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Uniform honeycombs in three-space

by Norman Johnson, 2003

Besides the generic term polytope, we have the more specific terms polygon, polyhedron, and polychoron for polytopes of two, three, and four dimensions. Likewise, honeycomb is the generic term for a polytopal covering of n-space, but one can speak more specifically of a partition of a line, a tessellation of a plane, or a cellulation of 3-space.

There is a standard name, a little simpler than George Olshevsky's panoploid honeycomb, for a honeycomb that fills n-space without gaps or overlaps, namely, a tiling. A tiling is convex if all the tiles are convex.

A number of uniform tilings of Euclidean 3-space were described by Alfredo Andreini (1905). Others were discovered by J. C. P. Miller and H. S. M. Coxeter (see Coxeter 1932, 1940, 1985). In 1959 I found another one (No. 23 in George's list), which I described in my 1966 Ph.D. thesis. So far as I know, the first appearance in the literature of the presumably complete list of 28 uniform tilings was a note by Branko Grunbaum (1994).

To my knowledge, there is as yet no proof that there are exactly 28 uniform tilings of Euclidean 3-space, but it seems highly likely that this is the case. All of these tilings have counterparts in higher-dimensional space.

Most of the uniform tilings of 3-space can be derived by Wythoff's construction either from the regular tiling of cubes or from the eleven uniform tilings of the plane, but there are also a few anomalous ones.

Here I describe each of the twenty-eight, giving an extended Schlafli symbol, the name I use, the number of cell polyhedra of each type at a vertex, and a cross reference to the list recently by George. Two tilings with alternative derivations are listed twice.

Cells

The thirteen uniform solids listed below occur as cell polyhedra of uniform tilings of 3-space. Some of these may occur with less than their full symmetry and are given in alternative forms.

{3,3} = T (tetrahedron, , ) or v{3, 1} = 3V (triangular pyramid) or s{2}h{} = 2Q (disphenoid, or , ) {3,4} = O (octahedron, , ) or s{3}h{} = 3Q (triangular antiprism, or , ) or v{4, 2} = 4X (square bipyramid, ) {4,3} = C (cube, ) or {4}×{} = 4P (square prism, , ) or t{2}×{} = t2P (rectangular prism, , ) t{3,3} = tT (truncated tetrahedron, ), t{3,4} = tO (truncated octahedron, ), t{4,3} = tC (truncated cube, ), r{3,4} = CO (cuboctahedron, ), rr{3,4} = rCO (rhombicuboctahedron, ), tr{3,4} = tCO (truncated cuboctahedron, ),	{3}×{} = 3P (triangular prism, , ), or v{1}×{} = v1P or u{1}×{} = u1P or w{2, 1} = 2W (fastigium, ) {6}×{} = 6P (hexagonal prism, , ) or t{3}×{} = t3P (ditrigonal prism, , ) t{4}×{} = t4P (ditetragonal prism, , ) t{6}×{} = t6P (dihexagonal prism, , )

Generalized truncations

Index	Name Symbols	Cells				Subtilings
[1]	Cubic cellulation {4,3,4},	8 C				Q
[2]	Rectified cubic cellulation r{4,3,4},	2 O	4 CO			rQ	DH
[3]	Truncated cubic cellulation t{4,3,4},	1 O	4 tC			tQ
[4]	Cantellated cubic cellulation e₂{4,3,4},	1 CO	2 rCO	2 4P
[1] REPEAT	Runcinated cubic cellulation e₃{4,3,4},	2 C	6 4P			rrQ
[5]	Bitruncated cubic cellulation 2t{4,3,4},	4 tO
[6]	Cantitruncated cubic cellulation e₂t{4,3,4},	1 tO	2 tCO	1 4P
[7]	Runcitruncated cubic cellulation e₃t{4,3,4},	1 tC	1 rCO	1 4P	4 t4P	tQ
[8]	Omnitruncated cubic cellulation e_2,3t{4,3,4},	2 tCO	2 t4P

Alternated tilings

Index	Name Symbols	Cells			Subtilings
[11]	Half cubic cellulation h{4,3,4}, =	8 T =	6 O		D
[12]	Runcic cubic cellulation h₃{4,3,4}, =	1 T =	3 rCO	1 C
[13]	Cantic cubic cellulation h₂{4,3,4}, =	2 tT =	2 tO	1 CO
[14]	Runcicantic cubic cellulation h_2,3{4,3,4}, =	1 tT =	2 tCO	1 tC
[15]	Quarter cubic cellulation q{4,3,4}, =	2 T =	6 tT		DH

Prismatic tilings

Index	Name Symbols	Cells			Subtilings
[9]	Tomo-square prismatic cellulation t{4,4}×{∞},	2 4P	4 t4P		tQ
[10]	Simo-square prismatic cellulation s{4,4}×{∞} or	4 4P	6 v1P		sQ
[16]	Hexagonal prismatic cellulation {6,3}×{∞},	6 6P			H
[17]	Trihexagonal prismatic cellulation r{3,6}×{∞},	4 3P	4 6P		DH	rQ
[18]	Triangular prismatic cellulation {3,6}×{∞},	12 3P			D	Q
[19]	Tomo-hexagonal prismatic cellulation t{6,3}×{∞},	2 4P	2 6P	2 t6P	tH
[20]	Rhombitrihexagonal prismatic cellulation rr{3,6}×{∞},	2 3P	2 6P	4 t2P	rDH
[21]	Tomo-trihexagonal prismatic cellulation tr{3,6}×{∞},	2 t3P	2 t6P	2 t2P	tDH
[22]	Simo-trihexagonal prismatic cellulation sr{3,6}×{∞},	2 3P	2 6P	6 u1P	sDH

Antiprismatic tilings

Index	Name Symbols	Cells		Subtilings		Image
[11] REPEAT	Tetragonal disphenoidal cellulation h{4,4}h{∞},	8 2Q	6 4X	rQ	hH
[26]	Triangular antiprismatic cellulation h{6,3}h{∞},	6 3Q	8 3V		hH

Elongated tilings

Index	Name Symbols	Cells			Subtilings		Image
[24]	Elongated antiprismatic prismatic cellulation s{∞}h₁{∞}×{∞} or	6 2W	4 4P		Q	D:e
[28]	Elongated triangular antiprismatic cellulation {3,6}h₁{∞},	4 3V	3 3Q	6 3P		D

Anomalous tilings

Index	Name Symbols	Cells			Subtilings
[23]	Parasquare fastigial cellulation {4,4}f{∞}	12 2W			Q
[25]	Elongated parasquare fastigial cellulation {4,4}f₁{∞}	6 2W	4 4P		Q
[27]	Elongated triangular gyroprismatic cellulation {3,6}g₁{∞}	4 3V	3 3Q	6 3P	D

References

A. Andreini 1905 "Sulle reti di poliedri regolari e semiregolari," Mem. Soc. Ital. Sci. (3) 14, 75-129.
H. S. M. Coxeter
- 1932 "The polytopes with regular-prismatic vertex figures, II," Proc. London Math. Soc. (2) 34, 126-189.
- 1940 "Regular and semi-regular polytopes, I," Math. Z. 46, 380-407.
- 1985 "Regular and semi-regular polytopes, II," Math. Z. 188, 559-591.
Norman Johnson The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966[2]
B. Grunbaum 1994 "Uniform tilings of 3-space," Geombinatorics 4, 49-56.