Quarter cubic honeycomb

Quarter cubic honeycomb
Error creating thumbnail:  File:HC A1-P1.png
Type	Uniform honeycomb
Family	Truncated simplectic honeycomb Quarter hypercubic honeycomb
Indexing	J25,33, A13 W10, G6
Schläfli symbol	t0,1{3[4]} or q{4,3,4}
Coxeter-Dynkin diagram	Template:CDD = Template:CDD = Template:CDD
Cell types	{3,3} File:Tetrahedron.png (3.6.6) File:Truncated tetrahedron.png
Face types	{3}, {6}
Vertex figure	File:T01 quarter cubic honeycomb verf.png (isosceles triangular antiprism)
Space group	FdTemplate:Overlinem (227)
Coxeter group	${\tilde{A}}_3$×22, [[3[4]]]
Dual	oblate cubille Cell: File:Oblate cubille cell.png (1/4 of rhombic dodecahedron)
Properties	vertex-transitive, edge-transitive

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 – is four times that of the cubic honeycomb.

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

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, which is the pyrochlore lattice.

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.

File:Tetrahedral-truncated tetrahedral honeycomb slab.png

Error creating thumbnail: trihexagonal tiling: Template:CDD

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	${\tilde{A}}_3$, [3[4]]	${\tilde{A}}_3$×2, [[3[4]]]
Space group	FTemplate:Overline3m (216)	FdTemplate:Overlinem (227)
Coloring	File:Quarter cubic honeycomb.png
Vertex figure	File:T01 quarter cubic honeycomb verf.png	File:T01 quarter cubic honeycomb verf2.png
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}.

File:Tiling Semiregular 3-6-3-6 Trihexagonal.svg Four sets of parallel planes of trihexagonal tilings exist throughout this honeycomb.

This honeycomb is one of five distinct uniform honeycombs constructed by the ${\tilde{A}}_3$ 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
FTemplate:Overline3m (216)	1o:2	a1 File:Scalene tetrahedron diagram.png	[3[4]]	Template:CDD	${\tilde{A}}_3$	(None)
FmTemplate:Overlinem (225)	2−:2	d2	<[3[4]]> ↔ [4,31,1]	Template:CDD ↔ Template:CDD	${\tilde{A}}_3$×21 ↔ ${\tilde{B}}_3$	Template:CDD 1,Template:CDD 2
FdTemplate:Overlinem (227)	2+:2	g2	Template:Brackets or [2+[3[4]]]	Template:CDD ↔ Template:CDD	${\tilde{A}}_3$×22	Template:CDD 3
PmTemplate:Overlinem (221)	4−:2	d4 File:Digonal disphenoid diagram.png	<2[3[4]]> ↔ [4,3,4]	Template:CDD ↔ Template:CDD	${\tilde{A}}_3$×41 ↔ ${\tilde{C}}_3$	Template:CDD 4
ITemplate:Overline (204)	8−o	r8 File:Regular tetrahedron diagram.png	[4[3[4]]]+ ↔ Template:Brackets	Template:CDD ↔ Template:CDD	½${\tilde{A}}_3$×8 ↔ ½${\tilde{C}}_3$×2	Template:CDD (*)
ImTemplate:Overlinem (229)	8o:2		[4[3[4]]] ↔ Template:Brackets		${\tilde{A}}_3$×8 ↔ ${\tilde{C}}_3$×2	Template:CDD 5

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

See also

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

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, Template:Isbn (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)
• Template:The Geometrical Foundation of Natural Structure (book)
• Template:Cite book
• 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, Template:Isbn [1]
  • (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
• Template:KlitzingPolytopes
• Uniform Honeycombs in 3-Space: 15-Batatoh

Template:Navbar-collapsible
Space	Family	${\tilde{A}}_{n-1}$	${\tilde{C}}_{n-1}$	${\tilde{B}}_{n-1}$	${\tilde{D}}_{n-1}$	${\tilde{G}}_2$ / ${\tilde{F}}_4$ / ${\tilde{E}}_{n-1}$
E2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21