In [[geometry]], a '''convex uniform honeycomb''' is a uniform space-filling [[tessellation]] in three-dimensional [[Euclidean space]] with non-overlapping convex [[uniform polyhedron|uniform polyhedral]] cells.

Twenty-eight such honeycombs exist:
* the familiar [[Convex uniform honeycomb#The_R4.2C_.5B4.2C3.2C4.5D_group_.28cubic.29|cubic honeycomb]] and 7 truncations thereof;
* the [[Convex uniform honeycomb#Alternated cubic forms|alternated cubic honeycomb]] and 4 truncations thereof;
* 10 prismatic forms based on the [[Convex uniform honeycomb#Stacked prismatic forms|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]].

== 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 complete list of 28.
* '''1994''': [[Branko Grünbaum]], in his paper ''Uniform tilings of 3-space'', also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and [[Convex uniform honeycomb#Gyrated and elongated forms|4 being missing]]. Grünbaum also states that [[I. Alexeyev]] of Russia also independently enumerated these forms around the same 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 polychoron]]s in 4-space).

Only 14 of the convex uniform polyhedra appear in these patterns:
* three of the five [[Platonic solid]]s,
* six of the thirteen [[Archimedean solid]]s, and
* five of the infinite family of [[prism (geometry)|prism]]s.

=== 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 tessellations'''.

The individual honeycombs are listed with names given to them by [[Norman Johnson]]. (Some of the terms used below are defined in [[Uniform polychoron#Geometric derivations]].)

For 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]runbaum(1-28).

== Tessellations listed by infinite Coxeter group families ==
[[Image:Coxeter-Dynkin 3-space groups.png|400px|thumb|Fundamental domains in a cubic element of three groups.]]

The fundamental infinite [[Coxeter group]]s for 3-space are:
# The R₄, [4,3,4], cubic, [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]] (8 unique forms plus one alternation)
# The S₄, h[4,3,4], alternated cubic, [[Image:CD_dot.png]][[Image:CD_3b.png]][[Image:CD_downbranch-00.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_dot.png]] (11 forms, 3 new)
# The P₄ cyclic group, [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]] (5 forms, one new)

In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with ''elongation'' and ''gyration'' operations.

The total unique honeycombs above are 18.

The prismatic stacks from infinite Coxeter groups for 3-space are:
# The R₃xW₂, [4,4]x[∞] prismatic group, [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]] (2 new forms)
# The V₃xW₂, [6,3]x[∞] prismatic group, [[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]] (7 unique forms)
# The P₃xW₂, [Δ]x[∞] prismatic group, [[Image:CD righttriangle-000.png]][[Image:CD_2.png]][[Image:CD_dot.png]][[Image:CD_infin.png]][[Image:CD_dot.png]] (No new forms)
# The W₂xW₂xW₂, [∞]x[∞]x[∞] prismatic group, [[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]] (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 R₄, [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.)

<br clear=right>
{|class="wikitable"
!rowspan=2|Reference<BR>Indices
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
!rowspan=2|Honeycomb name
! colspan=4|Cell counts/vertex<BR>and positions in cubic honeycomb<BR>
!
!
!
|-
!(0)
!(1)
!(2)
!(3)
!Solids<BR>(Partial)
!Frames<BR>(Perspective)
![[vertex figure]]
|-
|J_11,15<BR>A₁<BR>W₁<BR>G₂₂
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]<BR>t₀{4,3,4}
|align=center|[[cubic honeycomb|cubic]]
|align=center|&nbsp;
|align=center|&nbsp;
|align=center|&nbsp;
|align=center|8<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
| [[Image:Partial cubic honeycomb.png|75px]]
|[[Image:Cubic honeycomb.png|75px|]]
|[[Image:VF-cubic.png|75px]]<BR>[[octahedron]]
|-
|J_12,32<BR>A₁₅<BR>W₁₄<BR>G₇
|align=center|[[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]<BR>t₁{4,3,4}
|align=center|[[rectified cubic honeycomb|rectified cubic]]
|align=center|2<BR>[[Image:octahedron.png|30px]]<BR>[[octahedron|(3.3.3.3)]]
|align=center|&nbsp;
|align=center|&nbsp;
|align=center|4<BR>[[Image:cuboctahedron.png|30px]]<BR>[[cuboctahedron|(3.4.3.4)]]
|[[Image:Rectified cubic honeycomb.png|75px]]
|[[Image:Rectified cubic tiling.png|75px|]]
|[[Image:VF-rectified cubic.png|75px]]<BR>[[cuboid]]
|-
|J₁₃<BR>A₁₄<BR>W₁₅<BR>G₈
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]<BR>t_0,1{4,3,4}
|align=center|[[truncated cubic honeycomb|truncated cubic]]
|align=center|1<BR>[[Image:octahedron.png|30px]]<BR>[[octahedron|(3.3.3.3)]]
|align=center|&nbsp;
|align=center|&nbsp;
|align=center|4<BR>[[Image:truncated hexahedron.png|30px]]<BR>[[truncated cube|(3.8.8)]]
|[[Image:Truncated cubic honeycomb.png|75px]]
|[[Image:Truncated cubic tiling.png|75px]]
|[[Image:VF-truncated cubic.png|75px]]<BR>[[square pyramid]]
|-
|J₁₄<BR>A₁₇<BR>W₁₂<BR>G₉
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]<BR>t_0,2{4,3,4}
|align=center|[[cantellated cubic honeycomb|cantellated cubic]]
|align=center|1<BR>[[Image:cuboctahedron.png|30px]]<BR>[[cuboctahedron|(3.4.3.4)]]
|align=center|2<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
|align=center|&nbsp;
|align=center|2<BR>[[Image:small rhombicuboctahedron.png|30px]]<BR>[[small rhombicuboctahedron|(3.4.4.4)]]
|[[Image:Cantellated cubic honeycomb.jpg|75px]]
|[[Image:Cantellated cubic tiling.png|75px|]]
|[[Image:VF-cantellated cubic.png|75px]]<BR>[[wedge (geometry)|wedge]]
|-
|J_11,15
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]]<BR>t_0,3{4,3,4}
|align=center|'''[[Runcination (geometry)|runcinated]] cubic'''<BR>(same as regular [[cubic honeycomb|cubic]])
|align=center|1<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
|align=center|3<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
|align=center|3<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
|align=center|1<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
| [[Image:Runcinated cubic honeycomb.png|75px]]
|[[Image:Cubic tiling.png|75px|]]
|[[Image:VF-cubic.png|75px]]<BR>[[octahedron]]
|-
|J₁₆<BR>A₃<BR>W₂<BR>G₂₈
|align=center|[[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]<BR>t_1,2{4,3,4}
|align=center|[[bitruncated cubic honeycomb|bitruncated cubic]]
|align=center|2<BR>[[Image:truncated octahedron.png|30px]]<BR>[[truncated octahedron|(4.6.6)]]
|align=center|&nbsp;
|align=center|&nbsp;
|align=center|2<BR>[[Image:truncated octahedron.png|30px]]<BR>[[truncated octahedron|(4.6.6)]]
|[[Image:Bitruncated cubic honeycomb.png|75px]]
|[[Image:Bitruncated cubic tiling.png|75px|]]
|[[Image:VF-bitruncated cubic.png|75px]]<BR>isosceles [[tetrahedron]]
|-
|J₁₇<BR>A₁₈<BR>W₁₃<BR>G₂₅
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]<BR>t_0,1,2{4,3,4}
|align=center|[[cantitruncated cubic honeycomb|cantitruncated cubic]]
|align=center|1<BR>[[Image:truncated octahedron.png|30px]]<BR>[[truncated octahedron|(4.6.6)]]
|align=center|1<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
|align=center|&nbsp;
|align=center|2<BR>[[Image:great rhombicuboctahedron.png|30px]]<BR>[[great rhombicuboctahedron|(4.6.8)]]
|[[Image:Cantitruncated cubic honeycomb.jpg|75px]]
|[[Image:Cantitruncated cubic tiling.png|75px|]]
|[[Image:VF-cantitruncated cubic.png|75px]]<BR>irregular [[tetrahedron]]
|-
|J₁₈<BR>A₁₉<BR>W₁₉<BR>G₂₀
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]]<BR>t_0,1,3{4,3,4}
|align=center|[[runcitruncated cubic honeycomb|runcitruncated cubic]]
|align=center|1<BR>[[Image:small rhombicuboctahedron.png|30px]]<BR>[[small rhombicuboctahedron|(3.4.4.4)]]
|align=center|1<BR>[[Image:hexahedron.png|30px]]<BR>[[cube|(4.4.4)]]
|align=center|2<BR>[[Image:octagonal prism.png|30px]]<BR>[[octagonal prism|(4.4.8)]]
|align=center|1<BR>[[Image:truncated hexahedron.png|30px]]<BR>[[truncated cube|(3.8.8)]]
|[[Image:Runcitruncated cubic honeycomb.jpg|75px]]
|[[Image:Runcitruncated cubic tiling.png|75px|]]
|[[Image:VF-runcitruncated cubic.png|75px]]<br>oblique trapezoidal pyramid
|-
|J₁₉<BR>A₂₂<BR>W₁₈<BR>G₂₇
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]]<BR>t_0,1,2,3{4,3,4}
|align=center|[[omnitruncated cubic honeycomb|omnitruncated cubic]]
|align=center|1<BR>[[Image:great rhombicuboctahedron.png|30px]]<BR>[[great rhombicuboctahedron|(4.6.8)]]
|align=center|1<BR>[[Image:octagonal prism.png|30px]]<BR>[[octagonal prism|(4.4.8)]]
|align=center|1<BR>[[Image:octagonal prism.png|30px]]<BR>[[octagonal prism|(4.4.8)]]
|align=center|1<BR>[[Image:great rhombicuboctahedron.png|30px]]<BR>[[great rhombicuboctahedron|(4.6.8)]]
|[[Image:Omnitruncated cubic honeycomb.jpg|75px]]
|[[Image:Omnitruncated cubic tiling.png|75px|]]
|[[Image:VF-omnitruncated cubic.png|75px]]<BR>irregular [[tetrahedron]]
|-
|-
|J_21,31,51<BR>A₂<BR>W₉<BR>G₁
|align=center|[[Image:CDW_hole.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]<BR>h₀{4,3,4}
|align=center|[[Tetrahedral-octahedral honeycomb|alternated cubic]]
|align=center|6<BR>[[Image:Octahedron.png|30px]]<BR>[[Octahedron|3.3.3.3]]
|align=center|&nbsp;
|align=center|&nbsp;
|align=center|8<BR>[[Image:Tetrahedron.png|30px]]<BR>[[Tetrahedron|3.3.3]]
|[[Image:Tetrahedral-octahedral honeycomb.png|75px]]
|[[Image:Alternated cubic tiling.png|75px|]]
|[[Image:VF-alternated cubic.png|75px]]<BR>[[cuboctahedron]]
|}

=== S₄, h[4,3,4], [4,3^1,1] group ===

The S₄ group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.

The honeycombs from this group are called ''alternated cubic'' because the first form can be seen as a ''cubic honeycomb'' with alternate vertices cubic cells to tetrahedra, and creating octahedron cells in the deleted 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.

{|class="wikitable"
!rowspan=2|Referenced<BR>indices
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
!rowspan=2|Honeycomb name
!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)
!(1)
!(0')
!(3)
|-
|J_21,31,51<BR>A₂<BR>W₉<BR>G₁
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-00.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_dot.png]]
|align=center|[[Tetrahedral-octahedral honeycomb|alternated cubic]]
|align=center|&nbsp;
|align=center|&nbsp;
|align=center|[[Image:Octahedron.png|30px]] (6)<BR>[[Octahedron|3.3.3.3]]
|align=center|[[Image:Tetrahedron.png|30px]](8)<BR>[[Tetrahedron|3.3.3]]
|[[Image:Tetrahedral-octahedral honeycomb.png|75px]]
|[[Image:Alternated cubic tiling.png|75px|]]
|[[Image:VF-alternated cubic.png|75px]]<BR>[[cuboctahedron]]
|-
|J_22,34<BR>A₂₁<BR>W₁₇<BR>G₁₀
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-10.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_dot.png]]
|align=center|[[truncated alternated cubic honeycomb|truncated alternated cubic]]
|align=center|&nbsp;
|align=center|[[Image:Cuboctahedron.png|30px]] (1)<BR>[[cuboctahedron|3.4.3.4]]
|align=center|[[Image:Truncated octahedron.png|30px]] (2)<BR>[[truncated octahedron|4.6.6]]
|align=center|[[Image:Truncated tetrahedron.png|30px]] (2)<BR>[[Truncated tetrahedron|3.6.6]]
||[[Image:Truncated alternated cubic honeycomb.jpg|75px]]
|[[Image:Truncated alternated cubic tiling.png|75px|]]
|[[Image:VF-Truncated alternated cubic.png|75px]]
|-
|J_12,32<BR>A₁₅<BR>W₁₄<BR>G₇
|align=center|[[Image:CD_dot.png]][[Image:CD_3b.png]][[Image:CD_downbranch-10.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_dot.png]]
|align=center|[[rectified cubic honeycomb|rectified cubic]]<BR>(rectified alternate cubic)
|align=center|[[Image:cuboctahedron.png|30px]] (2)<BR>[[cuboctahedron|(3.4.3.4)]]
|align=center|&nbsp;
|align=center|[[Image:cuboctahedron.png|30px]] (2)<BR>[[cuboctahedron|(3.4.3.4)]]
|align=center|[[Image:Uniform polyhedron-33-t1.png|30px]] (2)<BR>[[octahedron|(3.3.3.3)]]
|[[Image:Rectified_cubic_honeycomb4.png|75px]]
|[[Image:Rectified cubic tiling.png|75px|]]
|[[Image:VF-rectified cubic.png|75px]]<BR>[[cuboid]]
|-
|J_12,32<BR>A₁₅<BR>W₁₄<BR>G₇
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-01.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_dot.png]]
|align=center|[[rectified cubic honeycomb|rectified cubic]]<BR>(cantellated alternate cubic)
|align=center|[[Image:octahedron.png|30px]] (1)<BR>[[octahedron|(3.3.3.3)]]
|align=center|&nbsp;
|align=center|[[Image:octahedron.png|30px]] (1)<BR>[[octahedron|(3.3.3.3)]]
|align=center|[[Image:Uniform polyhedron-33-t02.png|30px]] (4)<BR>[[cuboctahedron|(3.4.3.4)]]
|[[Image:Rectified_cubic_honeycomb3.png|75px]]
|[[Image:Rectified cubic tiling.png|75px|]]
|[[Image:VF-rectified cubic.png|75px]]<BR>[[cuboid]]
|-
|J₁₆<BR>A₃<BR>W₂<BR>G₂₈
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-11.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_dot.png]]
|align=center|[[bitruncated cubic honeycomb|bitruncated cubic]]<BR>(cantitruncated alternate cubic)
|align=center|[[Image:truncated octahedron.png|30px]] (1)<BR>[[truncated octahedron|(4.6.6)]]
|align=center|&nbsp;
|align=center|[[Image:truncated octahedron.png|30px]] (1)<BR>[[truncated octahedron|(4.6.6)]]
|align=center|[[Image:Uniform polyhedron-33-t012.png|30px]] (2)<BR>[[truncated octahedron|(4.6.6)]]
|[[Image:Bitruncated cubic honeycomb.png|75px]]
|[[Image:Bitruncated cubic tiling.png|75px|]]
|[[Image:VF-bitruncated cubic.png|75px]]<BR>isosceles [[tetrahedron]]
|-
|J₁₃<BR>A₁₄<BR>W₁₅<BR>G₈
|align=center|[[Image:CD_dot.png]][[Image:CD_3b.png]][[Image:CD_downbranch-10.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_ring.png]]
|align=center|[[truncated cubic honeycomb|truncated cubic]]<BR>(bicantellated alternate cubic)
|align=center|[[Image:truncated hexahedron.png|30px]] (2)<BR>[[truncated cube|(3.8.8)]]
|align=center|&nbsp;
|align=center|[[Image:truncated hexahedron.png|30px]] (2)<BR>[[truncated cube|(3.8.8)]]
|align=center|[[Image:Uniform polyhedron-33-t1.png|30px]] (1)<BR>[[octahedron|(3.3.3.3)]]
|[[Image:Truncated cubic honeycomb2.png|75px]]
|[[Image:Truncated cubic tiling.png|75px]]
|[[Image:VF-truncated cubic.png|75px]]<BR>[[square pyramid]]
|-
|J_11,15<BR>A₁<BR>W₁<BR>G₂₂
|align=center|[[Image:CD_dot.png]][[Image:CD_3b.png]][[Image:CD_downbranch-00.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_ring.png]]
|align=center|[[cubic honeycomb|cubic]]<BR>(trirectified alternate cubic)
|align=center|[[Image:hexahedron.png|30px]] (4)<BR>[[cube|(4.4.4)]]
|align=center|&nbsp;
|align=center|[[Image:hexahedron.png|30px]] (4)<BR>[[cube|(4.4.4)]]
|align=center|&nbsp;
|[[Image:Bicolor cubic honeycomb.png|75px]]
|[[Image:Cubic tiling.png|75px|]]
|[[Image:VF-cubic.png|75px]]<BR>[[octahedron]]
|-
|J₂₃<BR>A₁₆<BR>W₁₁<BR>G₅
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-00.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_ring.png]]
|align=center|[[runcinated alternated cubic honeycomb|runcinated alternated cubic]]
|align=center|[[Image:hexahedron.png|30px]] (1)<BR>[[cube]]
|align=center|&nbsp;
|align=center|[[Image:Small rhombicuboctahedron.png|30px]] (3)<BR>[[rhombicuboctahedron|3.4.4.4]]
|align=center|[[Image:tetrahedron.png|30px]] (1)<BR>[[tetrahedron|3.3.3]]
|[[Image:Runcinated alternated cubic honeycomb.jpg|75px]]
|[[Image:Runcinated alternated cubic tiling.png|75px|]]
|[[Image:VF-runcinated alternated cubic.png|75px]]
|-
|J₁₄<BR>A₁₇<BR>W₁₂<BR>G₉
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-01.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_ring.png]]
|align=center|[[cantellated cubic honeycomb|cantellated cubic]]<BR>(runcicantellated alternate cubic)
|align=center|[[Image:small rhombicuboctahedron.png|30px]] (1)<BR>[[small rhombicuboctahedron|(3.4.4.4)]]
|align=center|[[Image:Tetragonal prism.png|30px]] (2)<BR>[[cube|(4.4.4)]]
|align=center|[[Image:small rhombicuboctahedron.png|30px]] (1)<BR>[[small rhombicuboctahedron|(3.4.4.4)]]
|align=center|[[Image:Uniform_polyhedron-33-t02.png|30px]] (1)<BR>[[cuboctahedron|(3.4.3.4)]]
|[[Image:Cantellated cubic honeycomb.jpg|75px]]
|[[Image:Cantellated cubic tiling.png|75px|]]
|[[Image:VF-cantellated cubic.png|75px]]<BR>[[wedge (geometry)|wedge]]
|-
|J₂₄<BR>A₂₀<BR>W₁₆<BR>G₂₁
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-10.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_ring.png]]
|align=center|[[cantitruncated alternated cubic honeycomb|cantitruncated alternated cubic]]<BR>(or runcitruncated alternate cubic)
|align=center|&nbsp;
|align=center|[[Image:truncated hexahedron.png|30px]] (1)<BR>[[truncated cube|3.8.8]]
|align=center|[[Image:Great rhombicuboctahedron.png|30px]](2)<BR>[[truncated cuboctahedron|4.6.8]]
|align=center|[[Image:truncated tetrahedron.png|30px]] (1)<BR>[[truncated tetrahedron|3.6.6]]
|[[Image:Cantitruncated alternated cubic honeycomb.jpg|75px]]
|[[Image:Cantitruncated alternated cubic tiling.png|75px|]]
|[[Image:VF-cantitruncated alternated cubic.png|75px]]
|-
|J₁₇<BR>A₁₈<BR>W₁₃<BR>G₂₅
|align=center|[[Image:CD_ring.png]][[Image:CD_3b.png]][[Image:CD_downbranch-11.png]][[Image:CD_3b.png]][[Image:CD_4.png]][[Image:CD_ring.png]]
|align=center|[[cantitruncated cubic honeycomb|cantitruncated cubic]]<BR>(omnitruncated alternated cubic)
|align=center|[[Image:great rhombicuboctahedron.png|30px]] (1)<BR>[[great rhombicuboctahedron|(4.6.8)]]
|align=center|[[Image:Uniform polyhedron 222-t012.png|30px]] (1)<BR>[[cube|(4.4.4)]]
|align=center|[[Image:great rhombicuboctahedron.png|30px]] (1)<BR>[[great rhombicuboctahedron|(4.6.8)]]
|align=center|[[Image:Uniform polyhedron-33-t012.png|30px]](1)<BR>[[truncated octahedron|(4.6.6)]]
|[[Image:Cantitruncated cubic honeycomb.jpg|75px]]
|[[Image:Cantitruncated cubic tiling.png|75px|]]
|[[Image:VF-cantitruncated cubic.png|75px]]<BR>irregular [[tetrahedron]]
|}

=== P₄ group ===

There are 5 forms constructed from the [[p-group|P₄ group]], only the ''quarter cubic honeycomb'' is unique.

{|class="wikitable"
!rowspan=2|Referenced<BR>indices
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
!rowspan=2|Honeycomb name
!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)
!(1)
!(2)
!(3)
|-
|J_21,31,51<BR>A₂<BR>W₉<BR>G₁
|align=center|[[Image:CD_p4-1000.png]]
|align=center|[[Tetrahedral-octahedral honeycomb|alternated cubic]]
|align=center|&nbsp;
|align=center|[[Image:Uniform polyhedron-33-t0.png|30px]] (4)<BR>[[Tetrahedron|3.3.3]]
|align=center|[[Image:Uniform polyhedron-33-t1.png|30px]] (6)<BR>[[Octahedron|3.3.3.3]]
|align=center|[[Image:Uniform polyhedron-33-t2.png|30px]] (4)<BR>[[Tetrahedron|3.3.3]]
|[[Image:Tetrahedral-octahedral honeycomb2.png|75px]]
|[[Image:Alternated cubic tiling.png|75px|]]
|[[Image:VF-alternated cubic.png|75px]]<BR>[[cuboctahedron]]
|-
|J_12,32<BR>A₁₅<BR>W₁₄<BR>G₇
|align=center|[[Image:CD_p4-1010.png]]
|align=center|[[rectified cubic honeycomb|rectified cubic]]
|align=center|[[Image:Uniform polyhedron-33-t02.png|30px]] (2)<BR>[[cuboctahedron|(3.4.3.4)]]
|align=center|[[Image:Uniform polyhedron-33-t1.png|30px]] (1)<BR>[[octahedron|(3.3.3.3)]]
|align=center|[[Image:Uniform polyhedron-33-t02.png|30px]] (2)<BR>[[cuboctahedron|(3.4.3.4)]]
|align=center|[[Image:Uniform polyhedron-33-t1.png|30px]] (1)<BR>[[octahedron|(3.3.3.3)]]
|[[Image:Rectified_cubic_honeycomb2.png|75px]]
|[[Image:Rectified cubic tiling.png|75px|]]
|[[Image:VF-rectified cubic.png|75px]]<BR>[[cuboid]]
|-
|J_25,33<BR>A₁₃<BR>W₁₀<BR>G₆
|align=center|[[Image:CD_p4-1100.png]]
|align=center|[[quarter cubic honeycomb|quarter cubic]]
|align=center|[[Image:Tetrahedron.png|30px]] (1)<BR>[[tetrahedron|3.3.3]]
|align=center|[[Image:Tetrahedron.png|30px]] (1)<BR>[[tetrahedron|3.3.3]]
|align=center|[[Image:Truncated tetrahedron.png|30px]] (3)<BR>[[truncated tetrahedron|3.6.6]]
|align=center|[[Image:Truncated tetrahedron.png|30px]] (3)<BR>[[truncated tetrahedron|3.6.6]]
|[[Image:Quarter cubic honeycomb.png|75px]]
|[[Image:Bitruncated alternated cubic tiling.png|75px|]]
|[[Image:VF-bitruncated alternated cubic.png|75px]]
|-
|J_22,34<BR>A₂₁<BR>W₁₇<BR>G₁₀
|align=center|[[Image:CD_p4-1110.png]]
|align=center|[[truncated alternated cubic honeycomb|truncated alternated cubic]]
|align=center|[[Image:Truncated tetrahedron.png|30px]] (1)<BR>[[Truncated tetrahedron|3.6.6]]
|align=center|[[Image:Uniform polyhedron-33-t02.png|30px]] (1)<BR>[[cuboctahedron|3.4.3.4]]
|align=center|[[Image:Truncated tetrahedron.png|30px]] (1)<BR>[[Truncated tetrahedron|3.6.6]]
|align=center|[[Image:Uniform polyhedron-33-t012.png|30px]] (2)<BR>[[truncated octahedron|4.6.6]]
||[[Image:Truncated alternated cubic honeycomb.jpg|75px]]
|[[Image:Truncated alternated cubic tiling.png|75px|]]
|[[Image:VF-Truncated alternated cubic.png|75px]]
|-
|J₁₆<BR>A₃<BR>W₂<BR>G₂₈
|align=center|[[Image:CD_p4-1111.png]]
|align=center|[[bitruncated cubic honeycomb|bitruncated cubic]]
|align=center|[[Image:Uniform polyhedron-33-t012.png|30px]] (1)<BR>[[truncated octahedron|(4.6.6)]]
|align=center|[[Image:Uniform polyhedron-33-t012.png|30px]] (1)<BR>[[truncated octahedron|(4.6.6)]]
|align=center|[[Image:Uniform polyhedron-33-t012.png|30px]] (1)<BR>[[truncated octahedron|(4.6.6)]]
|align=center|[[Image:Uniform polyhedron-33-t012.png|30px]] (1)<BR>[[truncated octahedron|(4.6.6)]]
|[[Image:Bitruncated cubic honeycomb4.png|75px]]
|[[Image:Bitruncated cubic tiling.png|75px|]]
|[[Image:VF-bitruncated cubic.png|75px]]<BR>isosceles [[tetrahedron]]
|}

=== Gyrated and elongated forms ===
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
|h{4,3,4}:g
|align=center|[[gyrated alternated cubic honeycomb|gyrated alternated cubic]]
|align=center|[[tetrahedron]] (8)<BR>[[octahedron]] (6)
|[[Image:Gyrated alternated cubic.jpg|75px]]
|[[Image:Gyrated alternated cubic.png|75px|]]
|[[Image:VF-gyrated alternated cubic.png|75px]]
|-
|J₆₁<BR>A_?<BR>G₃
|h{4,3,4}:ge
|align=center|[[Gyroelongated alternated cubic honeycomb|gyroelongated alternated cubic]]
|align=center|[[triangular prism]] (6)<BR>[[tetrahedron]] (4)<BR>[[octahedron]] (3)
|[[Image:Gyroelongated alternated cubic honeycomb.png|75px]]
|[[Image:Gyroelongated alternated cubic tiling.png|75px|]]
| -
|-
|J₆₂<BR>A_?<BR>G₄
|h{4,3,4}:e
|align=center|[[Elongated alternated cubic honeycomb|elongated alternated cubic]]
|align=center|[[triangular prism]] (6)<BR>[[tetrahedron]] (4)<BR>[[octahedron]] (3)
|[[Image:Elongated alternated cubic honeycomb.png|75px|]]
|[[Image:Elongated alternated cubic tiling.png|75px|]]
|[[Image:VF-extended alternated cubic.png|75px]]
|-
|J₆₃<BR>A_?<BR>G₁₂
|{3,6}:g x {&infin;}
|align=center|[[Gyrated triangular prismatic honeycomb|gyrated triangular prismatic]]
|align=center|[[triangular prism]] (12)
|[[Image:Gyrated triangular prismatic honeycomb.png|75px|]]
|[[Image:Gyrated triangular prismatic tiling.png|75px|]]
|[[Image:VF-gyrated prismatic triangular.png|75px]]
|-
|J₆₄<BR>A_?<BR>G₁₅
|{3,6}:ge x {&infin;}
|align=center|[[gyroelongated triangular prismatic honeycomb|gyroelongated triangular prismatic]]
|align=center|[[triangular prism]] (6)<BR>[[cube]] (4)
|[[Image:Gyroelongated triangular prismatic honeycomb.png|75px|]]
|[[Image:Gyroelongated triangular prismatic tiling.png|75px|]]
|[[Image:VF-prismatic extended triangular.png|75px]]
|}

=== 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 R₃xW₂, [4,4] x [&infin;], prismatic group ====

There's only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form.

{|class="wikitable"
!Indices
![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
!Honeycomb name
!Plane<BR>tiling
!Solids<BR>(Partial)
!Tiling
|-
|J_11,15<BR>A₁<BR>G₂₂
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>{4,4} x {&infin;}
|align=center|[[Cubic honeycomb|Cubic]]<BR>(Square prismatic)
|[[Square tiling|(4.4.4.4)]]
|[[Image:Partial cubic honeycomb.png|80px]]
|[[Image:Uniform_tiling_44-t0.png|80px]]
|-
|J₄₅<BR>A₆<BR>G₂₄
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t_0,1{4,4} x {&infin;}
|align=center|[[Truncated square prismatic honeycomb|Truncated/Bitruncated square prismatic]]
|[[Truncated square tiling|(4.8.8)]]
|[[Image:Truncated square prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_44-t01.png|80px]]
|-
|J_11,15<BR>A₁<BR>G₂₂
|align=center|[[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t₁{4,4} x {&infin;}
|align=center|[[Cubic honeycomb|Cubic]]<BR>(Rectified square prismatic)
|[[Square tiling|(4.4.4.4)]]
|[[Image:Square prismatic 2-color honeycomb.png|80px]]
|[[Image:Uniform_tiling_44-t1.png|80px]]
|-
|J_11,15<BR>A₁<BR>G₂₂
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t_0,2{4,4} x {&infin;}
|align=center|[[Cubic honeycomb|Cubic]]<BR>(Cantellated square prismatic)
|[[Square tiling|(4.4.4.4)]]
|[[Image:Partial cubic honeycomb.png|80px]]
|[[Image:Uniform_tiling_44-t02.png|80px]]
|-
|J₄₅<BR>A₆<BR>G₂₄
|align=center|[[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_4.png]][[Image:CDW_ring.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t_0,1,2{4,4} x {&infin;}
|align=center|[[Truncated square prismatic honeycomb|Truncated square prismatic]]<BR>(Omnitruncated square prismatic)
|[[Truncated square tiling|(4.8.8)]]
|[[Image:Truncated square prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_44-t012.png|80px]]
|-
|J₄₄<BR>A₁₁<BR>G₁₄
|align=center|[[Image:CDW_hole.png]][[Image:CDW_4.png]][[Image:CDW_hole.png]][[Image:CDW_4.png]][[Image:CDW_hole.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>s{4,4} x {&infin;}
|align=center|[[Snub square prismatic honeycomb|Snub square prismatic]]
|[[Snub square tiling|(3.3.4.3.4)]]
|[[Image:Snub square prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_44-snub.png|80px]]
|}

==== The V₃xW₂, [6,3] x [&infin;] prismatic group ====
{|class="wikitable"
!Indices
![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
!Honeycomb name
!Plane<BR>tiling
!Solids<BR>(Partial)
!Tiling
|-
|J₄₂<BR>A₅<BR>G₂₆
|align=center|[[Image:CDW_ring.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t₀{6,3} x {&infin;}
|align=center|[[Hexagonal prismatic honeycomb|Hexagonal prismatic]]
|[[hexagonal tiling|(6³)]]
|[[Image:Hexagonal prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-t0.png|80px]]
|-
|J₄₆<BR>A₇<BR>G₁₉
|align=center|[[Image:CDW_ring.png]][[Image:CDW_6.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t_0,1{6,3} x {&infin;}
|align=center|[[Truncated hexagonal prismatic honeycomb|Truncated hexagonal prismatic]]
|[[Truncated hexagonal tiling|(3.12.12)]]
|[[Image:Truncated hexagonal prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-t01.png|80px]]
|-
|J₄₃<BR>A₈<BR>G₁₈
|align=center|[[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t₁{6,3} x {&infin;}
|align=center|[[Triangular-hexagonal prismatic honeycomb|Trihexagonal prismatic]]
|[[Trihexagonal tiling|(3.6.3.6)]]
|[[Image:Triangular-hexagonal prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-t1.png|80px]]
|-
|J₄₂<BR>A₅<BR>G₂₆
|align=center|[[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t_1,2{6,3} x {&infin;}
|align=center|''Truncated triangular prismatic''<BR>[[Hexagonal prismatic honeycomb|Hexagonal prismatic]]
|[[Hexagonal tiling|(6.6.6)]]
||[[Image:Truncated triangular prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-t12.png|80px]]
|-
|J₄₁<BR>A₄<BR>G₁₁
|align=center|[[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t₂{6,3} x {&infin;}
|align=center|[[Triangular prismatic honeycomb|Triangular prismatic]]
|[[triangular tiling|(3⁶)]]
|[[Image:Triangular prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-t2.png|80px]]
|-
|J₄₇<BR>A₉<BR>G₁₆
|align=center|[[Image:CDW_ring.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t_0,2{6,3} x {&infin;}
|align=center|[[Rhombitriangular-hexagonal prismatic honeycomb|Rhombi-trihexagonal prismatic]]
|[[Small rhombitrihexagonal tiling|(3.4.6.4)]]
|[[Image:Rhombitriangular-hexagonal prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-t02.png|80px]]
|-
|J₄₉<BR>A₁₀<BR>G₂₃
|align=center|[[Image:CDW_ring.png]][[Image:CDW_6.png]][[Image:CDW_ring.png]][[Image:CDW_3.png]][[Image:CDW_ring.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>t_0,1,2{6,3} x {&infin;}
|align=center|[[Omnitruncated triangular-hexagonal prismatic honeycomb|Omnitruncated trihexagonal prismatic]]
|[[Great rhombitrihexagonal tiling|(4.6.12)]]
|[[Image:Omnitruncated triangular-hexagonal prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-t012.png|80px]]
|-
|J₄₈<BR>A₁₂<BR>G₁₇
|align=center|[[Image:CDW_hole.png]][[Image:CDW_6.png]][[Image:CDW_hole.png]][[Image:CDW_3.png]][[Image:CDW_hole.png]][[Image:CDW_2.png]][[Image:CDW_ring.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]]<BR>s{6,3} x {&infin;}
|align=center|[[Snub triangular-hexagonal prismatic honeycomb|Snub trihexagonal prismatic]]
|[[Snub hexagonal tiling|(3.3.3.3.6)]]
|[[Image:Snub triangular-hexagonal prismatic honeycomb.png|80px]]
|[[Image:Uniform_tiling_63-snub.png|80px]]
|-
|J₆₅<BR>A_11'<BR>G₁₃
|{3,6}:e x {&infin;}
|align=center|[[elongated triangular prismatic honeycomb|elongated triangular prismatic]]
|align=center|[[elongated triangular tiling|3.3.3.4.4]]
|[[Image:Elongated triangular prismatic honeycomb.png|80px]]
|[[Image:Tile 33344.svg|80px]]
|}

==Examples==
All 28 of these tessellations are found in [[crystal]] arrangements.

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]
[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.
-->

== References ==
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
* [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
* {{cite book | first=Robert | last=Williams | authorlink=Robert Williams | title=The Geometrical Foundation of Natural Structure: A Source Book of Design | publisher=Dover Publications, Inc | year=1979 | id=ISBN 0-486-23729-X }}
* {{cite book | first=Keith | last=Critchlow | authorlink=Keith Critchlow | title=Order in Space: A design source book | publisher=Viking Press| year=1970 | id=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]], ''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 (1905) 75–129.
* [[Duncan MacLaren Young Sommerville|D. M. Y. Sommerville]], ''An Introduction to the Geometry of '''n''' Dimensions.'' New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes

==External links==
* {{mathworld | title = Honeycomb | urlname = Honeycomb}}
*[http://polyhedra.doskey.com/UniformHoneycombs.html Uniform Honeycombs in 3-Space] VRML models
*[http://web.ukonline.co.uk/polyhedra/honeycombs/honeycombs.htm Elementary Honeycombs]
* [http://arxiv.org/PS_cache/math/pdf/9906/9906034.pdf Uniform partitions of 3-space, their relatives and embedding] PDF, 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

[[Category:Honeycombs (geometry)]]

[[eo:Konveksa uniforma kaheligo de eŭklida 3-spaco]]