Rhombitrihexagonal tiling

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Short description: Semiregular tiling of the Euclidean plane

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.[1] 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 CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.png, 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, CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png.

Symmetry [6,3], (*632) [6,3+], (3*3)
Name Rhombitrihexagonal Cantic snub triangular Snub triangular
Image Rhombitrihexagonal tiling uniform coloring.png
Uniform face coloring
Rhombitrihexagonal tiling snub edge coloring.png
Uniform edge coloring
Rhombitrihexagonal tiling snub edge coloring nonuniform.png
Nonuniform geometry
Snub triangular tiling with rhombitrihexagonal coloring.png
Limit
Schläfli
symbol
rr{3,6} s2{3,6} s{3,6}
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png

Examples

Wallpaper group-p6m-4.jpg
From The Grammar of Ornament (1856)
Kensington board.svg
The game Kensington
Semi-regular-floor-3464.JPG
Floor tiling, Archeological Museum of Seville, Sevilla, Spain
Nîmes-Temple de Diane-6.jpg
The Temple of Diana in Nîmes, France
0 Mosaïque de Castel Guido - Pal. Massimo 1.JPG
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.[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
1-uniform 6.png
3.4.6.4
Regular hexagon.svg
Triangular tiling vertfig.png
2-uniform 18.png
3.3.4.3.4 & 36
D Inset to CH.gif
to CH
Dual Tilings

3.4.6.4
Regular dodecagon.svg
Hexagonal cupola flat.svg

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.

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.)

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.

Deltoidal trihexagonal tiling

Deltoidal trihexagonal tiling
Tiling small rhombi 3-6 dual simple.svg
TypeDual semiregular tiling
Faceskite
Coxeter diagramCDel node f1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node f1.png
Symmetry groupp6m, [6,3], (*632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronRhombitrihexagonal tiling
Face configurationV3.4.6.4
Tiling small rhombi 3-6 dual face.svg
Propertiesface-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 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.

P5 dual.png

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]

Isohedral variations
Symmetry p6m, [6,3], (*632) p31m, [6,3+], (3*3)
Form Isohedral tiling p4-41.png Isohedral tiling p4-40b.png Isohedral tiling p4-40.png
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.

P3 hull.png

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.

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 Inscribedstar.svg Isohedral tiling p4-53.png Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
Configuration V4.4.4.4 V6.4.3.4

See also

  • Tilings of regular polygons
  • List of uniform tilings

Notes

  1. 1.0 1.1 Conway, 2008, p288 table
  2. Ring Cycles a Jacks Chain variation
  3. Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications 17: 147–165. doi:10.1016/0898-1221(89)90156-9. https://www.beloit.edu/computerscience/faculty/chavey/catalog/. 
  4. "Uniform Tilings". http://www.uwgb.edu/dutchs/symmetry/uniftil.htm. 
  5. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern B
  6. Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles", Mathematics Magazine 84 (4): 283–289, doi:10.4169/math.mag.84.4.283 .
  7. Weisstein, Eric W.. "Dual tessellation". http://mathworld.wolfram.com/DualTessellation.html.  (See comparative overlay of this tiling and its dual)
  8. Tilings and Patterns

References