|thumb|Fundamental domains in a cubic element of three groups. 380px|thumb|Family correspondences The fundamental infinite Coxeter groups for 3-space are: # The <math>{\tilde{C}}_3</math>, [4,3,4], cubic, {{CDD|node|4|node|3|node|4|node}} (8 unique forms plus one alternation) # The <math>{\tilde{B}}_3</math>, [4,3^1,1], alternated cubic, {{CDD|nodes|split2|node|4|node}} (11 forms, 3 new) # The <math>{\tilde{A}}_3</math> cyclic group, [(3,3,3,3)] or [3^[4]], {{CDD|branch|3ab|branch}} (5 forms, one new)

There is a correspondence between all three families. Removing one mirror from <math>{\tilde{C}}_3</math> produces <math>{\tilde{B}}_3</math>, and removing one mirror from <math>{\tilde{B}}_3</math> produces <math>{\tilde{A}}_3</math>. 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 <math>{\tilde{C}}_2</math>×<math>{\tilde{I}}_1</math>, [4,4,2,∞] prismatic group, {{CDD|node|4|node|4|node|2|node|infin|node}} (2 new forms) # The <math>{\tilde{G}}_2</math>×<math>{\tilde{I}}_1</math>, [6,3,2,∞] prismatic group, {{CDD|node|6|node|3|node|2|node|infin|node}} (7 unique forms) # The <math>{\tilde{A}}_2</math>×<math>{\tilde{I}}_1</math>, [(3,3,3),2,∞] prismatic group, {{CDD|node|split1|branch|2|node|infin|node}} (No new forms) # The <math>{\tilde{I}}_1</math>×<math>{\tilde{I}}_1</math>×<math>{\tilde{I}}_1</math>, [∞,2,∞,2,∞] prismatic group, {{CDD|node|infin|node|2|node|infin|node|2|node|infin|node}} (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}}

{{Clear|right}} {|class="wikitable" style="text-align:center;" |+ [4,3,4], space group Pm{{overline|3}}m (221) !rowspan=2|Reference<br/>Indices !rowspan=2|Honeycomb name<br/>Coxeter diagram<br/>and Schläfli symbol ! colspan=6|Cell counts/vertex<br/>and positions in cubic honeycomb<br/> !rowspan=2|Frames<br/>(Perspective) !rowspan=2|Vertex figure !rowspan=2|Dual cell |- !(0)<br/>{{CDD|node|3|node|4|node}} !(1)<br/>{{CDD|node|2|node|4|node}} !(2)<br/>{{CDD|node|4|node|2|node}} !(3)<br/>{{CDD|node|4|node|3|node}} !Alt !Solids<br/>(Partial) |- |J_11,15<br/>A₁<br/>W₁<br/>G₂₂<br/>&delta;₄ |cubic (chon)<br/>{{CDD|node_1|4|node|3|node|4|node}} <br/>t₀{4,3,4}<br/>{4,3,4} |&nbsp; |&nbsp; |&nbsp; |(8)<br/>30px<br/>(4.4.4) |&nbsp; | 75px |75px |75px<br/>octahedron | 80px<br/>Cube, {{CDD|node_f1|3|node|4|node}} |- |J_12,32<br/>A₁₅<br/>W₁₄<br/>G₇<br/>O₁ |rectified cubic (rich)<br/>{{CDD|node|4|node_1|3|node|4|node}} <br/>t₁{4,3,4}<br/>r{4,3,4} |(2)<br/>30px<br/>(3.3.3.3) |&nbsp; |&nbsp; |(4)<br/>30px<br/>(3.4.3.4) |&nbsp; |75px |75px |75px<br/>cuboid |80px<BR/>Square bipyramid<BR/>{{CDD|node_f1|2|node_f1|4|node}} |- |J₁₃<br/>A₁₄<br/>W₁₅<br/>G₈<br/>t₁&delta;₄<br/>O₁₅ |truncated cubic (tich)<br/>{{CDD|node_1|4|node_1|3|node|4|node}} <br/>t_0,1{4,3,4}<br/>t{4,3,4} |(1)<br/>30px<br/>(3.3.3.3) |&nbsp; |&nbsp; |(4)<br/>30px<br/>(3.8.8) |&nbsp; |75px |75px |75px<br/>square pyramid |80px<BR/>Isosceles square pyramid |- |J₁₄<br/>A₁₇<br/>W₁₂<br/>G₉<br/>t_0,2&delta;₄<br/>O₁₄ |cantellated cubic (srich)<br/>{{CDD|node_1|4|node|3|node_1|4|node}} <br/>t_0,2{4,3,4}<br/>rr{4,3,4} |(1)<br/>30px<br/>(3.4.3.4) |(2)<br/>30px<br/>(4.4.4) |&nbsp; |(2)<br/>30px<br/>(3.4.4.4) |&nbsp; |75px |75px |75px<br/>oblique triangular prism |80px<BR/>Triangular bipyramid |- |J₁₇<br/>A₁₈<br/>W₁₃<br/>G₂₅<br/>t_0,1,2&delta;₄<br/>O₁₇ |cantitruncated cubic (grich)<br/>{{CDD|node_1|4|node_1|3|node_1|4|node}} <br/>t_0,1,2{4,3,4}<br/>tr{4,3,4} |(1)<br/>30px<br/>(4.6.6) |(1)<br/>30px<br/>(4.4.4) |&nbsp; |(2)<br/>30px<br/>(4.6.8) |&nbsp; |75px |75px |75px<br/>irregular tetrahedron |80px<BR/>Triangular pyramidille |- |J₁₈<br/>A₁₉<br/>W₁₉<br/>G₂₀<br/>t_0,1,3&delta;₄<br/>O₁₉ |runcitruncated cubic (prich)<br/>{{CDD|node_1|4|node_1|3|node|4|node_1}}<br/>t_0,1,3{4,3,4} |(1)<br/>30px<br/>(3.4.4.4) |(1)<br/>30px<br/>(4.4.4) |(2)<br/>30px<br/>(4.4.8) |(1)<br/>30px<br/>(3.8.8) |&nbsp; |75px |75px |75px<br/>oblique trapezoidal pyramid |80px <BR/>Square quarter pyramidille |- valign=top BGCOLOR="#d0f0f0" |J_21,31,51<br/>A₂<br/>W₉<br/>G₁<br/>h&delta;₄<br/>O₂₁ |alternated cubic (octet)<br/>{{CDD|node_h1|4|node|3|node|4|node}}<br/>h{4,3,4} |&nbsp; |&nbsp; |&nbsp; |(8)<br/>30px<br/>(3.3.3) |(6)<br/>30px<br/>(3.3.3.3) |76px |75px |75px<br/>cuboctahedron |80px<BR/>Dodecahedrille |- valign=top BGCOLOR="#d0f0f0" |J_22,34<br/>A₂₁<br/>W₁₇<br/>G₁₀<br/>h₂&delta;₄<br/>O₂₅ |Cantic cubic (tatoh)<br/>{{CDD|node_h1|4|node|3|node_1|4|node}} ↔ {{CDD|nodes_10ru|split2|node_1|4|node}} |(1)<br/>30px(3.4.3.4) |&nbsp; | |(2)<br/>30px(3.6.6) |(2)<br/>30px(4.6.6) |75px |75px |60px<br/>rectangular pyramid |80px<BR/>Half oblate octahedrille |- valign=top BGCOLOR="#d0f0f0" |J₂₃<br/>A₁₆<br/>W₁₁<br/>G₅<br/>h₃&delta;₄<br/>O₂₆ |Runcic cubic (sratoh)<br/>{{CDD|node_h1|4|node|3|node|4|node_1}} ↔ {{CDD|nodes_10ru|split2|node|4|node_1}} |(1)<br>30px<br/>(4.4.4) |&nbsp; | |(1)<br>30px<br/>(3.3.3) |(3)<br>30px<br/>(3.4.4.4) |75px |75px |60px<br/>tapered triangular prism |80px<BR/>Quarter cubille |- valign=top BGCOLOR="#d0f0f0" |J₂₄<br/>A₂₀<br/>W₁₆<br/>G₂₁<br/>h_2,3&delta;₄<br/>O₂₈ |Runcicantic cubic (gratoh)<br/>{{CDD|node_h1|4|node|3|node_1|4|node_1}} ↔ {{CDD|nodes_10ru|split2|node_1|4|node_1}} |(1)<br>30px<br/>(3.8.8) |&nbsp; | |(1)<br>30px<br/>(3.6.6) |(2)<br>30px<br/>(4.6.8) |75px |75px |60px<br/>Irregular tetrahedron |80px<BR/>Half pyramidille |- valign=top BGCOLOR="#d0f0f0" |Nonuniform_b |snub rectified cubic (serch)<br/>{{CDD|node_h|4|node_h|3|node_h|4|node}}<br/>sr{4,3,4} |(1)<br>30px<br/>(3.3.3.3.3)<br/>{{CDD|node_h|3|node_h|4|node}} |(1)<br>30px<br/>(3.3.3)<br/>{{CDD|node_h|2|node_h|4|node}} |&nbsp; |(2)<br>30px<br/>(3.3.3.3.4)<br/>{{CDD|node_h|4|node_h|3|node_h}} |(4)<br>30px<br/>(3.3.3) |75px | ||75px<br/>Irr. tridiminished icosahedron

|- valign=top BGCOLOR="#d0f0f0" |Nonuniform |Cantic snub cubic (casch)<br/>{{CDD|node_1|4|node_h|3|node_h|4|node}}<br/>2s₀{4,3,4} |(1)<br>30px<br/>(3.3.3.3.3)<br/>{{CDD|node_h|3|node_h|4|node}} | | |(2)<br>30px<br/>(3.4.4.4)<br/>{{CDD|node_h|3|node_h|4|node_1}} |(3)<br>30px<br>(3.4.4) | | | |- valign=top BGCOLOR="#d0f0f0" |Nonuniform |Runcicantic snub cubic (rusch)<br/>{{CDD|node_h|4|node_1|3|node|4|node_h}} |(1)<br>30px<br>(3.4.3.4) |(2)<br>30px<br>(4.4.4) |(1)<br>30px<br>(3.3.3) |(1)<br>30px<br>(3.6.6) |(3)<br>30px<br>Tricup | | | |- valign=top BGCOLOR="#d0f0f0" |Nonuniform |Runcic cantitruncated cubic (esch)<br/>{{CDD|node_h|4|node_h|3|node_h|4|node_1}} <br/>sr₃{4,3,4} |(1)<br>30px<br/>(3.3.3.3.4)<br/>{{CDD|node_h|4|node_h|3|node_h}} |(1)<br>30px<br/>(4.4.4)<br/>{{CDD|node_h|4|node_h|2|node_1}} |(1)<br>30px<br/>(4.4.4)<br/>{{CDD|node_h|2|node_h|4|node_1}} |(1)<br>30px<br/>(3.4.4.4)<br/>{{CDD|node_h|3|node_h|4|node_1}} |(3)<br>30px<br>(3.4.4) | | | |}

{|class="wikitable" |+ {{brackets|4,3,4}} honeycombs, space group Im{{overline|3}}m (229) |- !rowspan=2|Reference<br/>Indices !rowspan=2|Honeycomb name<br/>Coxeter diagram<br/>{{CDD|branch_c1|4a4b|nodeab_c2}}<br/>and Schläfli symbol !colspan=3|Cell counts/vertex<br/>and positions in cubic honeycomb<br/> !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|Vertex figure !rowspan=2|Dual cell |- !(0,3)<br/>{{CDD|node|3|node|4|node}}<br/>{{CDD|node|4|node|3|node}} !(1,2)<br/>{{CDD|node|2|node|4|node}}<br/>{{CDD|node|4|node|2|node}} !Alt |- BGCOLOR="#e0f0e0" |J_11,15<br/>A₁<br/>W₁<br/>G₂₂<br/>&delta;₄<br/>O₁ |'''runcinated cubic'''<br/>(same as regular cubic) (chon)<br/>{{CDD|branch|4a4b|nodes_11}}<br/>t_0,3{4,3,4} |(2)<br/>30px<br/>(4.4.4) |(6)<br/>30px<br/>(4.4.4) |&nbsp; | 75px |75px |75px<br/>octahedron | 80px<br/>Cube |- valign=top BGCOLOR="#e0f0e0" |J₁₆<br/>A₃<br/>W₂<br/>G₂₈<br/>t_1,2&delta;₄<br/>O₁₆ |bitruncated cubic (batch)<br/>{{CDD|branch_11|4a4b|nodes}} <br/>t_1,2{4,3,4}<br/>2t{4,3,4} |(4)<br/>30px<br/>(4.6.6) |&nbsp; |&nbsp; |75px |75px |75px<br/>(disphenoid) |80px<BR/>Oblate tetrahedrille |- valign=top BGCOLOR="#e0f0e0" |J₁₉<br/>A₂₂<br/>W₁₈<br/>G₂₇<br/>t_0,1,2,3&delta;₄<br/>O₂₀ |omnitruncated cubic (gippich)<br/>{{CDD|branch_11|4a4b|nodes_11}}<br/>t_0,1,2,3{4,3,4} |(2)<br/>30px<br/>(4.6.8) |(2)<br/>30px<br/>(4.4.8) |&nbsp; |75px |75px |75px<br/>irregular tetrahedron |80px<BR/>Eighth pyramidille |- valign=top BGCOLOR="#d0f0f0" |J_21,31,51<br/>A₂<br/>W₉<br/>G₁<br/>h&delta;₄<br/>O₂₇ |Quarter cubic honeycomb (cytatoh)<br/>{{CDD|branch|4a4b|nodes_h1h1}}<br/>ht₀ht₃{4,3,4} |(2)<br/>30px<br/>(3.3.3) |(6)<br/>30px<br/>(3.6.6) | |76px |75px |75px<br/>elongated triangular antiprism

|80px<BR/>Oblate cubille |- valign=top BGCOLOR="#d0f0f0" |J_21,31,51<br/>A₂<br/>W₉<br/>G₁<br/>h&delta;₄<br/>O₂₁ |Alternated runcinated cubic (octet)<br/>(same as alternated cubic)<br/>{{CDD|branch|4a4b|nodes_hh}}<br/>ht_0,3{4,3,4} |(2)<br/>30px<br/>(3.3.3) |(6)<br/>30px<br/>(3.3.3) |(6)<br/>30px<br/>(3.3.3.3) |76x76px |75px |75px<br/>cuboctahedron

|- valign=top BGCOLOR="#d0f0f0" |Nonuniform |Biorthosnub cubic honeycomb (gabreth)<br>{{CDD|branch_11|4a4b|nodes_hh}}<br/>2s_0,3{(4,2,4,3)} |(2)<br>30px<br>(4.6.6) |(2)<br>30px<br/>(4.4.4) |(2)<br>30px<br/>(4.4.6) | | | |- valign=top BGCOLOR="#d0f0f0" |Nonuniform_a |Alternated bitruncated cubic (bisch)<br/>{{CDD|branch_hh|4a4b|nodes}}<br/>h2t{4,3,4} |30px (4)<br/>(3.3.3.3.3) |&nbsp; |30px (4)<br/>(3.3.3) |75px | ||75px |80px |- valign=top BGCOLOR="#d0f0f0" |Nonuniform |Cantic bisnub cubic (cabisch)<br>{{CDD|branch_hh|4a4b|nodes_11}}<br/>2s_0,3{4,3,4} |(2)<br>30px<br/>(3.4.4.4) |(2)<br>30px<br/>(4.4.4) |(2)<br>30px<br/>(4.4.4) | | | |- valign=top BGCOLOR="#d0f0f0" |Nonuniform_c |Alternated omnitruncated cubic (snich)<br/>{{CDD|branch_hh|4a4b|nodes_hh}}<br/>ht_0,1,2,3{4,3,4} |(2)<br>30px<br/>(3.3.3.3.4) |(2)<br>30px<br/>(3.3.3.4) |(4)<br>30px<br/>(3.3.3) |&nbsp; | ||75px |}

=== B̃₃, [4,3^1,1] group === The <math>{\tilde{B}}_3</math>, [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⁺,4,3^1,1], [4,(3^1,1)⁺], and [4,3^1,1]⁺. 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}}

{|class="wikitable" |+ [4,3^1,1] uniform honeycombs, space group Fm{{overline|3}}m (225) |- !rowspan=2|Referenced<br/>indices !rowspan=2|Honeycomb name<br/>Coxeter diagrams !colspan=4|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|vertex figure |- !(0)<br/>{{CDD|nodea|3a|nodea|4a|nodea}} !(1)<br/>{{CDD|nodea|2|nodeb|2|nodea}} !(0')<br/>{{CDD|nodea|3a|nodea|4a|nodea}} !(3)<br/>{{CDD|nodea|3a|branch}} |- |J_21,31,51<br/>A₂<br/>W₉<br/>G₁<br/>h&delta;₄<br/>O₂₁ |Alternated cubic (octet)<br/>{{CDD|nodes_10ru|split2|node|4|node}} ↔ {{CDD|node_h1|4|node|3|node|4|node}} |&nbsp; |&nbsp; |30px (6)<br/>(3.3.3.3) |30px(8)<br/>(3.3.3) |76px |75px |60px<br/>cuboctahedron |- |J_22,34<br/>A₂₁<br/>W₁₇<br/>G₁₀<br/>h₂&delta;₄<br/>O₂₅ |Cantic cubic (tatoh)<br/>{{CDD|nodes_10ru|split2|node_1|4|node}} ↔ {{CDD|node_h1|4|node|3|node_1|3|node}} |30px (1)<br/>(3.4.3.4) |&nbsp; |30px (2)<br/>(4.6.6) |30px (2)<br/>(3.6.6) ||75px |75px |60px<br/>rectangular pyramid |- |J₂₃<br/>A₁₆<br/>W₁₁<br/>G₅<br/>h₃&delta;₄<br/>O₂₆ |Runcic cubic (sratoh)<br/>{{CDD|nodes_10ru|split2|node|4|node_1}} ↔ {{CDD|node_h1|4|node|3|node|4|node_1}} |30px (1)<br/>cube |&nbsp; |30px (3)<br/>(3.4.4.4) |30px (1)<br/>(3.3.3) |75px |75px |60px<br/>tapered triangular prism |- |J₂₄<br/>A₂₀<br/>W₁₆<br/>G₂₁<br/>h_2,3&delta;₄<br/>O₂₈ |Runcicantic cubic (gratoh)<br/>{{CDD|nodes_10ru|split2|node_1|4|node_1}} ↔ {{CDD|node_h1|4|node|3|node_1|3|node_1}} |30px (1)<br/>(3.8.8) |&nbsp; |30px(2)<br/>(4.6.8) |30px (1)<br/>(3.6.6) |75px |75px |60px<br/>Irregular tetrahedron |}

{|class="wikitable" |+ <[4,3^1,1]> uniform honeycombs, space group Pm{{overline|3}}m (221) |- !rowspan=2|Referenced<br/>indices !rowspan=2|Honeycomb name<br/>Coxeter diagrams<br/>{{CDD|nodeab_c1|split2|node_c2|4|node_c3}} ↔ {{CDD|node_h0|4|node_c1|3|node_c2|4|node_c3}} !colspan=4|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|vertex figure |- !(0,0')<br/>{{CDD|nodea|3a|nodea|4a|nodea}} !(1)<br/>{{CDD|nodea|2|nodeb|2|nodea}} !(3)<br/>{{CDD|nodea|3a|branch}} !Alt |- BGCOLOR="#e0f0e0" |J_11,15<br/>A₁<br/>W₁<br/>G₂₂<br/>&delta;₄<br/>O₁ |Cubic (chon)<br/>{{CDD|nodes|split2|node|4|node_1}} ↔ {{CDD|node_h0|4|node|3|node|3|node_1}} |30px (8)<br/>(4.4.4) |&nbsp; |&nbsp; |&nbsp; |75px |75px |60px<br/>octahedron

|- BGCOLOR="#e0f0e0" |rowspan=2|J_12,32<br/>A₁₅<br/>W₁₄<br/>G₇<br/>t₁&delta;₄<br/>O₁₅ |Rectified cubic (rich)<br/>{{CDD|nodes|split2|node_1|4|node}} ↔ {{CDD|node_h0|4|node|3|node_1|4|node}} |30px (4)<br/>(3.4.3.4) |&nbsp; |30px (2)<br/>(3.3.3.3) |&nbsp; |75px |rowspan=2|75px |60px<br/>cuboid |- BGCOLOR="#e0f0e0" |Rectified cubic (rich)<br/>{{CDD|nodes_11|split2|node|4|node}} ↔ {{CDD|node_h0|4|node_1|3|node|4|node}} |30px (2)<br/>(3.3.3.3) |&nbsp; |30px (4)<br/>(3.4.3.4) |&nbsp; |75px |60px<br/>cuboid |- BGCOLOR="#e0f0e0" |J₁₃<br/>A₁₄<br/>W₁₅<br/>G₈<br/>t_0,1&delta;₄<br/>O₁₄ |Truncated cubic (tich)<br/>{{CDD|nodes|split2|node_1|4|node_1}} ↔ {{CDD|node_h0|4|node|3|node_1|4|node_1}} |30px (4)<br/>(3.8.8) |&nbsp; |30px (1)<br/>(3.3.3.3) |&nbsp; |75px |75px |60px<br/>square pyramid |- BGCOLOR="#e0f0e0" |J₁₄<br/>A₁₇<br/>W₁₂<br/>G₉<br/>t_0,2&delta;₄<br/>O₁₇ |Cantellated cubic (srich)<br/>{{CDD|nodes_11|split2|node|4|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node|4|node_1}} |30px (2)<br/>(3.4.4.4) |30px (2)<br/>(4.4.4) |30px (1)<br/>(3.4.3.4) |&nbsp; |75px |75px |60px<br/>obilique triangular prism |- BGCOLOR="#e0f0e0" |J₁₆<br/>A₃<br/>W₂<br/>G₂₈<br/>t_0,2&delta;₄<br/>O₁₆ |Bitruncated cubic (batch)<br/>{{CDD|nodes_11|split2|node_1|4|node}} ↔ {{CDD|node_h0|4|node_1|3|node_1|4|node}} |30px (2)<br/>(4.6.6) |&nbsp; |30px (2)<br/>(4.6.6) |&nbsp; |75px |75px |60px<br/>isosceles tetrahedron |- BGCOLOR="#e0f0e0" |J₁₇<br/>A₁₈<br/>W₁₃<br/>G₂₅<br/>t_0,1,2&delta;₄<br/>O₁₈ |Cantitruncated cubic (grich)<br/>{{CDD|nodes_11|split2|node_1|4|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node_1|4|node_1}} |30px (2)<br/>(4.6.8) |30px (1)<br/>(4.4.4) |30px(1)<br/>(4.6.6) |&nbsp; |75px |75px |60px<br/>irregular tetrahedron |- BGCOLOR="#d0f0f0" |J_21,31,51<br/>A₂<br/>W₉<br/>G₁<br/>h&delta;₄<br/>O₂₁ |Alternated cubic (octet)<br/>{{CDD|node_h1|4|node|split1|nodes}} ↔ {{CDD|node_1|split1|nodes|split2|node}} |30px (8)<br/>(3.3.3) |&nbsp; |&nbsp; |30px (6)<br/>(3.3.3.3) |75px |75px |60px<br/>cuboctahedron

|- BGCOLOR="#d0f0f0" |J_22,34<br/>A₂₁<br/>W₁₇<br/>G₁₀<br/>h₂&delta;₄<br/>O₂₅ |Cantic cubic (tatoh)<br/>{{CDD|node_h1|4|node|split1|nodes_11}} ↔ {{CDD|node_1|split1|nodes_11|split2|node}} |30px (2)<br/>(3.6.6) |&nbsp; |30px (1)<br/>(3.4.3.4) |30px (2)<br/>(4.6.6) ||75px |75px |60px<br/>rectangular pyramid

|- BGCOLOR="#d0f0f0" |Nonuniform_a |Alternated bitruncated cubic (bisch)<br/>{{CDD|nodes_hh|split2|node_h|4|node}} ↔ {{CDD|node_h0|4|node_h|3|node_h|4|node}} |30px (2)<br/>(3.3.3.3.3) |&nbsp; |30px (2)<br/>(3.3.3.3.3) |30px (4)<br/>(3.3.3) | | |60px |- BGCOLOR="#d0f0f0" |Nonuniform_b |Alternated cantitruncated cubic (serch)<br/>{{CDD|nodes_hh|split2|node_h|4|node_h}} ↔ {{CDD|node_h0|4|node_h|3|node_h|4|node_h}} |30px (2)<br/>(3.3.3.3.4) |30px (1)<br/>(3.3.3) |30px (1)<br/>(3.3.3.3.3) |30px (4)<br/>(3.3.3) |75px | |60px<br/>Irr. tridiminished icosahedron |}

=== Ã₃, [3^[4]] group === There are 5 forms<ref>[http://mathworld.wolfram.com/Necklace.html], A000029 6-1 cases, skipping one with zero marks</ref> constructed from the <math>{\tilde{A}}_3</math>, [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}}

{|class="wikitable" |+ {{Brackets|3^{{{Bracket|4}}}}} uniform honeycombs, space group Fd{{overline|3}}m (227) |- !rowspan=2|Referenced<br/>indices !rowspan=2|Honeycomb name<br/>Coxeter diagrams<br/>{{CDD|branch_c1-2|3ab|branch_c1-2}} !colspan=2|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|vertex figure |- !(0,1)<br/>{{CDD|nodeb|3b|branch}} !(2,3)<br/>{{CDD|branch|3a|nodea}} |- |J_25,33<br/>A₁₃<br/>W₁₀<br/>G₆<br/>q&delta;₄<br/>O₂₇ |quarter cubic (cytatoh)<br/>{{CDD|branch_10r|3ab|branch_10l}} ↔ {{CDD|node_h1|4|node|3|node|4|node_h1}}<br/>q{4,3,4} |30px (2)<br/>(3.3.3) |30px (6)<br/>(3.6.6) |75px |75px |75px<br/>triangular antiprism |}

{|class="wikitable" |+ <[3^[4]]> ↔ [4,3^1,1] uniform honeycombs, space group Fm{{overline|3}}m (225) |- !rowspan=2|Referenced<br/>indices !rowspan=2|Honeycomb name<br/>Coxeter diagrams<br/>{{CDD|node_c3|split1|nodeab_c1-2|split2|node_c3}} ↔ {{CDD|node|3|node_c3|split1|nodeab_c1-2}} !colspan=3|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|vertex figure |- !0 !(1,3) !2 |- BGCOLOR="#e0f0e0" |J_21,31,51<br/>A₂<br/>W₉<br/>G₁<br/>h&delta;₄<br/>O₂₁ |alternated cubic (octet)<br/>{{CDD|node_1|split1|nodes|split2|node}} ↔ {{CDD|nodes_10ru|split2|node|4|node}} ↔ {{CDD|node_h1|4|node|3|node|4|node}}<br/>h{4,3,4} | |30px (8)<br/>(3.3.3) |30px (6)<br/>(3.3.3.3) |75px |75px |75px<br/>cuboctahedron |- BGCOLOR="#e0f0e0" |J_22,34<br/>A₂₁<br/>W₁₇<br/>G₁₀<br/>h₂&delta;₄<br/>O₂₅ |cantic cubic (tatoh)<br/>{{CDD|node_1|split1|nodes_11|split2|node}} ↔ {{CDD|nodes_10ru|split2|node_1|4|node}} ↔ {{CDD|node_h1|4|node|3|node_1|4|node}}<br/>h₂{4,3,4} |30px (2)<br/>(3.6.6) |30px (1)<br/>(3.4.3.4) |30px (2)<br/>(4.6.6) |75px |75px |75px<br/>Rectangular pyramid |}

{|class="wikitable" |+ [2[3^[4]<nowiki>]]</nowiki> ↔ [4,3,4] uniform honeycombs, space group Pm{{overline|3}}m (221) |- !rowspan=2|Referenced<br/>indices !rowspan=2|Honeycomb name<br/>Coxeter diagrams<br/>{{CDD|node_c1|split1|nodeab_c2|split2|node_c1}} ↔ {{CDD|node|4|node_c1|3|node_c2|4|node}} !colspan=2|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|vertex figure |- !(0,2)<br/>{{CDD|nodeb|3b|branch}} !(1,3)<br/>{{CDD|branch|3b|nodeb}} |- BGCOLOR="#a0f0a0" |J_12,32<br/>A₁₅<br/>W₁₄<br/>G₇<br/>t₁&delta;₄<br/>O₁ |rectified cubic (rich)<br/>{{CDD|node_1|split1|nodes|split2|node_1}} ↔ {{CDD|nodes|split2|node_1|4|node}} ↔ {{CDD|nodes_11|split2|node|4|node}} ↔ {{CDD|node|4|node|3|node_1|4|node}}<br/>r{4,3,4} |30px (2)<br/>(3.4.3.4) |30px (1)<br/>(3.3.3.3) |75px |75px |75px<br/>cuboid

|}

{|class="wikitable" |+ [4[3^[4]<nowiki>]]</nowiki> ↔ {{brackets|4,3,4}} uniform honeycombs, space group Im{{overline|3}}m (229) |- !rowspan=2|Referenced<br/>indices !rowspan=2|Honeycomb name<br/>Coxeter diagrams<br/>{{CDD|node_c1|split1|nodeab_c1|split2|node_c1}} ↔ {{CDD|nodeab_c1|split2|node_c1|4|node_h0}} ↔ {{CDD|node_h0|4|node_c1|3|node_c1|4|node_h0}} !colspan=2|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|vertex figure |- !(0,1,2,3)<br/>{{CDD|node|3|node|3|node}} !Alt |- BGCOLOR="#60f060" |J₁₆<br/>A₃<br/>W₂<br/>G₂₈<br/>t_1,2&delta;₄<br/>O₁₆ |bitruncated cubic (batch)<br/>{{CDD|node_1|split1|nodes_11|split2|node_1}} ↔ {{CDD|nodes_11|split2|node_1|4|node_h0}} ↔ {{CDD|node_h0|4|node_1|3|node_1|4|node_h0}}<br/>2t{4,3,4} |30px (4)<br/>(4.6.6) | |75px |75px |75px<br/>isosceles tetrahedron |- BGCOLOR="#d0f0f0" |Nonuniform_a |Alternated cantitruncated cubic (bisch)<br/>{{CDD|node_h|split1|nodes_hh|split2|node_h}} ↔ {{CDD|nodes_hh|split2|node_h|4|node_h0}} ↔ {{CDD|node_h0|4|node_h|3|node_h|4|node_h0}}<br/>h2t{4,3,4} |30px (4)<br/>(3.3.3.3.3) |30px (4)<br/>(3.3.3) |&nbsp; | |75px

|}

=== 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.

{|class="wikitable" !Referenced<br/>indices !symbol !Honeycomb name !cell types (# at each vertex) !Solids<br/>(Partial) !Frames<br/>(Perspective) !vertex figure |- |J₅₂<br/>A_2'<br/>G₂<br/>O₂₂ |h{4,3,4}:g |align=center|gyrated alternated cubic (gytoh) |align=center|tetrahedron (8)<br/>octahedron (6) |70px |100px |80px<br/> triangular orthobicupola |- |J₆₁<br/>A_?<br/>G₃<br/>O₂₄ |h{4,3,4}:ge |align=center|gyroelongated alternated cubic (gyetoh) |align=center|triangular prism (6)<br/>tetrahedron (4)<br/>octahedron (3) |70px |100px |rowspan=2|80px |- |J₆₂<br/>A_?<br/>G₄<br/>O₂₃ |h{4,3,4}:e |align=center|elongated alternated cubic (etoh) |align=center|triangular prism (6)<br/>tetrahedron (4)<br/>octahedron (3) |70px |80px |- |J₆₃<br/>A_?<br/>G₁₂<br/>O₁₂ |{3,6}:g × {∞} |align=center|gyrated triangular prismatic (gytoph) |align=center|triangular prism (12) |70px |100px |80px |- |J₆₄<br/>A_?<br/>G₁₅<br/>O₁₃ |{3,6}:ge × {∞} |align=center|gyroelongated triangular prismatic (gyetaph) |align=center|triangular prism (6)<br/>cube (4) |70px |100px |80px |}

=== 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(&infin;), [4,4,2,&infin;], 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.

{|class="wikitable" !Indices !Coxeter-Dynkin<br/>and Schläfli<br/>symbols !Honeycomb name !Plane<br/>tiling !Solids<br/>(Partial) !Tiling |- |rowspan=3|J_11,15<br/>A₁<br/>G₂₂ |align=center|{{CDD|node_1|4|node|4|node|2|node_1|infin|node}} <br/>{4,4}×{∞} |rowspan=3 align=center|Cubic<br/>(Square prismatic) (chon) |rowspan=3|(4.4.4.4) |rowspan=3|80px |50px |- BGCOLOR="#e0f0e0" |align=center|{{CDD|node|4|node_1|4|node|2|node_1|infin|node}} <br/>r{4,4}×{∞} |50px |- BGCOLOR="#e0f0e0" |align=center|{{CDD|node_1|4|node|4|node_1|2|node_1|infin|node}} <br/>rr{4,4}×{∞} |50px |- |rowspan=2|J₄₅<br/>A₆<br/>G₂₄ |align=center|{{CDD|node_1|4|node_1|4|node|2|node_1|infin|node}} <br/>t{4,4}×{∞} |rowspan=2 align=center|Truncated/Bitruncated square prismatic (tassiph) |rowspan=2|(4.8.8) |rowspan=2|80px |50px |- BGCOLOR="#e0f0e0" |align=center|{{CDD|node_1|4|node_1|4|node_1|2|node_1|infin|node}} <br/>tr{4,4}×{∞} |50px |- BGCOLOR="#d0f0f0" |J₄₄<br/>A₁₁<br/>G₁₄ |align=center|{{CDD|node_h|4|node_h|4|node_h|2|node_1|infin|node}} <br/>sr{4,4}×{∞} |align=center|Snub square prismatic (sassiph) |(3.3.4.3.4) |80px |50px |- BGCOLOR="#d0f0f0" |Nonuniform |align=center|{{CDD|node_h|4|node_h|4|node_h|2x|node_h|infin|node}}<br/>ht_0,1,2,3{4,4,2,∞} | | | | |}

==== The G̃₂xĨ₁(&infin;), [6,3,2,&infin;] prismatic group ==== {|class="wikitable" !Indices !Coxeter-Dynkin<br/>and Schläfli<br/>symbols !Honeycomb name !Plane<br/>tiling !Solids<br/>(Partial) !Tiling |- |J₄₁<br/>A₄<br/>G₁₁ |{{CDD|node|6|node|3|node_1|2|node_1|infin|node}} <br/>{3,6} × {∞} |Triangular prismatic (tiph) |(3⁶) |60px |60px |- |rowspan=2|J₄₂<br/>A₅<br/>G₂₆ |{{CDD|node_1|6|node|3|node|2|node_1|infin|node}} <br/>{6,3} × {∞} |rowspan=2 align=center|Hexagonal prismatic (hiph) |rowspan=2|(6³) |60px |60px |- |{{CDD|node|6|node_1|3|node_1|2|node_1|infin|node}} <br/>t{3,6} × {∞} ||60px |60px |- |J₄₃<br/>A₈<br/>G₁₈ |{{CDD|node|6|node_1|3|node|2|node_1|infin|node}} <br/>r{6,3} × {∞} |Trihexagonal prismatic (thiph) |(3.6.3.6) |60px |60px |- |J₄₆<br/>A₇<br/>G₁₉ |{{CDD|node_1|6|node_1|3|node|2|node_1|infin|node}} <br/>t{6,3} × {∞} |Truncated hexagonal prismatic (thaph) |(3.12.12) |60px |60px |- |J₄₇<br/>A₉<br/>G₁₆ |{{CDD|node_1|6|node|3|node_1|2|node_1|infin|node}} <br/>rr{6,3} × {∞} |Rhombi-trihexagonal prismatic (srothaph) |(3.4.6.4) |60px |60px |- BGCOLOR="#d0f0f0" |J₄₈<br/>A₁₂<br/>G₁₇ |{{CDD|node_h|6|node_h|3|node_h|2|node_1|infin|node}} <br/>sr{6,3} × {∞} |Snub hexagonal prismatic (snathaph) |(3.3.3.3.6) |60px |60px |- |J₄₉<br/>A₁₀<br/>G₂₃ |{{CDD|node_1|6|node_1|3|node_1|2|node_1|infin|node}} <br/>tr{6,3} × {∞} |truncated trihexagonal prismatic (grothaph) |(4.6.12) |60px |60px |- BGCOLOR="#d0f0f0" |J₆₅<br/>A_11'<br/>G₁₃ |{{CDD|node|infin|node_h|2x|node_h|infin|node_1|2|node_1|infin|node}} <br/>{3,6}:e × {∞} |elongated triangular prismatic (etoph) |(3.3.3.4.4) |60px |60px |- BGCOLOR="#d0f0f0" |rowspan=2|J₅₂<br/>A_2'<br/>G₂ |{{CDD|node|3|node|6|node_h|2x|node_h|infin|node}}<br/>h3t{3,6,2,∞} |rowspan=2|gyrated tetrahedral-octahedral (gytoh) |rowspan=2|(3⁶) |rowspan=2|60px |rowspan=2|60px |- BGCOLOR="#d0f0f0" |{{CDD|node|6|node_h|3|node_h|2x|node_h|infin|node}}<br/>s2r{3,6,2,∞} |- BGCOLOR="#d0f0f0" |Nonuniform |{{CDD|node_h|3|node_h|6|node_h|2x|node_h|infin|node}}<br/>ht_0,1,2,3{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. {| class=wikitable style="text-align:center;" !Coxeter group !Extended<br/>symmetry !colspan=2|Honeycombs !Chiral<br/>extended<br/>symmetry !colspan=2|Alternation honeycombs

|- |rowspan=4|[4,3,4]<br/>{{CDD|node|4|node|3|node|4|node}}||[4,3,4]<br/>{{CDD|node_c1|4|node_c2|3|node_c3|4|node_c4}}||6 | {{CDD|node_1|4|node|3|node|4|node}}₂₂ | {{CDD|node|4|node_1|3|node|4|node}}₇ | {{CDD|node_1|4|node_1|3|node|4|node}}₈<br/>{{CDD|node_1|4|node|3|node_1|4|node}}₉ | {{CDD|node_1|4|node_1|3|node_1|4|node}}₂₅ | {{CDD|node_1|4|node_1|3|node|4|node_1}}₂₀ |[1⁺,4,3⁺,4,1⁺]||(2) |{{CDD|node_h1|4|node|3|node|4|node}}₁ | {{CDD|node_h|4|node_h|3|node_h|4|node}}_b |- BGCOLOR="#e0f0e0" |[2⁺[4,3,4<nowiki>]]</nowiki><br/>{{CDD|node_c1|4|node|3|node|4|node_c1}} = {{CDD|node_c1|4|node|3|node|4|node}}||(1) |{{CDD|node_1|4|node|3|node|4|node_1}} ₂₂ |[2⁺[(4,3⁺,4,2⁺)<nowiki>]]</nowiki>||(1) |{{CDD|branch|4a4b|branch_hh|label2}}₁ | {{CDD|branch|4a4b|nodes_hh}}₆

|- |[2⁺[4,3,4<nowiki>]]</nowiki><br/>{{CDD|node_c1|4|node_c2|3|node_c2|4|node_c1}}||1 |{{CDD|node|4|node_1|3|node_1|4|node}}₂₈ |[2⁺[(4,3⁺,4,2⁺)<nowiki>]]</nowiki>||(1) |{{CDD|node|4|node_h|3|node_h|4|node}}_a

|- |[2⁺[4,3,4<nowiki>]]</nowiki><br/>{{CDD|node_c1|4|node_c2|3|node_c2|4|node_c1}}||2 |{{CDD|node_1|4|node_1|3|node_1|4|node_1}}₂₇

|[2⁺[4,3,4<nowiki>]]</nowiki>⁺||(1) |{{CDD|node_h|4|node_h|3|node_h|4|node_h}}_c

|- |rowspan=3|[4,3^1,1]<br/>{{CDD|node|4|node|split1|nodes}} ||[4,3^1,1]<br/>{{CDD|node_c3|4|node_c4|split1|nodeab_c1-2}}||4 |{{CDD|node|4|node|split1|nodes_10lu}}₁ | {{CDD|node_1|4|node|split1|nodes_10lu}}₇ | {{CDD|node|4|node_1|split1|nodes_10lu}}₁₀ | {{CDD|node_1|4|node_1|split1|nodes_10lu}}₂₈ |colspan=3| |- BGCOLOR="#e0f0e0" align=center |rowspan=2|[1[4,3^1,1<nowiki>]]</nowiki>=[4,3,4]<br/>{{CDD|node_c1|4|node_c2|split1|nodeab_c3}} = {{CDD|node_c1|4|node_c2|3|node_c3|4|node_h0}}||rowspan=2|(7) |rowspan=2|{{CDD|node_1|4|node|split1|nodes}}₂₂ | {{CDD|node|4|node_1|split1|nodes}}₇ | {{CDD|node_1|4|node_1|split1|nodes}}₂₂ | {{CDD|node|4|node|split1|nodes_11}}₇ | {{CDD|node_1|4|node|split1|nodes_11}}₉ | {{CDD|node|4|node_1|split1|nodes_11}}₂₈ | {{CDD|node_1|4|node_1|split1|nodes_11}}₂₅ |[1[1⁺,4,3^1,1<nowiki>]]</nowiki>⁺||(2) |{{CDD|node_h1|4|node|split1|nodes}}₁ | {{CDD|node_h1|4|node|split1|nodes_10lu}}₆ | {{CDD|node|4|node_h|split1|nodes_hh}}_a |- BGCOLOR="#e0f0e0" align=center |[1[4,3^1,1<nowiki>]]</nowiki>⁺<br/>=[4,3,4]⁺||(1) |{{CDD|node_h|4|node_h|split1|nodes_hh}}_b

|- |rowspan=5|[3^[4]]<br/>{{CDD|branch|3ab|branch}}||[3^[4]] |colspan=5|(none) |- ||[2⁺[3^[4]<nowiki>]]</nowiki><br/>{{CDD|branch_c1|3ab|branch_c2}} || 1 | {{CDD|branch_11|3ab|branch}}₆ |colspan=3| |- BGCOLOR="#e0f0e0" align=center ||[1[3^[4]<nowiki>]]</nowiki>=[4,3^1,1]<br/>{{CDD|node_c3|split1|nodeab_c1-2|split2|node_c3}} = {{CDD|node_h0|3|node_c3|split1|nodeab_c1-2}} || (2) |{{CDD|node_1|split1|nodes|split2|node}}₁ | {{CDD|node_1|split1|nodes_11|split2|node}}₁₀ |colspan=3| |- BGCOLOR="#e0f0e0" ||[2[3^[4]<nowiki>]]</nowiki>=[4,3,4]<br/>{{CDD|node_c1|split1|nodeab_c2|split2|node_c1}} = {{CDD|node_h0|4|node_c1|3|node_c2|4|node_h0}} || (1) | {{CDD|node_1|split1|nodes|split2|node_1}}₇ |colspan=3| |- BGCOLOR="#e0f0e0" align=center |[(2⁺,4)[3^[4]<nowiki>]]</nowiki>=[2⁺[4,3,4<nowiki>]]</nowiki><br/>{{CDD|branch_c1|3ab|branch_c1}} = {{CDD|node_h0|4|node_c1|3|node_c1|4|node_h0}} ||(1) | {{CDD|branch_11|3ab|branch_11}}₂₈ |[(2⁺,4)[3^[4]<nowiki>]]</nowiki>⁺<br/>= [2⁺[4,3,4<nowiki>]]</nowiki>⁺ |(1)|| {{CDD|branch_hh|3ab|branch_hh}}_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). [http://tabletoptelephone.com/~hopspage/Fuller.html] [http://members.cruzio.com/~devarco/energy.htm] [https://web.archive.org/web/20050113123708/http://www.n55.dk/manuals/DISCUSSIONS/OTHER_TEXTS/CM_TEXT.html] [http://www.cjfearnley.com/fuller-faq-2.html]. Octet trusses are now among the most common types of truss used in construction. <!-- FIXME: move this discussion of octet truss to Buckminster Fuller or perhaps octet truss, leaving behind a link to where it went. -->

== Frieze forms == If cells are allowed to be uniform tilings, more uniform honeycombs can be defined:

Families: *<math>{\tilde{C}}_2</math>×<math>A_1</math>: [4,4,2] {{CDD|node|4|node|4|node|2|node}} ''Cubic slab honeycombs'' (3 forms) *<math>{\tilde{G}}_2</math>×<math>A_1</math>: [6,3,2] {{CDD|node|6|node|3|node|2|node}} ''Tri-hexagonal slab honeycombs'' (8 forms) * <math>{\tilde{A}}_2</math>×<math>A_1</math>: [(3,3,3),2] {{CDD|node|split1|branch|2|node}} ''Triangular slab honeycombs'' (No new forms) *<math>{\tilde{I}}_1</math>×<math>A_1</math>×<math>A_1</math>: [∞,2,2] {{CDD|node|infin|node|2|node|2|node}} = {{CDD|node|infin|node|2|node|4|node}} ''Cubic column honeycombs'' (1 form) *<math>I_2(p)</math>×<math>{\tilde{I}}_1</math>: [p,2,∞] {{CDD|node|p|node|2|node|infin|node}} ''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) * <math>{\tilde{I}}_1</math>×<math>{\tilde{I}}_1</math>×<math>A_1</math>: [∞,2,∞,2] = [4,4,2] - {{CDD|node|infin|node|2|node|infin|node|2|node}} = {{CDD|node|4|node|4|node|2|node}} (Same as cubic slab honeycomb family)

{| class=wikitable style="text-align:center; width:540px;" |+ Examples (partially drawn) |- !Cubic slab honeycomb<br/>{{CDD|node_1|4|node|4|node|2|node_1}} !Alternated hexagonal slab honeycomb<br/>{{CDD|node_h|2x|node_h|6|node|3|node}} !Trihexagonal slab honeycomb<br/>{{CDD|node|6|node_1|3|node|2|node_1}} |- |180px |180px |180px |- |180px<br/>(4) 4³: cube<br/>(1) 4⁴: square tiling |180px<br/>(4) 3³: tetrahedron<br/>(3) 3⁴: octahedron<br/>(1) 3⁶: triangular tiling |180px<br/>(2) 3.4.4: triangular prism<br/>(2) 4.4.6: hexagonal prism<br/>(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''.<ref>{{cite journal | last=Gosset | first=Thorold | authorlink = Thorold Gosset | title = On the regular and semi-regular figures in space of ''n'' dimensions | journal = Messenger of Mathematics | volume = 29 | pages = 43&ndash;48 | year = 1900}}</ref>

== 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.<ref>{{Cite web|url=http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform|title = Polytope-tree}}</ref>

{| class=wikitable style="text-align:center; width:600px;" |+ Euclidean honeycomb scaliforms !colspan=3|Frieze slabs !Prismatic stacks |- !s₃{2,6,3}, {{CDD|node_h|2x|node_h|6|node|3|node_1}} !s₃{2,4,4}, {{CDD|node_h|2x|node_h|4|node|4|node_1}} !s{2,4,4}, {{CDD|node_h|2x|node_h|4|node|4|node}} !3s₄{4,4,2,&infin;}, {{CDD|node|4|node|4|node_h|2x|node_h|infin|node_1}} |- |200px |200px |200px |200px |- ! 40px 40px 40px ! 40px 40px 40px ! 40px 40px 40px ! 40px 40px 40px |- valign=top |200px<br/>(1) 3.4.3.4: triangular cupola<br/>(2) 3.4.6: triangular cupola<br/>(1) 3.3.3.3: octahedron<br/>(1) 3.6.3.6: trihexagonal tiling |200px<br/>(1) 3.4.4.4: square cupola<br/>(2) 3.4.8: square cupola<br/>(1) 3.3.3: tetrahedron<br/>(1) 4.8.8: truncated square tiling |200px<br/>(1) 3.3.3.3: square pyramid<br/>(4) 3.3.4: square pyramid<br/>(4) 3.3.3: tetrahedron<br/>(1) 4.4.4.4: square tiling |200px<br/>(1) 3.3.3.3: square pyramid<br/>(4) 3.3.4: square pyramid<br/>(4) 3.3.3: tetrahedron<br/>(4) 4.4.4: cube |}

== Hyperbolic forms == [[File:Hyperbolic orthogonal dodecahedral honeycomb.png|thumb|The order-4 dodecahedral honeycomb, {5,3,4} in perspective]] [[File:Hyperbolic 3d hexagonal tiling.png|thumb|The paracompact hexagonal tiling honeycomb, {6,3,3}, in perspective]] {{main|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] : {{CDD|node|3|node|5|node|3|node}} - 9 forms * [5,3,4] : {{CDD|node|5|node|3|node|4|node}} - 15 forms * [5,3,5] : {{CDD|node|5|node|3|node|5|node}} - 9 forms * [5,3^1,1] : {{CDD|nodes|split2|node|5|node}} - 11 forms (7 overlap with [5,3,4] family, 4 are unique) * [(4,3,3,3)] : {{CDD|label4|branch|3ab|branch}} - 9 forms * [(4,3,4,3)] : {{CDD|label4|branch|3ab|branch|label4}} - 6 forms * [(5,3,3,3)] : {{CDD|label5|branch|3ab|branch}} - 9 forms * [(5,3,4,3)] : {{CDD|label5|branch|3ab|branch|label4}} - 9 forms * [(5,3,5,3)] : {{CDD|label5|branch|3ab|branch|label5}} - 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|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: {| class=wikitable style="text-align:center;" |+ Simplectic hyperbolic paracompact group summary !Type !Coxeter groups !Unique honeycomb count |- !Linear graphs |{{CDD|node|6|node|3|node|3|node}} | {{CDD|node|4|node|4|node|3|node}} | {{CDD|node|6|node|3|node|4|node}} | {{CDD|node|6|node|3|node|5|node}} | {{CDD|node|4|node|4|node|4|node}} | {{CDD|node|3|node|6|node|3|node}} | {{CDD|node|6|node|3|node|6|node}} |4×15+6+8+8 = 82 |- !Tridental graphs | {{CDD|node|3|node|split1-44|nodes}} | {{CDD|node|6|node|split1|nodes}} | {{CDD|node|4|node|split1-44|nodes}} |4+4+0 = 8 |- !Cyclic graphs | {{CDD|label6|branch|3ab|branch|2}} | {{CDD|label6|branch|3ab|branch|label4}} | {{CDD|label4|branch|4-4|branch}} | {{CDD|label6|branch|3ab|branch|label5}} | {{CDD|label6|branch|3ab|branch|label6}} | {{CDD|label4|branch|4-4|branch|label4}} | {{CDD|node|split1-44|nodes|split2|node}} | {{CDD|node|split1|branch|split2|node}} | {{CDD|branch|splitcross|branch}} |4×9+5+1+4+1+0 = 47 |- !Loop-n-tail graphs |{{CDD|node|3|node|split1|branch}} | {{CDD|node|4|node|split1|branch}} | {{CDD|node|5|node|split1|branch}} | {{CDD|node|6|node|split1|branch}} |4+4+4+2 = 14 |}

== References == <references/> * 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 * {{The Geometrical Foundation of Natural Structure (book)}} (Chapter 5: Polyhedra packing and space filling) * {{cite book | first=Keith | last=Critchlow | author-link=Keith Critchlow | title=Order in Space: A design source book | publisher=Viking Press| year=1970 | 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] ** (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 [https://web.archive.org/web/20140429195143/http://media.accademiaxl.it/memorie/Serie3_T14.pdf] * 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 * {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7 }} Chapter 5. Joining polyhedra * [https://books.google.com/books?id=nVx-tu596twC&q=space-filling+packings&pg=PA54 Crystallography of Quasicrystals: Concepts, Methods and Structures] by Walter Steurer, Sofia Deloudi (2009), p.&nbsp;54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry

==External links== {{Commons category|Uniform tilings of Euclidean 3-space}} * {{mathworld | title = Honeycomb | urlname = Honeycomb}} *[http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space] VRML models *[http://www.orchidpalms.com/polyhedra/honeycombs/honeycombs.htm Elementary Honeycombs] Vertex transitive space filling honeycombs with non-uniform cells. * [https://arxiv.org/abs/math/9906034 Uniform partitions of 3-space, their relatives and embedding], 1999 *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra] *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra *[http://octettruss.kilu.de/ octet truss animation] *[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183540634 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.)] * {{KlitzingPolytopes|flat.htm|3D|Euclidean tesselations}} * {{OEIS|A242941}}

{{Honeycombs}}

{{DEFAULTSORT:Convex Uniform Honeycomb}} Category:Honeycombs (geometry)