Editing Convex uniform honeycomb (section)

== 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>