Rhombitrihexagonal tiling

Rhombitrihexagonal tiling
Type	Semiregular tiling
Vertex configuration	3.4.6.4
Schläfli symbol	t0,2{6,3}
Wythoff symbol	3 | 6 2
Coxeter diagram
Symmetry	p6m, [6,3], (*632)
Rotation symmetry	p6, [6,3]+, (632)
Bowers acronym	Rothat
Dual	Deltoidal trihexagonal tiling
Properties	Vertex-transitive
Vertex figure: 3.4.6.4

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of t_0,2{3, 6}.

Conway calls it a rhombihexadeltille. It can be considered a cantellated or expanded hexagonal tiling by Johnson's operational language.

There are 3 regular and 8 semiregular tilings in the plane.

Uniform colorings

There is only one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.)

Related polyhedra and tilings

There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling). Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

Uniform hexagonal/triangular tilings

Symmetry: [6,3], (*632)							[6,3]+, (632)	[1+,6,3], (*333)	[6,3+], (3*3)
{6,3}	t0,1{6,3}	t1{6,3}	t1,2{6,3}	t2{6,3}	t0,2{6,3}	t0,1,2{6,3}	s{6,3}	h{6,3}	h1,2{6,3}
Uniform duals
V6.6.6	V3.12.12	V3.6.3.6	V6.6.6	V3.3.3.3.3.3	V3.4.12.4	V.4.6.12	V3.3.3.3.6	V3.3.3.3.3.3

This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

Dimensional family of expanded polyhedra and tilings: 3.4.n.4

Symmetry *n32 [n,3]	Spherical				Planar	Hyperbolic...
	*232 [2,3] D3h	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Expanded figure	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4	3.4.6.4	3.4.7.4	3.4.8.4	3.4.∞.4
Coxeter Schläfli	t0,2{2,3}	t0,2{3,3}	t0,2{4,3}	t0,2{5,3}	t0,2{6,3}	t0,2{7,3}	t0,2{8,3}	t0,2{∞,3}
Deltoidal figure	V3.4.2.4	V3.4.3.4	V3.4.4.4	V3.4.5.4	V3.4.6.4	V3.4.7.4	V3.4.8.4	V3.4.∞.4
Coxeter

The hexagonal cupola contains the pattern of this tiling, but closes it into a degenerate polygon with a dodecagon base.

Family of cupolae

2	3	4	5	6
Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)

Circle packing

The Rhombitrihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (kissing number). The gap inside each hexagon allows for one circle, for a denser packing with kissing number 5.

Gallery

An ornamental version

Nonuniform pattern (with rectangles)

The game Kensington

See also

• Tilings of regular polygons
• List of uniform tilings

References

• Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1.  (Chapter 2.1: Regular and uniform tilings, p. 58-65)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  p40
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings.
• Weisstein, Eric W., "Uniform tessellation", MathWorld.
• Weisstein, Eric W., "Semiregular tessellation", MathWorld.
• Richard Klitzing, 2D Euclidean tilings, x3o6x - rothat - O8