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Convex uniform honeycomb

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

Twenty-eight such honeycombs exist:

the familiar cubic honeycomb and 7 truncations thereof;
the alternated cubic honeycomb and 4 truncations thereof;
10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb);
5 modifications of some of the above by elongation and/or gyration.

They can be considered the three-dimensional analogue to the uniform tilings of the plane.

History

1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions, including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra.
1905: Alfredo Andreini enumerated 25 of these tessellations.
1991: Norman Johnson's manuscript Uniform Polytopes identified the 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 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 polychorons in 4-space).

Only 14 of the convex uniform polyhedra appear in these patterns:

three of the five Platonic solids,
six of the thirteen Archimedean solids, and
five of the infinite family of prisms.

Names

This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (non-regular) polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations.

The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in Uniform 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

The fundamental infinite Coxeter groups for 3-space are:

The B^~₃, [4,3,4], cubic, (8 unique forms plus one alternation)
The C^~₃, [4,3^1,1], alternated cubic, (11 forms, 3 new)
The A^~₃ cyclic group, (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 B^~₂xI^~₁, [4,4]x[∞] prismatic group, (2 new forms)
The H^~₂xI^~₁, [6,3]x[∞] prismatic group, (7 unique forms)
The A^~₂xI^~₁, [Δ]x[∞] prismatic group, (No new forms)
The I^~₁xI^~₁xI^~₁, [∞]x[∞]x[∞] prismatic group, (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.

Noncompact forms

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

Families:

B^~₂xA₁: [4,4]x[] Cubic prismatic slab honeycomb (3 forms)
H^~₂xA₁: [6,3]x[] Tri-hexagonal prismatic slab honeycomb (8 forms)
A^~₂xA₁: [Δ]x[] triangular prismatic slab (No new forms)
I^~₁xA₁xA₁: [∞]x[]x[] = Cubic column honeycomb (1 form)
I₂(p)xI^~₁: [p]x[∞] Prismatic column honeycomb
I^~₁xI^~₁xA₁: [∞]x[∞]x[] = [4,4]x[] - = (Same as cubic slab honeycomb family)

Examples (partially drawn): Cubic slab honeycomb and Alternated hexagonal slab honeycomb.

Hyperbolic forms

There are 9 families of compact uniform honeycombs in hyperbolic 3-space:

[3,5,3] (Order-3 icosahedral honeycomb)
[5,3,4] (Order-5 cubic honeycomb/Order-4 dodecahedral honeycomb)
[5,3,5] (Order-5 dodecahedral honeycomb)
[5,3^1,1] *****

The B^~₃, [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.)

Reference Indices	Coxeter-Dynkin and Schläfli symbols	Honeycomb name	Cell counts/vertex and positions in cubic honeycomb
Reference Indices	Coxeter-Dynkin and Schläfli symbols	Honeycomb name	(0)	(1)	(2)	(3)	Solids (Partial)	Frames (Perspective)	vertex figure
J_11,15 A₁ W₁ G₂₂	t₀{4,3,4}	cubic				8 (4.4.4)			octahedron
J_12,32 A₁₅ W₁₄ G₇	t₁{4,3,4}	rectified cubic	2 (3.3.3.3)			4 (3.4.3.4)			cuboid
J₁₃ A₁₄ W₁₅ G₈	t_0,1{4,3,4}	truncated cubic	1 (3.3.3.3)			4 (3.8.8)			square pyramid
J₁₄ A₁₇ W₁₂ G₉	t_0,2{4,3,4}	cantellated cubic	1 (3.4.3.4)	2 (4.4.4)		2 (3.4.4.4)			wedge
J_11,15	t_0,3{4,3,4}	runcinated cubic (same as regular cubic)	1 (4.4.4)	3 (4.4.4)	3 (4.4.4)	1 (4.4.4)			octahedron
J₁₆ A₃ I^~₁(∞) G₂₈	t_1,2{4,3,4}	bitruncated cubic	2 (4.6.6)			2 (4.6.6)			isosceles tetrahedron
J₁₇ A₁₈ W₁₃ G₂₅	t_0,1,2{4,3,4}	cantitruncated cubic	1 (4.6.6)	1 (4.4.4)		2 (4.6.8)			irregular tetrahedron
J₁₈ A₁₉ W₁₉ G₂₀	t_0,1,3{4,3,4}	runcitruncated cubic	1 (3.4.4.4)	1 (4.4.4)	2 (4.4.8)	1 (3.8.8)			oblique trapezoidal pyramid
J₁₉ A₂₂ W₁₈ G₂₇	t_0,1,2,3{4,3,4}	omnitruncated cubic	1 (4.6.8)	1 (4.4.8)	1 (4.4.8)	1 (4.6.8)			irregular tetrahedron
J_21,31,51 A₂ W₉ G₁	h₀{4,3,4}	alternated cubic	6 3.3.3.3			8 3.3.3			cuboctahedron

C^~₄, h[4,3,4], [4,3^1,1] group

The C^~₄ 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 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.

Referenced indices	Coxeter-Dynkin diagram	Honeycomb name	Cells by location (and count around each vertex)				Solids (Partial)	Frames (Perspective)	vertex figure
Referenced indices	Coxeter-Dynkin diagram	Honeycomb name	(0)	(1)	(0')	(3)	Solids (Partial)	Frames (Perspective)	vertex figure
J_21,31,51 A₂ W₉ G₁	align=center	alternated cubic			(6) 3.3.3.3	(8) 3.3.3			cuboctahedron
J_22,34 A₂₁ W₁₇ G₁₀	align=center	truncated alternated cubic		(1) 3.4.3.4	(2) 4.6.6	(2) 3.6.6
J_12,32 A₁₅ W₁₄ G₇	align=center	rectified cubic (rectified alternate cubic)	(2) (3.4.3.4)		(2) (3.4.3.4)	(2) (3.3.3.3)			cuboid
J_12,32 A₁₅ W₁₄ G₇	align=center	rectified cubic (cantellated alternate cubic)	(1) (3.3.3.3)		(1) (3.3.3.3)	(4) (3.4.3.4)			cuboid
J₁₆ A₃ I^~₁(∞) G₂₈	align=center	bitruncated cubic (cantitruncated alternate cubic)	(1) (4.6.6)		(1) (4.6.6)	(2) (4.6.6)			isosceles tetrahedron
J₁₃ A₁₄ W₁₅ G₈	align=center	truncated cubic (bicantellated alternate cubic)	(2) (3.8.8)		(2) (3.8.8)	(1) (3.3.3.3)			square pyramid
J_11,15 A₁ W₁ G₂₂	align=center	cubic (trirectified alternate cubic)	(4) (4.4.4)		(4) (4.4.4)				octahedron
J₂₃ A₁₆ W₁₁ G₅	align=center	runcinated alternated cubic	(1) cube		(3) 3.4.4.4	(1) 3.3.3
J₁₄ A₁₇ W₁₂ G₉	align=center	cantellated cubic (runcicantellated alternate cubic)	(1) (3.4.4.4)	(2) (4.4.4)	(1) (3.4.4.4)	(1) (3.4.3.4)			wedge
J₂₄ A₂₀ W₁₆ G₂₁	align=center	cantitruncated alternated cubic (or runcitruncated alternate cubic)		(1) 3.8.8	(2) 4.6.8	(1) 3.6.6
J₁₇ A₁₈ W₁₃ G₂₅	align=center	cantitruncated cubic (omnitruncated alternated cubic)	(1) (4.6.8)	(1) (4.4.4)	(1) (4.6.8)	(1) (4.6.6)			irregular tetrahedron

A^~₃ group

There are 5 forms constructed from the A^~₃ group, only the quarter cubic honeycomb is unique.

Referenced indices	Coxeter-Dynkin diagram	Honeycomb name	Cells by location (and count around each vertex)				Solids (Partial)	Frames (Perspective)	vertex figure
Referenced indices	Coxeter-Dynkin diagram	Honeycomb name	(0)	(1)	(2)	(3)	Solids (Partial)	Frames (Perspective)	vertex figure
J_21,31,51 A₂ W₉ G₁	align=center	alternated cubic		(4) 3.3.3	(6) 3.3.3.3	(4) 3.3.3			cuboctahedron
J_12,32 A₁₅ W₁₄ G₇	align=center	rectified cubic	(2) (3.4.3.4)	(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.3.3.3)			cuboid
J_25,33 A₁₃ W₁₀ G₆	align=center	quarter cubic	(1) 3.3.3	(1) 3.3.3	(3) 3.6.6	(3) 3.6.6
J_22,34 A₂₁ W₁₇ G₁₀	align=center	truncated alternated cubic	(1) 3.6.6	(1) 3.4.3.4	(1) 3.6.6	(2) 4.6.6
J₁₆ A₃ I^~₁(∞) G₂₈	align=center	bitruncated cubic	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.6)			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.

Referenced indices	symbol	Honeycomb name	cell types (# at each vertex)	vertex figure
J₅₂ A_2' G₂	h{4,3,4}:g	gyrated alternated cubic	tetrahedron (8) octahedron (6)	triangular orthobicupola
J₆₁ A_? G₃	h{4,3,4}:ge	gyroelongated alternated cubic	triangular prism (6) tetrahedron (4) octahedron (3)	-
J₆₂ A_? G₄	h{4,3,4}:e	elongated alternated cubic	triangular prism (6) tetrahedron (4) octahedron (3)
J₆₃ A_? G₁₂	{3,6}:g x {∞}	gyrated triangular prismatic	triangular prism (12)
J₆₄ A_? G₁₅	{3,6}:ge x {∞}	gyroelongated triangular prismatic	triangular prism (6) cube (4)

Prismatic stacks

Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The vertex figure of each is an irregular bipyramid whose faces are isosceles triangles.

The B^~₂xI^~₁(∞), [4,4] x [∞], 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.

Indices	Coxeter-Dynkin and Schläfli symbols	Honeycomb name	Plane tiling
J_11,15 A₁ G₂₂	{4,4} x {∞}	Cubic (Square prismatic)	(4.4.4.4)
J₄₅ A₆ G₂₄	t_0,1{4,4} x {∞}	Truncated/Bitruncated square prismatic	(4.8.8)
J_11,15 A₁ G₂₂	t₁{4,4} x {∞}	Cubic (Rectified square prismatic)	(4.4.4.4)
J_11,15 A₁ G₂₂	t_0,2{4,4} x {∞}	Cubic (Cantellated square prismatic)	(4.4.4.4)
J₄₅ A₆ G₂₄	t_0,1,2{4,4} x {∞}	Truncated square prismatic (Omnitruncated square prismatic)	(4.8.8)
J₄₄ A₁₁ G₁₄	s{4,4} x {∞}	Snub square prismatic	(3.3.4.3.4)

The H^~₂xI^~₁(∞), [6,3] x [∞] prismatic group

Indices	Coxeter-Dynkin and Schläfli symbols	Honeycomb name	Plane tiling
J₄₂ A₅ G₂₆	t₀{6,3} x {∞}	Hexagonal prismatic	(6³)
J₄₆ A₇ G₁₉	t_0,1{6,3} x {∞}	Truncated hexagonal prismatic	(3.12.12)
J₄₃ A₈ G₁₈	t₁{6,3} x {∞}	Trihexagonal prismatic	(3.6.3.6)
J₄₂ A₅ G₂₆	t_1,2{6,3} x {∞}	Truncated triangular prismatic Hexagonal prismatic	(6.6.6)
J₄₁ A₄ G₁₁	t₂{6,3} x {∞}	Triangular prismatic	(3⁶)
J₄₇ A₉ G₁₆	t_0,2{6,3} x {∞}	Rhombi-trihexagonal prismatic	(3.4.6.4)
J₄₉ A₁₀ G₂₃	t_0,1,2{6,3} x {∞}	Omnitruncated trihexagonal prismatic	(4.6.12)
J₄₈ A₁₂ G₁₇	s{6,3} x {∞}	Snub trihexagonal prismatic	(3.3.3.3.6)
J₆₅ A_11' G₁₃	{3,6}:e x {∞}	elongated triangular prismatic	3.3.3.4.4

Examples

All 28 of these tessellations are found in crystal arrangements.

The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s). Octet trusses are now among the most common types of truss used in construction.

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 Uniform Polytopes, Manuscript (1991)
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.
Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. 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

(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)

A. Andreini, 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.
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