Rhombitrihexagonal tiling

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Rhombitrihexagonal tiling

Type Semiregular tiling
Vertex configuration
3.4.6.4
Schläfli symbol rr{6,3} or ${\displaystyle r{\begin{Bmatrix}6\\3\end{Bmatrix}}}$
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.[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 3 regular and 8 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

 An ornamental version Nonuniform pattern (with rectangles) The game Kensington Church floor tiling, Sevilla, Spain The tiling can be replaced by circular edges, centered on the hexagons. In quilting it is call Jacks chain.[2]

Related polyhedra and tilings

There is one related 2-uniform tilings, having hexagons dissected into 6 triangles.[3][4]

 3.4.6.4 3.3.4.3.4 & 36

The rhombitrihexagonal tiling is related to the truncated trihexagonal tiling by replacing some of the hexagons and surrounding squares and triangles with dodecagons:

 3.4.6.4 4.6.12

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 4 other circles in the packing (kissing number).[5] The translational lattice domain (red rhombus) contains 6 distinct circles. The gap inside each hexagon allows for one circle, related to a 2-uniform tiling with the hexagons divided into 6 triangles.

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 8 forms, 7 which are 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
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

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 7 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 quadrillaterals. Ignoring the face colors below, the fully symmetry is p6m, and the lower symmetry is p31m with 3 mirrors meeting at a point, and 3-fold rotation points.[8]

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.

*n42 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

Notes

1. ^ a b 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.
4. ^ 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, arXiv:, doi:10.4169/math.mag.84.4.283, MR 2843659.
7. ^ (See comparative overlay of this tiling and its dual)
8. ^ Tilings and Patterns

References

• Grünbaum, Branko ; and 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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings.