Rhombille tiling

Rhombille tiling
Type	Dual semiregular tiling
Faces	30-60 rhombus
Face configuration	V3.6.3.6
Symmetry group	p6m, [6,3], *632 p3m1, [3[3]], *333
Dual	Trihexagonal tiling
Properties	edge-transitive face-transitive

In geometry, the rhombille tiling is a tessellation of identical 60° rhombi on the Euclidean plane. There are two types of vertices, one with three rhombi and one with six rhombi.

It can be seen as a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. The diagonals of each rhomb are in the ratio 1:√3.

Dual tiling

This is the dual of the trihexagonal tiling.

Geometric variations

It can be considered an isometric projection view of a set of cubes and was used in the game Q*bert in this way. It is also used as a floor or wall tiling, sometimes with one fatter rhombus (or square) and two more narrow rhombi. [2] [3] [4]

Colorings

As a face-transitive dual uniform tiling all the rhombi are the same, but with larger symmetry fundamental domains, there are many possible colorings, including these five:

Two colors		Three colors

(2 colors) *632	(3 colors) *333
The single colored rhombus tiling has *632 symmetry, but vertices can be colored with alternating colors on the inner points leading to a *333 symmetry.

Related polyhedra and tilings

This tiling is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.

Polyhedra			Euclidean tiling	Hyperbolic tiling
[3,3]	[4,3]	[5,3]	[6,3]	[7,3]	[8,3]
Cube	Rhombic dodecahedron	Rhombic triacontahedron	Rhombille

Similarly it relates to the infinite series of 3-color tilings with the face configurations V3.2n.3.2n, the first a polyhedron, second this one in the Euclidean plane, and the rest in the hyperbolic plane.

V3.4.3.4 (Drawn as a net)

V3.6.3.6 Euclidean plane tiling Rhombille tiling

V3.8.3.8 Hyperbolic plane tiling (Drawn in a Poincaré disk model)

Other rhombic tilings

It has the same vertex arrangement as two other simple rhombic tilings, and the triangular tiling. The translational rhombic and zig-zag rhombic tilings, which are topologically equivalent to the square tiling, can be constructed from rhombi or parallelograms of any angle and lengths, while the rhombille tiling is limited to 60 degrees.

Lattice points

Triangular

Translational rhombic

Zig-zag rhombic

Rhombille

A variation of snub square tiling with merged triangles

The nonperiodic Penrose tilings consist of two types of rhombi, whose acute angles are 36 and 72 degrees.

This periodic tiling uses two types of Penrose tiles and matches the rhombille topologically

See also

• List of uniform tilings

References

• Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1.  (Chapter 2.1: Regular and uniform tilings, p.58-65) (Page 476, Tilings by polygons, #42 of 56 polygonal isohedral types by quadrangles)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  p38