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, .
The tiling can be replaced by circular edges, centered on the hexagons as an overlapping circles grid. In quilting it is called Jacks chain.[2]
There is one related 2-uniform tiling, having hexagons dissected into six triangles.[3][4] 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).[5] The translational lattice domain (red rhombus) contains six distinct circles.
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.)
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
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 calls it a tetrille.[1] 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.[6]
The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling.[7] 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.
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.[8]
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.
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.
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN0-486-23729-X. p40
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN978-1-56881-220-5[1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings.
