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[1] 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.

Contents

Dual tiling

This is the dual of the trihexagonal tiling.[2]

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

Notes

  1. ^ 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, p288 table)
  2. ^ Weisstein, Eric W., "Dual tessellation" from MathWorld.

References