View source for Convex uniform honeycomb

{{Short description|Spatial tiling of convex uniform polyhedra}}
[[File:Tetrahedral-octahedral honeycomb.png|320px|thumb|The ''alternated cubic honeycomb'' is one of 28 space-filling uniform tessellations in Euclidean 3-space, composed of alternating yellow [[tetrahedron|tetrahedra]] and red [[octahedron|octahedra]].]]
In [[geometry]], a '''convex uniform honeycomb''' is a [[uniform polytope|uniform]] [[tessellation]] which fills three-dimensional [[Euclidean space]] with non-overlapping [[convex polyhedron|convex]] [[uniform polyhedron|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 [[#Prismatic_stacks|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 [[List of uniform planar tilings|uniform tilings of the plane]].

The [[Voronoi diagram]] of any [[Lattice (group)|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 solid]]s) 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 (mathematician)|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 [[Convex uniform honeycomb#Nonwythoffian forms (gyrated and elongated)|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-polytope]]s 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 solid]]s (the [[tetrahedron]], [[cube]], and [[octahedron]]),
* six of the thirteen [[Archimedean solid]]s (the ones with reflective tetrahedral or octahedral symmetry), and
* five of the infinite family of [[prism (geometry)|prism]]s (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 solid]]s. Recently [[John Horton Conway|Conway]] has suggested naming the set as the '''Architectonic tessellations''' and the dual honeycombs as the '''[[Catoptric tessellation]]s'''.

The individual honeycombs are listed with names given to them by [[Norman Johnson (mathematician)|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;<sub>4</sub> for a [[cubic honeycomb]], h&delta;<sub>4</sub> for an [[alternated cubic honeycomb]], q&delta;<sub>4</sub> 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) ==
[[File:Coxeter-Dynkin 3-space groups.png|380px|thumb|Fundamental domains in a cubic element of three groups.]]
[[File:Coxeter diagram affine rank4 correspondence.png|380px|thumb|Family correspondences]]
The fundamental infinite [[Coxeter group]]s 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<sup>1,1</sup>], 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<sup>[4]</sup>], {{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 C̃<sub>3</sub>, [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<sup>+</sup>,4,3,4], [(4,3,4,2<sup>+</sup>)], [4,3<sup>+</sup>,4], and [4,3,4]<sup>+</sup>, 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#Extended for uniform 4-polytopes and 3-space honeycombs|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<sub>11,15</sub><br/>A<sub>1</sub><br/>W<sub>1</sub><br/>G<sub>22</sub><br/>&delta;<sub>4</sub>
|[[cubic honeycomb|cubic]] (chon)<br/>{{CDD|node_1|4|node|3|node|4|node}} <br/>t<sub>0</sub>{4,3,4}<br/>{4,3,4}
|&nbsp;
|&nbsp;
|&nbsp;
|(8)<br/>[[File:hexahedron.png|30px]]<br/>[[cube|(4.4.4)]]
|&nbsp;
| [[File:Partial cubic honeycomb.png|75px]]
|[[File:Cubic honeycomb.png|75px]]
|[[File:Cubic honeycomb verf.svg|75px]]<br/>[[octahedron]]
| [[File:Cubic full domain.png|80px]]<br/>[[Cube]], {{CDD|node_f1|3|node|4|node}}
|- 
|J<sub>12,32</sub><br/>A<sub>15</sub><br/>W<sub>14</sub><br/>G<sub>7</sub><br/>O<sub>1</sub>
|[[rectified cubic honeycomb|rectified cubic]] (rich)<br/>{{CDD|node|4|node_1|3|node|4|node}} <br/>t<sub>1</sub>{4,3,4}<br/>r{4,3,4}
|(2)<br/>[[File:octahedron.png|30px]]<br/>[[octahedron|(3.3.3.3)]]
|&nbsp;
|&nbsp;
|(4)<br/>[[File:cuboctahedron.png|30px]]<br/>[[cuboctahedron|(3.4.3.4)]]
|&nbsp;
|[[File:Rectified cubic honeycomb.png|75px]]
|[[File:Rectified cubic tiling.png|75px]]
|[[File:Rectified cubic honeycomb verf.png|75px]]<br/>[[cuboid]]
|[[File:Cubic square bipyramid.png|80px]]<BR/>[[Square bipyramid]]<BR/>{{CDD|node_f1|2|node_f1|4|node}}
|- 
|J<sub>13</sub><br/>A<sub>14</sub><br/>W<sub>15</sub><br/>G<sub>8</sub><br/>t<sub>1</sub>&delta;<sub>4</sub><br/>O<sub>15</sub>
|[[truncated cubic honeycomb|truncated cubic]] (tich)<br/>{{CDD|node_1|4|node_1|3|node|4|node}} <br/>t<sub>0,1</sub>{4,3,4}<br/>t{4,3,4}
|(1)<br/>[[File:octahedron.png|30px]]<br/>[[octahedron|(3.3.3.3)]]
|&nbsp;
|&nbsp;
|(4)<br/>[[File:truncated hexahedron.png|30px]]<br/>[[truncated cube|(3.8.8)]]
|&nbsp;
|[[File:Truncated cubic honeycomb.png|75px]]
|[[File:Truncated cubic tiling.png|75px]]
|[[File:Truncated cubic honeycomb verf.png|75px]]<br/>[[square pyramid]]
|[[File:Cubic square pyramid.png|80px]]<BR/>Isosceles [[square pyramid]]
|- 
|J<sub>14</sub><br/>A<sub>17</sub><br/>W<sub>12</sub><br/>G<sub>9</sub><br/>t<sub>0,2</sub>&delta;<sub>4</sub><br/>O<sub>14</sub>
|[[Cantellated cubic honeycomb|cantellated cubic]] (srich)<br/>{{CDD|node_1|4|node|3|node_1|4|node}} <br/>t<sub>0,2</sub>{4,3,4}<br/>rr{4,3,4}
|(1)<br/>[[File:cuboctahedron.png|30px]]<br/>[[cuboctahedron|(3.4.3.4)]]
|(2)<br/>[[File:hexahedron.png|30px]]<br/>[[cube|(4.4.4)]]
|&nbsp;
|(2)<br/>[[File:small rhombicuboctahedron.png|30px]]<br/>[[rhombicuboctahedron|(3.4.4.4)]]
|&nbsp;
|[[File:Cantellated cubic honeycomb.jpg|75px]]
|[[File:Cantellated cubic tiling.png|75px]]
|[[File:Cantellated cubic honeycomb verf.png|75px]]<br/>oblique [[triangular prism]]
|[[File:Quarter oblate octahedrille cell.png|80px]]<BR/>[[Triangular bipyramid]]
|- 
|J<sub>17</sub><br/>A<sub>18</sub><br/>W<sub>13</sub><br/>G<sub>25</sub><br/>t<sub>0,1,2</sub>&delta;<sub>4</sub><br/>O<sub>17</sub>
|[[cantitruncated cubic honeycomb|cantitruncated cubic]] (grich)<br/>{{CDD|node_1|4|node_1|3|node_1|4|node}} <br/>t<sub>0,1,2</sub>{4,3,4}<br/>tr{4,3,4}
|(1)<br/>[[File:truncated octahedron.png|30px]]<br/>[[truncated octahedron|(4.6.6)]]
|(1)<br/>[[File:hexahedron.png|30px]]<br/>[[cube|(4.4.4)]]
|&nbsp;
|(2)<br/>[[File:Great rhombicuboctahedron.png|30px]]<br/>[[Truncated cuboctahedron|(4.6.8)]]
|&nbsp;
|[[File:Cantitruncated Cubic Honeycomb.svg|75px]]
|[[File:Cantitruncated cubic tiling.png|75px]]
|[[File:Cantitruncated cubic honeycomb verf.png|75px]]<br/>irregular [[tetrahedron]]
|[[File:Triangular pyramidille cell1.png|80px]]<BR/>[[Triangular pyramidille]]
|- 
|J<sub>18</sub><br/>A<sub>19</sub><br/>W<sub>19</sub><br/>G<sub>20</sub><br/>t<sub>0,1,3</sub>&delta;<sub>4</sub><br/>O<sub>19</sub>
|[[runcitruncated cubic honeycomb|runcitruncated cubic]] (prich)<br/>{{CDD|node_1|4|node_1|3|node|4|node_1}}<br/>t<sub>0,1,3</sub>{4,3,4}
|(1)<br/>[[File:small rhombicuboctahedron.png|30px]]<br/>[[rhombicuboctahedron|(3.4.4.4)]]
|(1)<br/>[[File:hexahedron.png|30px]]<br/>[[cube|(4.4.4)]]
|(2)<br/>[[File:octagonal prism.png|30px]]<br/>[[octagonal prism|(4.4.8)]]
|(1)<br/>[[File:truncated hexahedron.png|30px]]<br/>[[truncated cube|(3.8.8)]]
|&nbsp;
|[[File:Runcitruncated cubic honeycomb.jpg|75px]]
|[[File:Runcitruncated cubic tiling.png|75px]]
|[[File:Runcitruncated cubic honeycomb verf.png|75px]]<br/>oblique trapezoidal pyramid
|[[File:Square quarter pyramidille cell.png|80px]] <BR/>[[Square quarter pyramidille]] 
|- valign=top BGCOLOR="#d0f0f0"
|J<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>h&delta;<sub>4</sub><br/>O<sub>21</sub>
|[[Tetrahedral-octahedral honeycomb|alternated cubic]] (octet)<br/>{{CDD|node_h1|4|node|3|node|4|node}}<br/>h{4,3,4}
|&nbsp;
|&nbsp;
|&nbsp;
|(8)<br/>[[File:Tetrahedron.png|30px]]<br/>[[Tetrahedron|(3.3.3)]]
|(6)<br/>[[File:Octahedron.png|30px]]<br/>[[Octahedron|(3.3.3.3)]]
|[[File:Tetrahedral-octahedral honeycomb.png|76px]]
|[[File:Alternated cubic tiling.png|75px]]
|[[File:Alternated cubic honeycomb verf.svg|75px]]<br/>[[cuboctahedron]]
|[[File:Dodecahedrille cell.png|80px]]<BR/>[[Dodecahedrille]]
|- valign=top BGCOLOR="#d0f0f0"
|J<sub>22,34</sub><br/>A<sub>21</sub><br/>W<sub>17</sub><br/>G<sub>10</sub><br/>h<sub>2</sub>&delta;<sub>4</sub><br/>O<sub>25</sub>
|[[Cantic cubic honeycomb|Cantic cubic]] (tatoh)<br/>{{CDD|node_h1|4|node|3|node_1|4|node}} ↔ {{CDD|nodes_10ru|split2|node_1|4|node}}
|(1)<br/>[[File:Cuboctahedron.png|30px]][[cuboctahedron|(3.4.3.4)]]
|&nbsp;
|
|(2)<br/>[[File:Truncated tetrahedron.png|30px]][[Truncated tetrahedron|(3.6.6)]]
|(2)<br/>[[File:Truncated octahedron.png|30px]][[truncated octahedron|(4.6.6)]]
|[[File:Truncated Alternated Cubic Honeycomb.svg|75px]]
|[[File:Truncated alternated cubic tiling.png|75px]]
|[[File:Truncated alternated cubic honeycomb verf.png|60px]]<br/>rectangular pyramid
|[[File:Half oblate octahedrille cell.png|80px]]<BR/>[[Half oblate octahedrille]]
|- valign=top BGCOLOR="#d0f0f0"
|J<sub>23</sub><br/>A<sub>16</sub><br/>W<sub>11</sub><br/>G<sub>5</sub><br/>h<sub>3</sub>&delta;<sub>4</sub><br/>O<sub>26</sub>
|[[Runcic cubic honeycomb|Runcic cubic]] (sratoh)<br/>{{CDD|node_h1|4|node|3|node|4|node_1}} ↔ {{CDD|nodes_10ru|split2|node|4|node_1}}
|(1)<br>[[File:hexahedron.png|30px]]<br/>[[cube|(4.4.4)]]
|&nbsp;
|
|(1)<br>[[File:tetrahedron.png|30px]]<br/>[[tetrahedron|(3.3.3)]]
|(3)<br>[[File:small rhombicuboctahedron.png|30px]]<br/>[[rhombicuboctahedron|(3.4.4.4)]]
|[[File:Runcinated alternated cubic honeycomb.jpg|75px]]
|[[File:Runcinated alternated cubic tiling.png|75px]]
|[[File:Runcinated alternated cubic honeycomb verf.png|60px]]<br/>tapered [[triangular prism]]
|[[File:Quarter cubille cell.png|80px]]<BR/>[[Quarter cubille]]
|- valign=top BGCOLOR="#d0f0f0"
|J<sub>24</sub><br/>A<sub>20</sub><br/>W<sub>16</sub><br/>G<sub>21</sub><br/>h<sub>2,3</sub>&delta;<sub>4</sub><br/>O<sub>28</sub>
|[[Runcicantic cubic honeycomb|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>[[File:truncated hexahedron.png|30px]]<br/>[[truncated cube|(3.8.8)]]
|&nbsp;
|
|(1)<br>[[File:truncated tetrahedron.png|30px]]<br/>[[truncated tetrahedron|(3.6.6)]]
|(2)<br>[[File:Great rhombicuboctahedron.png|30px]]<br/>[[truncated cuboctahedron|(4.6.8)]]
|[[File:Cantitruncated alternated cubic honeycomb.png|75px]]
|[[File:Cantitruncated alternated cubic tiling.png|75px]]
|[[File:Runcitruncated alternate cubic honeycomb verf.png|60px]]<br/>Irregular [[tetrahedron]]
|[[File:Half pyramidille cell.png|80px]]<BR/>[[Half pyramidille]]
|- valign=top BGCOLOR="#d0f0f0"
|Nonuniform<sub>b</sub>
|[[snub rectified cubic honeycomb|snub rectified cubic]] (serch)<br/>{{CDD|node_h|4|node_h|3|node_h|4|node}}<br/>sr{4,3,4}
|(1)<br>[[File:Uniform polyhedron-43-h01.svg|30px]]<br/>[[Regular icosahedron|(3.3.3.3.3)]]<br/>{{CDD|node_h|3|node_h|4|node}}
|(1)<br>[[File:tetrahedron.png|30px]]<br/>[[tetrahedron|(3.3.3)]]<br/>{{CDD|node_h|2|node_h|4|node}}
|&nbsp;
|(2)<br>[[File:snub hexahedron.png|30px]]<br/>[[snub cube|(3.3.3.3.4)]]<br/>{{CDD|node_h|4|node_h|3|node_h}}
|(4)<br>[[File:tetrahedron.png|30px]]<br/>[[tetrahedron|(3.3.3)]]
|[[File:Alternated cantitruncated cubic honeycomb.png|75px]]
|
||[[File:Alternated cantitruncated cubic honeycomb verf.png|75px]]<br/>Irr. [[tridiminished icosahedron]]

|- valign=top BGCOLOR="#d0f0f0"
|Nonuniform
|[[Cantic snub cubic honeycomb|Cantic snub cubic]] (casch)<br/>{{CDD|node_1|4|node_h|3|node_h|4|node}}<br/>2s<sub>0</sub>{4,3,4}
|(1)<br>[[File:Uniform polyhedron-43-h01.svg|30px]]<br/>[[icosahedron#snub octahedron|(3.3.3.3.3)]]<br/>{{CDD|node_h|3|node_h|4|node}}
|
|
|(2)<br>[[File:Rhombicuboctahedron uniform edge coloring.png|30px]]<br/>[[rhombicuboctahedron|(3.4.4.4)]]<br/>{{CDD|node_h|3|node_h|4|node_1}}
|(3)<br>[[File:Triangular prism.png|30px]]<br>[[triangular prism|(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>[[File:Cuboctahedron.png|30px]]<br>[[cuboctahedron|(3.4.3.4)]]
|(2)<br>[[File:Cube rotorotational symmetry.png|30px]]<br>[[cube|(4.4.4)]]
|(1)<br>[[File:Tetrahedron.png|30px]]<br>[[tetrahedron|(3.3.3)]]
|(1)<br>[[File:Truncated tetrahedron.png|30px]]<br>[[truncated tetrahedron|(3.6.6)]]
|(3)<br>[[File:Triangular cupola.png|30px]]<br>[[triangular cupola|Tricup]]
|
|
|
|- valign=top BGCOLOR="#d0f0f0"
|Nonuniform
|[[Runcic cantitruncated cubic honeycomb|Runcic cantitruncated cubic]] (esch)<br/>{{CDD|node_h|4|node_h|3|node_h|4|node_1}} <br/>sr<sub>3</sub>{4,3,4}
|(1)<br>[[File:snub hexahedron.png|30px]]<br/>[[snub cube|(3.3.3.3.4)]]<br/>{{CDD|node_h|4|node_h|3|node_h}}
|(1)<br>[[File:Tetragonal prism.png|30px]]<br/>(4.4.4)<br/>{{CDD|node_h|4|node_h|2|node_1}}
|(1)<br>[[File:Cube rotorotational symmetry.png|30px]]<br/>[[cube|(4.4.4)]]<br/>{{CDD|node_h|2|node_h|4|node_1}}
|(1)<br>[[File:Rhombicuboctahedron uniform edge coloring.png|30px]]<br/>[[rhombicuboctahedron|(3.4.4.4)]]<br/>{{CDD|node_h|3|node_h|4|node_1}}
|(3)<br>[[File:Triangular prism.png|30px]]<br>[[triangular prism|(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#Extended for uniform 4-polytopes and 3-space honeycombs|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<sub>11,15</sub><br/>A<sub>1</sub><br/>W<sub>1</sub><br/>G<sub>22</sub><br/>&delta;<sub>4</sub><br/>O<sub>1</sub>
|'''[[Runcination (geometry)|runcinated]] cubic'''<br/>(same as regular [[cubic honeycomb|cubic]]) (chon)<br/>{{CDD|branch|4a4b|nodes_11}}<br/>t<sub>0,3</sub>{4,3,4}
|(2)<br/>[[File:hexahedron.png|30px]]<br/>[[cube|(4.4.4)]]
|(6)<br/>[[File:hexahedron.png|30px]]<br/>[[cube|(4.4.4)]]
|&nbsp;
| [[File:Runcinated cubic honeycomb.png|75px]]
|[[File:Cubic honeycomb.png|75px]]
|[[File:Runcinated cubic honeycomb verf.png|75px]]<br/>[[octahedron]]
| [[File:Cubic full domain.png|80px]]<br/>[[Cube]]
|- valign=top BGCOLOR="#e0f0e0"
|J<sub>16</sub><br/>A<sub>3</sub><br/>W<sub>2</sub><br/>G<sub>28</sub><br/>t<sub>1,2</sub>&delta;<sub>4</sub><br/>O<sub>16</sub>
|[[Bitruncated cubic honeycomb|bitruncated cubic]] (batch)<br/>{{CDD|branch_11|4a4b|nodes}} <br/>t<sub>1,2</sub>{4,3,4}<br/>2t{4,3,4}
|(4)<br/>[[File:truncated octahedron.png|30px]]<br/>[[truncated octahedron|(4.6.6)]]
|&nbsp;
|&nbsp;
|[[File:Bitruncated cubic honeycomb.png|75px]]
|[[File:Bitruncated cubic tiling.png|75px]]
|[[File:Bitruncated cubic honeycomb verf.png|75px]]<br/>([[disphenoid]])
|[[File:Oblate tetrahedrille cell.png|80px]]<BR/>[[Oblate tetrahedrille]]
|- valign=top BGCOLOR="#e0f0e0"
|J<sub>19</sub><br/>A<sub>22</sub><br/>W<sub>18</sub><br/>G<sub>27</sub><br/>t<sub>0,1,2,3</sub>&delta;<sub>4</sub><br/>O<sub>20</sub>
|[[omnitruncated cubic honeycomb|omnitruncated cubic]] (gippich)<br/>{{CDD|branch_11|4a4b|nodes_11}}<br/>t<sub>0,1,2,3</sub>{4,3,4}
|(2)<br/>[[File:Great rhombicuboctahedron.png|30px]]<br/>[[Truncated cuboctahedron|(4.6.8)]]
|(2)<br/>[[File:octagonal prism.png|30px]]<br/>[[octagonal prism|(4.4.8)]]
|&nbsp;
|[[File:Omnitruncated cubic honeycomb.jpg|75px]]
|[[File:Omnitruncated cubic tiling.png|75px]]
|[[File:Omnitruncated cubic honeycomb verf.png|75px]]<br/>irregular [[tetrahedron]]
|[[File:Fundamental tetrahedron1.png|80px]]<BR/>[[Eighth pyramidille]]
|- valign=top BGCOLOR="#d0f0f0"
|J<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>h&delta;<sub>4</sub><br/>O<sub>27</sub>
|[[Quarter cubic honeycomb]] (cytatoh)<br/>{{CDD|branch|4a4b|nodes_h1h1}}<br/>ht<sub>0</sub>ht<sub>3</sub>{4,3,4}
|(2)<br/>[[File:Uniform polyhedron-33-t0.png|30px]]<br/>[[tetrahedron|(3.3.3)]]
|(6)<br/>[[File:Uniform polyhedron-33-t01.png|30px]]<br/>[[truncated tetrahedron|(3.6.6)]]
|
|[[File:quarter cubic honeycomb2.png|76px]]
|[[File:Bitruncated alternated cubic tiling.png|75px]]
|[[File:T01 quarter cubic honeycomb verf2.png|75px]]<br/>elongated [[triangular antiprism]]

|[[File:Oblate cubille cell.png|80px]]<BR/>[[Oblate cubille]]
|- valign=top BGCOLOR="#d0f0f0"
|J<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>h&delta;<sub>4</sub><br/>O<sub>21</sub>
|[[Alternated cubic honeycomb|Alternated runcinated cubic]] (octet)<br/>(same as alternated cubic)<br/>{{CDD|branch|4a4b|nodes_hh}}<br/>ht<sub>0,3</sub>{4,3,4}
|(2)<br/>[[File:Uniform polyhedron-33-t0.png|30px]]<br/>[[tetrahedron|(3.3.3)]]
|(6)<br/>[[File:Uniform polyhedron-33-t2.png|30px]]<br/>[[tetrahedron|(3.3.3)]]
|(6)<br/>[[File:Uniform polyhedron-33-t1.svg|30px]]<br/>[[octahedron|(3.3.3.3)]]
|[[File:Tetrahedral-octahedral honeycomb2.png|76px]]
|[[File:Alternated cubic tiling.png|75px]]
|[[File:Alternated cubic honeycomb verf.svg|75px]]<br/>[[cuboctahedron]]

|- valign=top BGCOLOR="#d0f0f0"
|Nonuniform
|[[Cubic honeycomb#Biorthosnub cubic honeycomb|Biorthosnub cubic honeycomb]] (gabreth)<br>{{CDD|branch_11|4a4b|nodes_hh}}<br/>2s<sub>0,3</sub>{(4,2,4,3)}
|(2)<br>[[File:Truncated octahedron.png|30px]]<br>[[truncated octahedron|(4.6.6)]]
|(2)<br>[[File:Cube rotorotational symmetry.png|30px]]<br/>[[cube|(4.4.4)]]
|(2)<br>[[File:Cantic snub hexagonal hosohedron2.png|30px]]<br/>[[hexagonal prism|(4.4.6)]]
|
|
|
|- valign=top BGCOLOR="#d0f0f0"
|Nonuniform<sub>a</sub>
|[[Bitruncated cubic honeycomb#Related polyhedra and honeycombs|Alternated bitruncated cubic]] (bisch)<br/>{{CDD|branch_hh|4a4b|nodes}}<br/>h2t{4,3,4}
|[[File:Uniform polyhedron-43-h01.svg|30px]] (4)<br/>[[Regular icosahedron|(3.3.3.3.3)]]
|&nbsp;
|[[File:tetrahedron.png|30px]] (4)<br/>[[tetrahedron|(3.3.3)]]
|[[File:Alternated bitruncated cubic honeycomb2.png|75px]]
|
||[[File:Alternated bitruncated cubic honeycomb verf.png|75px]]
|[[File:Ten-of-diamonds_decahedron_in_cube.png|80px]]
|- valign=top BGCOLOR="#d0f0f0"
|Nonuniform
|Cantic bisnub cubic (cabisch)<br>{{CDD|branch_hh|4a4b|nodes_11}}<br/>2s<sub>0,3</sub>{4,3,4}
|(2)<br>[[File:Rhombicuboctahedron uniform edge coloring.png|30px]]<br/>[[rhombicuboctahedron|(3.4.4.4)]]
|(2)<br>[[File:Tetragonal prism.png|30px]]<br/>(4.4.4)
|(2)<br>[[File:Cube rotorotational symmetry.png|30px]]<br/>[[cube|(4.4.4)]]
|
|
|
|- valign=top BGCOLOR="#d0f0f0"
|Nonuniform<sub>c</sub>
|[[Omnitruncated cubic honeycomb#Alternated omnitruncated cubic honeycomb|Alternated omnitruncated cubic]] (snich)<br/>{{CDD|branch_hh|4a4b|nodes_hh}}<br/>ht<sub>0,1,2,3</sub>{4,3,4}
|(2)<br>[[File:Snub hexahedron.png|30px]]<br/>[[Snub cube|(3.3.3.3.4)]]
|(2)<br>[[File:square antiprism.png|30px]]<br/>[[square antiprism|(3.3.3.4)]]
|(4)<br>[[File:tetrahedron.png|30px]]<br/>[[tetrahedron|(3.3.3)]]
|&nbsp;
|
||[[File:Snub cubic honeycomb verf.png|75px]]
|}

=== B̃<sub>3</sub>, [4,3<sup>1,1</sup>] 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<sup>+</sup>,4,3<sup>1,1</sup>], [4,(3<sup>1,1</sup>)<sup>+</sup>], and [4,3<sup>1,1</sup>]<sup>+</sup>. 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<sup>1,1</sup>] uniform honeycombs, [[space group]] Fm{{overline|3}}m (225)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s
!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<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>h&delta;<sub>4</sub><br/>O<sub>21</sub>
|[[Tetrahedral-octahedral honeycomb|Alternated cubic]] (octet)<br/>{{CDD|nodes_10ru|split2|node|4|node}} ↔ {{CDD|node_h1|4|node|3|node|4|node}}
|&nbsp;
|&nbsp;
|[[File:Octahedron.png|30px]] (6)<br/>[[Octahedron|(3.3.3.3)]]
|[[File:Tetrahedron.png|30px]](8)<br/>[[Tetrahedron|(3.3.3)]]
|[[File:Tetrahedral-octahedral honeycomb.png|76px]]
|[[File:Alternated cubic tiling.png|75px]]
|[[File:Alternated cubic honeycomb verf.svg|60px]]<br/>[[cuboctahedron]]
|- 
|J<sub>22,34</sub><br/>A<sub>21</sub><br/>W<sub>17</sub><br/>G<sub>10</sub><br/>h<sub>2</sub>&delta;<sub>4</sub><br/>O<sub>25</sub>
|[[Cantic cubic honeycomb|Cantic cubic]] (tatoh)<br/>{{CDD|nodes_10ru|split2|node_1|4|node}} ↔ {{CDD|node_h1|4|node|3|node_1|3|node}}
|[[File:Cuboctahedron.png|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]]
|&nbsp;
|[[File:Truncated octahedron.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]]
|[[File:Truncated tetrahedron.png|30px]] (2)<br/>[[Truncated tetrahedron|(3.6.6)]]
||[[File:Truncated Alternated Cubic Honeycomb.svg|75px]]
|[[File:Truncated alternated cubic tiling.png|75px]]
|[[File:Truncated alternated cubic honeycomb verf.png|60px]]<br/>rectangular pyramid
|- 
|J<sub>23</sub><br/>A<sub>16</sub><br/>W<sub>11</sub><br/>G<sub>5</sub><br/>h<sub>3</sub>&delta;<sub>4</sub><br/>O<sub>26</sub>
|[[Runcic cubic honeycomb|Runcic cubic]] (sratoh)<br/>{{CDD|nodes_10ru|split2|node|4|node_1}} ↔ {{CDD|node_h1|4|node|3|node|4|node_1}}
|[[File:hexahedron.png|30px]] (1)<br/>[[cube]]
|&nbsp;
|[[File:small rhombicuboctahedron.png|30px]] (3)<br/>[[rhombicuboctahedron|(3.4.4.4)]]
|[[File:tetrahedron.png|30px]] (1)<br/>[[tetrahedron|(3.3.3)]]
|[[File:Runcinated alternated cubic honeycomb.jpg|75px]]
|[[File:Runcinated alternated cubic tiling.png|75px]]
|[[File:Runcinated alternated cubic honeycomb verf.png|60px]]<br/>tapered [[triangular prism]]
|- 
|J<sub>24</sub><br/>A<sub>20</sub><br/>W<sub>16</sub><br/>G<sub>21</sub><br/>h<sub>2,3</sub>&delta;<sub>4</sub><br/>O<sub>28</sub>
|[[Runcicantic cubic honeycomb|Runcicantic cubic]] (gratoh)<br/>{{CDD|nodes_10ru|split2|node_1|4|node_1}} ↔ {{CDD|node_h1|4|node|3|node_1|3|node_1}}
|[[File:truncated hexahedron.png|30px]] (1)<br/>[[truncated cube|(3.8.8)]]
|&nbsp;
|[[File:Great rhombicuboctahedron.png|30px]](2)<br/>[[truncated cuboctahedron|(4.6.8)]]
|[[File:truncated tetrahedron.png|30px]] (1)<br/>[[truncated tetrahedron|(3.6.6)]]
|[[File:Cantitruncated alternated cubic honeycomb.png|75px]]
|[[File:Cantitruncated alternated cubic tiling.png|75px]]
|[[File:Runcitruncated alternate cubic honeycomb verf.png|60px]]<br/>Irregular [[tetrahedron]]
|}

{|class="wikitable"
|+ <[4,3<sup>1,1</sup>]> uniform honeycombs, [[space group]] Pm{{overline|3}}m (221)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<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<sub>11,15</sub><br/>A<sub>1</sub><br/>W<sub>1</sub><br/>G<sub>22</sub><br/>&delta;<sub>4</sub><br/>O<sub>1</sub>
|[[cubic honeycomb|Cubic]] (chon)<br/>{{CDD|nodes|split2|node|4|node_1}} ↔ {{CDD|node_h0|4|node|3|node|3|node_1}} 
|[[File:hexahedron.png|30px]] (8)<br/>[[cube|(4.4.4)]]
|&nbsp;
|&nbsp;
|&nbsp;
|[[File:Bicolor cubic honeycomb.png|75px]]
|[[File:Cubic tiling.png|75px]]
|[[File:Cubic honeycomb verf.svg|60px]]<br/>[[octahedron]]

|- BGCOLOR="#e0f0e0"
|rowspan=2|J<sub>12,32</sub><br/>A<sub>15</sub><br/>W<sub>14</sub><br/>G<sub>7</sub><br/>t<sub>1</sub>&delta;<sub>4</sub><br/>O<sub>15</sub>
|[[rectified cubic honeycomb|Rectified cubic]] (rich)<br/>{{CDD|nodes|split2|node_1|4|node}} ↔ {{CDD|node_h0|4|node|3|node_1|4|node}}
|[[File:cuboctahedron.png|30px]] (4)<br/>[[cuboctahedron|(3.4.3.4)]]
|&nbsp;
|[[File:Uniform polyhedron-33-t1.svg|30px]] (2)<br/>[[octahedron|(3.3.3.3)]]
|&nbsp;
|[[File:Rectified cubic honeycomb4.png|75px]]
|rowspan=2|[[File:Rectified cubic tiling.png|75px]]
|[[File:Rectified alternate cubic honeycomb verf.png|60px]]<br/>[[cuboid]]
|- BGCOLOR="#e0f0e0"
|[[rectified cubic honeycomb|Rectified cubic]] (rich)<br/>{{CDD|nodes_11|split2|node|4|node}} ↔ {{CDD|node_h0|4|node_1|3|node|4|node}}
|[[File:octahedron.png|30px]] (2)<br/>[[octahedron|(3.3.3.3)]]
|&nbsp;
|[[File:Uniform polyhedron-33-t02.svg|30px]] (4)<br/>[[cuboctahedron|(3.4.3.4)]]
|&nbsp;
|[[File:Rectified cubic honeycomb3.png|75px]]
|[[File:Cantellated alternate cubic honeycomb verf.png|60px]]<br/>[[cuboid]]
|- BGCOLOR="#e0f0e0"
|J<sub>13</sub><br/>A<sub>14</sub><br/>W<sub>15</sub><br/>G<sub>8</sub><br/>t<sub>0,1</sub>&delta;<sub>4</sub><br/>O<sub>14</sub>
|[[truncated cubic honeycomb|Truncated cubic]] (tich)<br/>{{CDD|nodes|split2|node_1|4|node_1}} ↔ {{CDD|node_h0|4|node|3|node_1|4|node_1}}
|[[File:truncated hexahedron.png|30px]] (4)<br/>[[truncated cube|(3.8.8)]]
|&nbsp;
|[[File:Uniform polyhedron-33-t1.svg|30px]] (1)<br/>[[octahedron|(3.3.3.3)]]
|&nbsp;
|[[File:Truncated cubic honeycomb2.png|75px]]
|[[File:Truncated cubic tiling.png|75px]]
|[[File:Bicantellated alternate cubic honeycomb verf.png|60px]]<br/>[[square pyramid]]
|- BGCOLOR="#e0f0e0"
|J<sub>14</sub><br/>A<sub>17</sub><br/>W<sub>12</sub><br/>G<sub>9</sub><br/>t<sub>0,2</sub>&delta;<sub>4</sub><br/>O<sub>17</sub>
|[[cantellated cubic honeycomb|Cantellated cubic]] (srich)<br/>{{CDD|nodes_11|split2|node|4|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node|4|node_1}}
|[[File:small rhombicuboctahedron.png|30px]] (2)<br/>[[rhombicuboctahedron|(3.4.4.4)]]
|[[File:Uniform polyhedron 222-t012.png|30px]] (2)<br/>[[cube|(4.4.4)]]
|[[File:Uniform polyhedron-33-t02.svg|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]]
|&nbsp;
|[[File:Cantellated cubic honeycomb.jpg|75px]]
|[[File:Cantellated cubic tiling.png|75px]]
|[[File:Runcicantellated alternate cubic honeycomb verf.png|60px]]<br/>obilique [[triangular prism]]
|- BGCOLOR="#e0f0e0"
|J<sub>16</sub><br/>A<sub>3</sub><br/>W<sub>2</sub><br/>G<sub>28</sub><br/>t<sub>0,2</sub>&delta;<sub>4</sub><br/>O<sub>16</sub>
|[[bitruncated cubic honeycomb|Bitruncated cubic]] (batch)<br/>{{CDD|nodes_11|split2|node_1|4|node}} ↔ {{CDD|node_h0|4|node_1|3|node_1|4|node}} 
|[[File:truncated octahedron.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]]
|&nbsp;
|[[File:Uniform polyhedron-33-t012.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]]
|&nbsp;
|[[File:Bitruncated cubic honeycomb3.png|75px]]
|[[File:Bitruncated cubic tiling.png|75px]]
|[[File:Cantitruncated alternate cubic honeycomb verf.png|60px]]<br/>isosceles [[tetrahedron]]
|- BGCOLOR="#e0f0e0"
|J<sub>17</sub><br/>A<sub>18</sub><br/>W<sub>13</sub><br/>G<sub>25</sub><br/>t<sub>0,1,2</sub>&delta;<sub>4</sub><br/>O<sub>18</sub>
|[[cantitruncated cubic honeycomb|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}}
|[[File:Great rhombicuboctahedron.png|30px]] (2)<br/>[[truncated cuboctahedron|(4.6.8)]]
|[[File:Uniform polyhedron 222-t012.png|30px]] (1)<br/>[[cube|(4.4.4)]]
|[[File:Uniform polyhedron-33-t012.png|30px]](1)<br/>[[truncated octahedron|(4.6.6)]]
|&nbsp;
|[[File:Cantitruncated Cubic Honeycomb.svg|75px]]
|[[File:Cantitruncated cubic tiling.png|75px]]
|[[File:Omnitruncated alternated cubic honeycomb verf.png|60px]]<br/>irregular [[tetrahedron]]
|- BGCOLOR="#d0f0f0"
|J<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>h&delta;<sub>4</sub><br/>O<sub>21</sub>
|[[Tetrahedral-octahedral honeycomb|Alternated cubic]] (octet)<br/>{{CDD|node_h1|4|node|split1|nodes}} ↔ {{CDD|node_1|split1|nodes|split2|node}}
|[[File:tetrahedron.png|30px]] (8)<br/>[[tetrahedron|(3.3.3)]]
|&nbsp;
|&nbsp;
|[[File:Octahedron.png|30px]] (6)<br/>[[Octahedron|(3.3.3.3)]]
|[[File:Tetrahedral-octahedral honeycomb2.png|75px]]
|[[File:Alternated cubic tiling.png|75px]]
|[[File:Alternated cubic honeycomb verf.svg|60px]]<br/>[[cuboctahedron]]

|- BGCOLOR="#d0f0f0"
|J<sub>22,34</sub><br/>A<sub>21</sub><br/>W<sub>17</sub><br/>G<sub>10</sub><br/>h<sub>2</sub>&delta;<sub>4</sub><br/>O<sub>25</sub>
|[[Cantic cubic honeycomb|Cantic cubic]] (tatoh)<br/>{{CDD|node_h1|4|node|split1|nodes_11}} ↔ {{CDD|node_1|split1|nodes_11|split2|node}}
|[[File:Truncated tetrahedron.png|30px]] (2)<br/>[[Truncated tetrahedron|(3.6.6)]]
|&nbsp;
|[[File:Cuboctahedron.png|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]]
|[[File:Truncated octahedron.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]]
||[[File:Truncated Alternated Cubic Honeycomb.svg|75px]]
|[[File:Truncated alternated cubic tiling.png|75px]]
|[[File:Truncated alternated cubic honeycomb verf.png|60px]]<br/>rectangular pyramid

|- BGCOLOR="#d0f0f0"
|Nonuniform<sub>a</sub>
|[[Bitruncated cubic honeycomb#Related polyhedra and honeycombs|Alternated bitruncated cubic]] (bisch)<br/>{{CDD|nodes_hh|split2|node_h|4|node}} ↔ {{CDD|node_h0|4|node_h|3|node_h|4|node}}
|[[File:Uniform polyhedron-43-h01.svg|30px]] (2)<br/>[[Regular icosahedron|(3.3.3.3.3)]]
|&nbsp;
|[[File:Uniform polyhedron-33-s012.svg|30px]] (2)<br/>[[Regular icosahedron|(3.3.3.3.3)]]
|[[File:tetrahedron.png|30px]] (4)<br/>[[tetrahedron|(3.3.3)]]
|
|
|[[File:Alternated bitruncated cubic honeycomb verf.png|60px]]
|- BGCOLOR="#d0f0f0"
|Nonuniform<sub>b</sub>
|[[Alternated cantitruncated cubic honeycomb|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}}
|[[File:snub hexahedron.png|30px]] (2)<br/>[[snub cube|(3.3.3.3.4)]]
|[[File:tetrahedron.png|30px]] (1)<br/>[[tetrahedron|(3.3.3)]]
|[[File:Uniform polyhedron-43-h01.svg|30px]] (1)<br/>[[Regular icosahedron|(3.3.3.3.3)]]
|[[File:tetrahedron.png|30px]] (4)<br/>[[tetrahedron|(3.3.3)]]
|[[File:Alternated cantitruncated cubic honeycomb.png|75px]]
|
|[[File:Alternated cantitruncated cubic honeycomb verf.png|60px]]<br/>Irr. [[tridiminished icosahedron]]
|}

=== Ã<sub>3</sub>, [3<sup>[4]</sup>] group ===
There are 5 forms<ref>[http://mathworld.wolfram.com/Necklace.html], [http://oeis.org/A000029 A000029] 6-1 cases, skipping one with zero marks</ref> constructed from the <math>{\tilde{A}}_3</math>, [3<sup>[4]</sup>] [[Coxeter group]], of which only the ''quarter cubic honeycomb'' is unique. There is one index 2 subgroup [3<sup>[4]</sup>]<sup>+</sup> which generates the snub form, which is not uniform, but included for completeness.

{{A3 honeycombs}}

{|class="wikitable"
|+ [[3<sup>[4]</sup>]] uniform honeycombs, [[space group]] Fd{{overline|3}}m (227)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<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<sub>25,33</sub><br/>A<sub>13</sub><br/>W<sub>10</sub><br/>G<sub>6</sub><br/>q&delta;<sub>4</sub><br/>O<sub>27</sub>
|[[quarter cubic honeycomb|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}
|[[File:Tetrahedron.png|30px]] (2)<br/>[[tetrahedron|(3.3.3)]]
|[[File:Truncated tetrahedron.png|30px]] (6)<br/>[[truncated tetrahedron|(3.6.6)]] 
|[[File:Quarter cubic honeycomb.png|75px]]
|[[File:Bitruncated alternated cubic tiling.png|75px]]
|[[File:t01 quarter cubic honeycomb verf.png|75px]]<br/>triangular antiprism
|}

{|class="wikitable"
|+ <[3<sup>[4]</sup>]> ↔ [4,3<sup>1,1</sup>] uniform honeycombs, [[space group]] Fm{{overline|3}}m (225)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<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<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>h&delta;<sub>4</sub><br/>O<sub>21</sub>
|[[Tetrahedral-octahedral honeycomb|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}
|
|[[File:Uniform polyhedron-33-t0.png|30px]] (8)<br/>[[Tetrahedron|(3.3.3)]]
|[[File:Uniform polyhedron-33-t1.svg|30px]] (6)<br/>[[Octahedron|(3.3.3.3)]]
|[[File:Tetrahedral-octahedral honeycomb2.png|75px]]
|[[File:Alternated cubic tiling.png|75px]]
|[[File:Alternated cubic honeycomb verf.svg|75px]]<br/>[[cuboctahedron]]
|- BGCOLOR="#e0f0e0"
|J<sub>22,34</sub><br/>A<sub>21</sub><br/>W<sub>17</sub><br/>G<sub>10</sub><br/>h<sub>2</sub>&delta;<sub>4</sub><br/>O<sub>25</sub>
|[[Cantic cubic honeycomb|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/>h<sub>2</sub>{4,3,4}
|[[File:Truncated tetrahedron.png|30px]] (2)<br/>[[Truncated tetrahedron|(3.6.6)]]
|[[File:Uniform polyhedron-33-t02.svg|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]]
|[[File:Uniform polyhedron-33-t012.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]]
|[[File:Truncated Alternated Cubic Honeycomb2.png|75px]]
|[[File:Truncated alternated cubic tiling.png|75px]]
|[[File:t012 quarter cubic honeycomb verf.png|75px]]<br/>Rectangular pyramid
|}

{|class="wikitable"
|+ [2[3<sup>[4]</sup>]] ↔ [4,3,4] uniform honeycombs, [[space group]] Pm{{overline|3}}m (221)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<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<sub>12,32</sub><br/>A<sub>15</sub><br/>W<sub>14</sub><br/>G<sub>7</sub><br/>t<sub>1</sub>&delta;<sub>4</sub><br/>O<sub>1</sub>
|[[rectified cubic honeycomb|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}
|[[File:Uniform polyhedron-33-t02.svg|30px]] (2)<br/>[[cuboctahedron|(3.4.3.4)]]
|[[File:Uniform polyhedron-33-t1.svg|30px]] (1)<br/>[[octahedron|(3.3.3.3)]]
|[[File:Rectified cubic honeycomb2.png|75px]]
|[[File:Rectified cubic tiling.png|75px]]
|[[File:t02 quarter cubic honeycomb verf.png|75px]]<br/>[[cuboid]]

|}

{|class="wikitable"
|+ [4[3<sup>[4]</sup>]] ↔ {{brackets|4,3,4}} uniform honeycombs, [[space group]] Im{{overline|3}}m (229)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<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"
|J<sub>16</sub><br/>A<sub>3</sub><br/>W<sub>2</sub><br/>G<sub>28</sub><br/>t<sub>1,2</sub>&delta;<sub>4</sub><br/>O<sub>16</sub>
|[[bitruncated cubic honeycomb|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}
|[[File:Uniform polyhedron-33-t012.png|30px]] (4)<br/>[[truncated octahedron|(4.6.6)]]
|
|[[File:Bitruncated cubic honeycomb2.png|75px]]
|[[File:Bitruncated cubic tiling.png|75px]]
|[[File:t0123 quarter cubic honeycomb verf.png|75px]]<br/>isosceles [[tetrahedron]]
|- BGCOLOR="#d0f0f0"
|Nonuniform<sub>a</sub>
|[[Bitruncated cubic honeycomb#Related polyhedra and honeycombs|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}
|[[File:Uniform polyhedron-33-s012.png|30px]] (4)<br/>[[Regular icosahedron|(3.3.3.3.3)]]
|[[File:Uniform polyhedron-33-t0.png|30px]] (4)<br/>[[tetrahedron|(3.3.3)]]
|&nbsp;
|
|[[File:Alternated bitruncated cubic honeycomb verf.png|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]]
|- 
|J<sub>52</sub><br/>A<sub>2'</sub><br/>G<sub>2</sub><br/>O<sub>22</sub>
|h{4,3,4}:g
|align=center|[[gyrated alternated cubic honeycomb|gyrated alternated cubic]] (gytoh)
|align=center|[[tetrahedron]] (8)<br/>[[octahedron]] (6)
|[[File:Gyrated alternated cubic honeycomb.png|70px]]
|[[File:Gyrated alternated cubic.png|100px]]
|[[File:Gyrated alternated cubic honeycomb verf.png|80px]]<br/> [[triangular orthobicupola]]
|- 
|J<sub>61</sub><br/>A<sub>?</sub><br/>G<sub>3</sub><br/>O<sub>24</sub>
|h{4,3,4}:ge
|align=center|[[Gyroelongated alternated cubic honeycomb|gyroelongated alternated cubic]] (gyetoh)
|align=center|[[triangular prism]] (6)<br/>[[tetrahedron]] (4)<br/>[[octahedron]] (3)
|[[File:Gyroelongated alternated cubic honeycomb.png|70px]]
|[[File:Gyroelongated alternated cubic tiling.png|100px]]
|rowspan=2|[[File:Gyroelongated alternated cubic honeycomb verf.png|80px]]
|- 
|J<sub>62</sub><br/>A<sub>?</sub><br/>G<sub>4</sub><br/>O<sub>23</sub>
|h{4,3,4}:e
|align=center|[[Elongated alternated cubic honeycomb|elongated alternated cubic]] (etoh)
|align=center|[[triangular prism]] (6)<br/>[[tetrahedron]] (4)<br/>[[octahedron]] (3)
|[[File:Elongated alternated cubic honeycomb.png|70px]]
|[[File:Elongated alternated cubic tiling.png|80px]]
|- 
|J<sub>63</sub><br/>A<sub>?</sub><br/>G<sub>12</sub><br/>O<sub>12</sub>
|{3,6}:g × {∞}
|align=center|[[Gyrated triangular prismatic honeycomb|gyrated triangular prismatic]] (gytoph)
|align=center|[[triangular prism]] (12)
|[[File:Gyrated triangular prismatic honeycomb.png|70px]]
|[[File:Gyrated triangular prismatic tiling.png|100px]]
|[[File:Gyrated triangular prismatic honeycomb verf.png|80px]]
|- 
|J<sub>64</sub><br/>A<sub>?</sub><br/>G<sub>15</sub><br/>O<sub>13</sub>
|{3,6}:ge × {∞}
|align=center|[[gyroelongated triangular prismatic honeycomb|gyroelongated triangular prismatic]] (gyetaph)
|align=center|[[triangular prism]] (6)<br/>[[cube]] (4)
|[[File:Gyroelongated triangular prismatic honeycomb.png|70px]]
|[[File:Gyroelongated triangular prismatic tiling.png|100px]]
|[[File:Gyroelongated alternated triangular prismatic honeycomb verf.png|80px]]
|}

=== Prismatic stacks ===
Eleven '''prismatic''' tilings are obtained by stacking the eleven [[tiling by regular polygons|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 triangle]]s.

==== The C̃<sub>2</sub>×Ĩ<sub>1</sub>(&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 diagram|Coxeter-Dynkin]]<br/>and [[Schläfli symbol#Extended for uniform 4-polytopes and 3-space honeycombs|Schläfli]]<br/>symbols
!Honeycomb name
!Plane<br/>tiling
!Solids<br/>(Partial)
!Tiling
|- 
|rowspan=3|J<sub>11,15</sub><br/>A<sub>1</sub><br/>G<sub>22</sub>
|align=center|{{CDD|node_1|4|node|4|node|2|node_1|infin|node}} <br/>{4,4}×{∞}
|rowspan=3 align=center|[[Cubic honeycomb|Cubic]]<br/>(Square prismatic) (chon)
|rowspan=3|[[Square tiling|(4.4.4.4)]]
|rowspan=3|[[File:Partial cubic honeycomb.png|80px]]
|[[File:Uniform tiling 44-t0.svg|50px]]
|- BGCOLOR="#e0f0e0"
|align=center|{{CDD|node|4|node_1|4|node|2|node_1|infin|node}} <br/>r{4,4}×{∞}
|[[File:Uniform tiling 44-t1.png|50px]]
|- BGCOLOR="#e0f0e0"
|align=center|{{CDD|node_1|4|node|4|node_1|2|node_1|infin|node}} <br/>rr{4,4}×{∞}
|[[File:Uniform tiling 44-t02.svg|50px]]
|- 
|rowspan=2|J<sub>45</sub><br/>A<sub>6</sub><br/>G<sub>24</sub>
|align=center|{{CDD|node_1|4|node_1|4|node|2|node_1|infin|node}} <br/>t{4,4}×{∞} 
|rowspan=2 align=center|[[Truncated square prismatic honeycomb|Truncated/Bitruncated square prismatic]] (tassiph)
|rowspan=2|[[Truncated square tiling|(4.8.8)]]
|rowspan=2|[[File:Truncated square prismatic honeycomb.png|80px]]
|[[File:Uniform tiling 44-t01.png|50px]]
|- BGCOLOR="#e0f0e0"
|align=center|{{CDD|node_1|4|node_1|4|node_1|2|node_1|infin|node}} <br/>tr{4,4}×{∞} 
|[[File:Uniform tiling 44-t012.png|50px]]
|- BGCOLOR="#d0f0f0"
|J<sub>44</sub><br/>A<sub>11</sub><br/>G<sub>14</sub>
|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 honeycomb|Snub square prismatic]] (sassiph)
|[[Snub square tiling|(3.3.4.3.4)]]
|[[File:Snub square prismatic honeycomb.png|80px]]
|[[File:Uniform tiling 44-snub.svg|50px]]
|- BGCOLOR="#d0f0f0"
|Nonuniform
|align=center|{{CDD|node_h|4|node_h|4|node_h|2x|node_h|infin|node}}<br/>ht<sub>0,1,2,3</sub>{4,4,2,∞}
|
|
|
|
|}

==== The G̃<sub>2</sub>xĨ<sub>1</sub>(&infin;), [6,3,2,&infin;] prismatic group ====
{|class="wikitable"
!Indices
![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<br/>and [[Schläfli symbol#Extended for uniform 4-polytopes and 3-space honeycombs|Schläfli]]<br/>symbols
!Honeycomb name
!Plane<br/>tiling
!Solids<br/>(Partial)
!Tiling
|- 
|J<sub>41</sub><br/>A<sub>4</sub><br/>G<sub>11</sub>
|{{CDD|node|6|node|3|node_1|2|node_1|infin|node}} <br/>{3,6} × {∞}
|[[Triangular prismatic honeycomb|Triangular prismatic]] (tiph)
|[[triangular tiling|(3<sup>6</sup>)]]
|[[File:Triangular prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-t2-red.svg|60px]]
|- 
|rowspan=2|J<sub>42</sub><br/>A<sub>5</sub><br/>G<sub>26</sub>
|{{CDD|node_1|6|node|3|node|2|node_1|infin|node}} <br/>{6,3} × {∞}
|rowspan=2 align=center|[[Hexagonal prismatic honeycomb|Hexagonal prismatic]] (hiph)
|rowspan=2|[[hexagonal tiling|(6<sup>3</sup>)]]
|[[File:Hexagonal prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-t0.svg|60px]]
|- 
|{{CDD|node|6|node_1|3|node_1|2|node_1|infin|node}} <br/>t{3,6} × {∞}
||[[File:Truncated triangular prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-t12.svg|60px]]
|- 
|J<sub>43</sub><br/>A<sub>8</sub><br/>G<sub>18</sub>
|{{CDD|node|6|node_1|3|node|2|node_1|infin|node}} <br/>r{6,3} × {∞}
|[[Trihexagonal prismatic honeycomb|Trihexagonal prismatic]] (thiph)
|[[Trihexagonal tiling|(3.6.3.6)]]
|[[File:Triangular-hexagonal prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-t1.png|60px]]
|- 
|J<sub>46</sub><br/>A<sub>7</sub><br/>G<sub>19</sub>
|{{CDD|node_1|6|node_1|3|node|2|node_1|infin|node}} <br/>t{6,3} × {∞}
|[[Truncated hexagonal prismatic honeycomb|Truncated hexagonal prismatic]] (thaph)
|[[Truncated hexagonal tiling|(3.12.12)]]
|[[File:Truncated hexagonal prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-t01.png|60px]]
|- 
|J<sub>47</sub><br/>A<sub>9</sub><br/>G<sub>16</sub>
|{{CDD|node_1|6|node|3|node_1|2|node_1|infin|node}} <br/>rr{6,3} × {∞}
|[[Rhombitrihexagonal prismatic honeycomb|Rhombi-trihexagonal prismatic]] (srothaph)
|[[Rhombitrihexagonal tiling|(3.4.6.4)]]
|[[File:Rhombitriangular-hexagonal prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-t02.png|60px]]
|- BGCOLOR="#d0f0f0"
|J<sub>48</sub><br/>A<sub>12</sub><br/>G<sub>17</sub>
|{{CDD|node_h|6|node_h|3|node_h|2|node_1|infin|node}} <br/>sr{6,3} × {∞}
|[[Snub hexagonal prismatic honeycomb|Snub hexagonal prismatic]] (snathaph)
|[[Snub hexagonal tiling|(3.3.3.3.6)]]
|[[File:Snub triangular-hexagonal prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-snub.png|60px]]
|- 
|J<sub>49</sub><br/>A<sub>10</sub><br/>G<sub>23</sub>
|{{CDD|node_1|6|node_1|3|node_1|2|node_1|infin|node}} <br/>tr{6,3} × {∞}
|[[truncated trihexagonal prismatic honeycomb|truncated trihexagonal prismatic]] (grothaph)
|[[Truncated trihexagonal tiling|(4.6.12)]]
|[[File:Omnitruncated triangular-hexagonal prismatic honeycomb.png|60px]]
|[[File:Uniform tiling 63-t012.svg|60px]]
|- BGCOLOR="#d0f0f0"
|J<sub>65</sub><br/>A<sub>11'</sub><br/>G<sub>13</sub>
|{{CDD|node|infin|node_h|2x|node_h|infin|node_1|2|node_1|infin|node}} <br/>{3,6}:e × {∞}
|[[elongated triangular prismatic honeycomb|elongated triangular prismatic]] (etoph)
|[[elongated triangular tiling|(3.3.3.4.4)]]
|[[File:Elongated triangular prismatic honeycomb.png|60px]]
|[[File:Tile 33344.svg|60px]]
|- BGCOLOR="#d0f0f0"
|rowspan=2|J<sub>52</sub><br/>A<sub>2'</sub><br/>G<sub>2</sub>
|{{CDD|node|3|node|6|node_h|2x|node_h|infin|node}}<br/>h3t{3,6,2,∞}
|rowspan=2|[[gyrated tetrahedral-octahedral honeycomb|gyrated tetrahedral-octahedral]] (gytoh)
|rowspan=2|[[triangular tiling|(3<sup>6</sup>)]]
|rowspan=2|[[File:Gyrated alternated cubic honeycomb.png|60px]]
|rowspan=2|[[File:Uniform tiling 63-t2-red.svg|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<sub>0,1,2,3</sub>{3,6,2,∞}
|
|
|
|
|}

=== Enumeration of Wythoff forms ===
All nonprismatic [[Wythoff construction]]s by Coxeter groups are given below, along with their [[alternation (geometry)|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
![[Goursat tetrahedron#Euclidean .28affine.29 3-space solutions|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}}<sub>22</sub> | {{CDD|node|4|node_1|3|node|4|node}}<sub>7</sub> | {{CDD|node_1|4|node_1|3|node|4|node}}<sub>8</sub><br/>{{CDD|node_1|4|node|3|node_1|4|node}}<sub>9</sub> | {{CDD|node_1|4|node_1|3|node_1|4|node}}<sub>25</sub> | {{CDD|node_1|4|node_1|3|node|4|node_1}}<sub>20</sub>
|[1<sup>+</sup>,4,3<sup>+</sup>,4,1<sup>+</sup>]||(2)
|{{CDD|node_h1|4|node|3|node|4|node}}<sub>1</sub> | {{CDD|node_h|4|node_h|3|node_h|4|node}}<sub>b</sub>
|- BGCOLOR="#e0f0e0"
|[2<sup>+</sup>[4,3,4]]<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}} <sub>22</sub>
|[2<sup>+</sup>[(4,3<sup>+</sup>,4,2<sup>+</sup>)]]||(1)
|{{CDD|branch|4a4b|branch_hh|label2}}<sub>1</sub> | {{CDD|branch|4a4b|nodes_hh}}<sub>6</sub>

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

|- 
|[2<sup>+</sup>[4,3,4]]<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}}<sub>27</sub>

|[2<sup>+</sup>[4,3,4]]<sup>+</sup>||(1)
|{{CDD|node_h|4|node_h|3|node_h|4|node_h}}<sub>c</sub>

|- 
|rowspan=3|[4,3<sup>1,1</sup>]<br/>{{CDD|node|4|node|split1|nodes}} ||[4,3<sup>1,1</sup>]<br/>{{CDD|node_c3|4|node_c4|split1|nodeab_c1-2}}||4
|{{CDD|node|4|node|split1|nodes_10lu}}<sub>1</sub> | {{CDD|node_1|4|node|split1|nodes_10lu}}<sub>7</sub> | {{CDD|node|4|node_1|split1|nodes_10lu}}<sub>10</sub> | {{CDD|node_1|4|node_1|split1|nodes_10lu}}<sub>28</sub>
|colspan=3|
|- BGCOLOR="#e0f0e0" align=center
|rowspan=2|[1[4,3<sup>1,1</sup>]]=[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}}<sub>22</sub> | {{CDD|node|4|node_1|split1|nodes}}<sub>7</sub> | {{CDD|node_1|4|node_1|split1|nodes}}<sub>22</sub> | {{CDD|node|4|node|split1|nodes_11}}<sub>7</sub> | {{CDD|node_1|4|node|split1|nodes_11}}<sub>9</sub> | {{CDD|node|4|node_1|split1|nodes_11}}<sub>28</sub> | {{CDD|node_1|4|node_1|split1|nodes_11}}<sub>25</sub> 
|[1[1<sup>+</sup>,4,3<sup>1,1</sup>]]<sup>+</sup>||(2)
|{{CDD|node_h1|4|node|split1|nodes}}<sub>1</sub> | {{CDD|node_h1|4|node|split1|nodes_10lu}}<sub>6</sub> | {{CDD|node|4|node_h|split1|nodes_hh}}<sub>a</sub>
|- BGCOLOR="#e0f0e0" align=center
|[1[4,3<sup>1,1</sup>]]<sup>+</sup><br/>=[4,3,4]<sup>+</sup>||(1)
|{{CDD|node_h|4|node_h|split1|nodes_hh}}<sub>b</sub>

|- 
|rowspan=5|[3<sup>[4]</sup>]<br/>{{CDD|branch|3ab|branch}}||[3<sup>[4]</sup>]
|colspan=5|(none)
|- 
||[2<sup>+</sup>[3<sup>[4]</sup>]]<br/>{{CDD|branch_c1|3ab|branch_c2}} || 1
| {{CDD|branch_11|3ab|branch}}<sub>6</sub>
|colspan=3|
|- BGCOLOR="#e0f0e0" align=center
||[1[3<sup>[4]</sup>]]=[4,3<sup>1,1</sup>]<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}}<sub>1</sub> | {{CDD|node_1|split1|nodes_11|split2|node}}<sub>10</sub>
|colspan=3|
|- BGCOLOR="#e0f0e0"
||[2[3<sup>[4]</sup>]]=[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}}<sub>7</sub>
|colspan=3|
|- BGCOLOR="#e0f0e0" align=center
|[(2<sup>+</sup>,4)[3<sup>[4]</sup>]]=[2<sup>+</sup>[4,3,4]]<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}}<sub>28</sub>
|[(2<sup>+</sup>,4)[3<sup>[4]</sup>]]<sup>+</sup><br/>= [2<sup>+</sup>[4,3,4]]<sup>+</sup>
|(1)|| {{CDD|branch_hh|3ab|branch_hh}}<sub>a</sub>

|}

===Examples===
The [[tetrahedral-octahedral honeycomb|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 [[Cell (mathematics)|cells]] are allowed to be [[uniform tiling]]s, 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 [[duoprism]]s: these look like a single infinite tower of p-gonal prisms, with the remaining space filled with [[apeirogonal prism]]s)
* <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}}
|-
|[[File:Cubic semicheck.png|180px]]
|[[File:Tetroctahedric semicheck.png|180px]]
|[[File:Trihexagonal prism slab honeycomb.png|180px]]
|-
|[[File:X4o4o2ox vertex figure.png|180px]]<br/>(4) 4<sup>3</sup>: [[cube]]<br/>(1) 4<sup>4</sup>: [[square tiling]]
|[[File:O6x3o2x vertex figure.png|180px]]<br/>(4) 3<sup>3</sup>: [[tetrahedron]]<br/>(3) 3<sup>4</sup>: [[octahedron]]<br/>(1) 3<sup>6</sup>: [[triangular tiling]]
|[[File:O3o6s2s vertex figure.png|180px]]<br/>(2) 3.4.4: [[triangular prism]]<br/>(2) 4.4.6: [[hexagonal prism]]<br/>(1) (3.6)<sup>2</sup>: [[trihexagonal tiling]]
|}

The first two forms shown above are [[semiregular polytope|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 polytope|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 solid]]s along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, [[pyramid (geometry)|pyramid]] and [[cupola (geometry)|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
|-
!s<sub>3</sub>{2,6,3}, {{CDD|node_h|2x|node_h|6|node|3|node_1}}
!s<sub>3</sub>{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}}
!3s<sub>4</sub>{4,4,2,&infin;}, {{CDD|node|4|node|4|node_h|2x|node_h|infin|node_1}}
|-
|[[File:Runcic snub 263 honeycomb.png|200px]]
|[[File:Runcic snub 244 honeycomb.png|200px]]
|[[File:Alternated cubic slab honeycomb.png|200px]]
|[[File:Elongated square antiprismatic celluation.png|200px]]
|-
! [[File:triangular cupola.png|40px]] [[File:octahedron.png|40px]] [[File:Uniform polyhedron-63-t1-1.svg|40px]]
! [[File:square cupola.png|40px]] [[File:tetrahedron.png|40px]] [[File:Uniform tiling 44-t01.png|40px]]
! [[File:square pyramid.png|40px]] [[File:tetrahedron.png|40px]] [[File:Uniform tiling 44-t0.svg|40px]]
! [[File:square pyramid.png|40px]] [[File:tetrahedron.png|40px]] [[File:hexahedron.png|40px]]
|- valign=top
|[[File:s2s6o3x vertex figure.png|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]]
|[[File:s2s4o4x vertex figure.png|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]]
|[[File:O4o4s2s vertex figure.png|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]]
|[[File:O4o4s2six vertex figure.png|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 space|hyperbolic 3-space]], generated as [[Wythoff construction]]s, and represented by ring permutations of the [[Coxeter-Dynkin diagram]]s 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<sup>1,1</sup>] : {{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 Horton Conway|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 (mathematician)|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)
* [[Alfredo Andreini|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]
* [[Duncan MacLaren Young Sommerville|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)]]