Quarter cubic honeycomb

The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – consists of four such units of the cubic honeycomb.

Quarter cubic honeycomb
Type	Uniform honeycomb
Family	Truncated simplectic honeycomb Quarter hypercubic honeycomb
Indexing[1]	J25,33, A13 W10, G6
Schläfli symbol	t0,1{3^[4]} or q{4,3,4}
Coxeter-Dynkin diagram	= =
Cell types	{3,3} (3.6.6)
Face types	{3}, {6}
Vertex figure	(isosceles triangular antiprism)
Space group	Fd3m (227)
Coxeter group	${\displaystyle {\tilde {A}}_{3}}$×22, [[3^[4]]]
Dual	oblate cubille Cell: (1/4 of rhombic dodecahedron)
Properties	vertex-transitive, edge-transitive

It is vertex-transitive with 6 truncated tetrahedra and 2 tetrahedra around each vertex.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

It is one of the 28 convex uniform honeycombs.

The faces of this honeycomb's cells form four families of parallel planes, each with a 3.6.3.6 tiling.

Its vertex figure is an isosceles antiprism: two equilateral triangles joined by six isosceles triangles.

John Horton Conway calls this honeycomb a truncated tetrahedrille, and its dual oblate cubille.

The vertices and edges represent a Kagome lattice in three dimensions.[2]

Construction

The quarter cubic honeycomb can be constructed in slab layers of truncated tetrahedra and tetrahedral cells, seen as two trihexagonal tilings. Two tetrahedra are stacked by a vertex and a central inversion. In each trihexagonal tiling, half of the triangles belong to tetrahedra, and half belong to truncated tetrahedra. These slab layers must be stacked with tetrahedra triangles to truncated tetrahedral triangles to construct the uniform quarter cubic honeycomb. Slab layers of hexagonal prisms and triangular prisms can be alternated for elongated honeycombs, but these are also not uniform.

trihexagonal tiling:

Symmetry

Cells can be shown in two different symmetries. The reflection generated form represented by its Coxeter-Dynkin diagram has two colors of truncated cuboctahedra. The symmetry can be doubled by relating the pairs of ringed and unringed nodes of the Coxeter-Dynkin diagram, which can be shown with one colored tetrahedral and truncated tetrahedral cells.

Two uniform colorings

Symmetry	${\displaystyle {\tilde {A}}_{3}}$, [3^[4]]	${\displaystyle {\tilde {A}}_{3}}$×2, [[3^[4]]]
Space group	F43m (216)	Fd3m (227)
Coloring
Vertex figure
Vertex figure symmetry	C3v [3] (*33) order 6	D3d [2^+,6] (2*3) order 12

Related polyhedra

The subset of hexagonal faces of this honeycomb contains a regular skew apeirohedron {6,6|3}.

Four sets of parallel planes of trihexagonal tilings exist throughout this honeycomb.

This honeycomb is one of five distinct uniform honeycombs[3] constructed by the ${\displaystyle {\tilde {A}}_{3}}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A3 honeycombs
Space group	Fibrifold	Square symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
F43m (216)	1^o:2	a1	[3^[4]]		${\displaystyle {\tilde {A}}_{3}}$	(None)
Fm3m (225)	2^−:2	d2	<[3^[4]]> ↔ [4,3^1,1]	↔	${\displaystyle {\tilde {A}}_{3}}$×21 ↔ ${\displaystyle {\tilde {B}}_{3}}$	1, 2
Fd3m (227)	2^+:2	g2	[[3^[4]]] or [2^+[3^[4]]]	↔	${\displaystyle {\tilde {A}}_{3}}$×22	3
Pm3m (221)	4^−:2	d4	<2[3^[4]]> ↔ [4,3,4]	↔	${\displaystyle {\tilde {A}}_{3}}$×41 ↔ ${\displaystyle {\tilde {C}}_{3}}$	4
I3 (204)	8^−o	r8	[4[3^[4]]]^+ ↔ [[4,3<sup>+</sup>,4]]	↔	½${\displaystyle {\tilde {A}}_{3}}$×8 ↔ ½${\displaystyle {\tilde {C}}_{3}}$×2	(*)
Im3m (229)	8^o:2		[4[3^[4]]] ↔ [[4,3,4]]		${\displaystyle {\tilde {A}}_{3}}$×8 ↔ ${\displaystyle {\tilde {C}}_{3}}$×2	5

C3 honeycombs
Space group	Fibrifold	Extended symmetry	Extended diagram	Order	Honeycombs
Pm3m (221)	4^−:2	[4,3,4]		×1	1, 2, 3, 4, 5, 6
Fm3m (225)	2^−:2	[1^+,4,3,4] ↔ [4,3^1,1]	↔	Half	7, 11, 12, 13
I43m (217)	4^o:2	[[(4,3,4,2^+)]]		Half × 2	(7),
Fd3m (227)	2^+:2	[[1^+,4,3,4,1^+]] ↔ [[3^[4]]]	↔	Quarter × 2	10,
Im3m (229)	8^o:2	[[4,3,4]]		×2	(1), 8, 9

The Quarter cubic honeycomb is related to a matrix of 3-dimensional honeycombs: q{2p,4,2q}

Euclidean/hyperbolic(paracompact/noncompact) quarter honeycombs q{p,3,q}
p \ q	4	6	8	... ∞
4	q{4,3,4} ↔ ↔	q{4,3,6} ↔ ↔	q{4,3,8} ↔	q{4,3,∞} ↔
6	q{6,3,4} ↔ ↔	q{6,3,6} ↔	q{6,3,8} ↔	q{6,3,∞} ↔
8	q{8,3,4} ↔	q{8,3,6} ↔	q{8,3,8} ↔	q{8,3,∞} ↔
... ∞	q{∞,3,4} ↔	q{∞,3,6} ↔	q{∞,3,8} ↔	q{∞,3,∞} ↔

See also

• Truncated simplectic honeycomb
• Triakis truncated tetrahedral honeycomb
• Architectonic and catoptric tessellation

References

1. For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).
2. "Physics Today article on the word kagome".
3. , OEIS sequence A000029 6-1 cases, skipping one with zero marks

• 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)
• 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
• Klitzing, Richard. "3D Euclidean Honeycombs x3x3o3o3*a - batatoh - O27".
• Uniform Honeycombs in 3-Space: 15-Batatoh

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E^2	Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
E^3	Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
E^4	Uniform 4-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E^5	Uniform 5-honeycomb	{3^[6]}	δ6	hδ6	qδ6
E^6	Uniform 6-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
E^7	Uniform 7-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
E^8	Uniform 8-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E^9	Uniform 9-honeycomb	{3^[10]}	δ10	hδ10	qδ10
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21