Rhombitrihexagonal tiling

Rhombitrihexagonal tiling
Type	Semiregular tiling
Vertex configuration	3.4.6.4
Schläfli symbol	rr{6,3} or $r{\begin{Bmatrix}6\\3\end{Bmatrix}}$
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

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 rr{3,6}.

John Conway calls it a rhombihexadeltille. It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling by Alicia Boole Stott's operational language.

There are three regular and eight 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.)

With edge-colorings there is a half symmetry form (3*3) orbifold notation. The hexagons can be considered as truncated triangles, t{3} with two types of edges. It has Coxeter diagram , Schläfli symbol s₂{3,6}. The bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, constructed as a snub triangular tiling, .

Symmetry	[6,3], (*632)	[6,3+], (3*3)
Name	Rhombitrihexagonal	Cantic snub triangular		Snub triangular
Image	Uniform face coloring	Uniform edge coloring	Nonuniform geometry	Limit
Schläfli symbol	rr{3,6}	s2{3,6}		s{3,6}
Coxeter diagram

Examples

From The Grammar of Ornament (1856)

The game Kensington

Floor tiling, Archeological Museum of Seville, Sevilla, Spain

The Temple of Diana in Nîmes, France

Roman floor mosaic in Castel di Guido

Related tilings

The tiling can be replaced by circular edges, centered on the hexagons as an overlapping circles grid. In quilting it is called Jacks chain.

There is one related 2-uniform tiling, having hexagons dissected into six triangles. The rhombitrihexagonal tiling is also related to the truncated trihexagonal tiling by replacing some of the hexagons and surrounding squares and triangles with dodecagons:

1-uniform	Dissection	2-uniform dissections
3.4.6.4		3.3.4.3.4 & 36	to CH
Dual Tilings
3.4.6.4		4.6.12	to 3

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 four other circles in the packing (kissing number). The translational lattice domain (red rhombus) contains six distinct circles.

Wythoff construction

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 eight forms, seven topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632)							[6,3]+ (632)	[6,3+] (3*3)
{6,3}	t{6,3}	r{6,3}	t{3,6}	{3,6}	rr{6,3}	tr{6,3}	sr{6,3}	s{3,6}
63	3.122	(3.6)2	6.6.6	36	3.4.6.4	4.6.12	3.3.3.3.6	3.3.3.3.3.3
Uniform duals
V63	V3.122	V(3.6)2	V63	V36	V3.4.6.4	V.4.6.12	V34.6	V36

Symmetry mutations

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.

*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paracomp.
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Figure
Config.	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

Deltoidal trihexagonal tiling

Deltoidal trihexagonal tiling
Type	Dual semiregular tiling
Faces	kite
Coxeter diagram
Symmetry group	p6m, [6,3], (*632)
Rotation group	p6, [6,3]+, (632)
Dual polyhedron	Rhombitrihexagonal tiling
Face configuration	V3.4.6.4
Properties	face-transitive

A 2023 discovered aperiodic monotile, solving the Einstein problem, is composed by a collection of 8 kites from the deltoidal trihexagonal tiling

The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. Conway called it a tetrille. The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling and a hexagonal tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.

The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling. Its faces are deltoids or kites.

Related polyhedra and tilings

It is one of seven dual uniform tilings in hexagonal symmetry, including the regular duals.

Dual uniform hexagonal/triangular tilings

Symmetry: [6,3], (*632)						[6,3]+, (632)
V63	V3.122	V(3.6)2	V36	V3.4.6.4	V.4.6.12	V34.6

This tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrilaterals. Ignoring the face colors below, the fully symmetry is p6m, and the lower symmetry is p31m with three mirrors meeting at a point, and threefold rotation points.

Isohedral variations

Symmetry	p6m, [6,3], (*632)	p31m, [6,3+], (3*3)
Form
Faces	Kite	Half regular hexagon	Quadrilaterals

This tiling is related to the trihexagonal tiling by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites.

The deltoidal trihexagonal tiling is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.

Symmetry mutations

This tiling is topologically related as a part of sequence of tilings with face configurations V3.4.n.4, and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

*n32 symmetry mutation of dual expanded tilings: V3.4.n.4

Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Figure Config.	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

Other deltoidal (kite) tiling

Other deltoidal tilings are possible.

Point symmetry allows the plane to be filled by growing kites, with the topology as a square tiling, V4.4.4.4, and can be created by crossing string of a dream catcher. Below is an example with dihedral hexagonal symmetry.

Another face transitive tiling with kite faces, also a topological variation of a square tiling and with face configuration V4.4.4.4. It is also vertex transitive, with every vertex containing all orientations of the kite face.

Symmetry	D6, [6], (*66)	pmg, [∞,(2,∞)+], (22*)	p6m, [6,3], (*632)
Tiling
Configuration	V4.4.4.4		V6.4.3.4

See also

• Tilings of regular polygons
• List of uniform tilings

References

• Grünbaum, Branko; 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-Strauss, 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.
• Klitzing, Richard. "2D Euclidean tilings x3o6x - rothat - O8".
• Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern N, Dual p. 77-76, pattern 2
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 116