Rhombitrihexagonal tiling

Rhombitrihexagonal tiling
Rhombitrihexagonal tiling
Type Semiregular tiling
Vertex configuration Tiling small rhombi 3-6 vertfig.svg
3.4.6.4
Schläfli symbol rr{6,3} or
Wythoff symbol 3 | 6 2
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
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 coloringsEdit

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
                 

ExamplesEdit

 
From The Grammar of Ornament (1856)
 
The game Kensington
 
Floor tiling, Archeological Museum of Seville, Sevilla, Spain
 
The Temple of Diana in Nîmes, France
 
Roman floor mosaic in Castel di Guido

Related tilingsEdit

 
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 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 packingEdit

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.

 

Wythoff constructionEdit

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.)

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+
(632)
[6,3+]
(3*3)
{6,3} t{6,3} r{6,3} t{3,6} {3,6} rr{6,3} tr{6,3} sr{6,3} s{3,6}
                                                     
                 
63 3.122 (3.6)2 6.6.6 36 3.4.6.4 4.6.12 3.3.3.3.6 3.3.3.3.3.3
Uniform duals
                 
V63 V3.122 V(3.6)2 V63 V36 V3.4.6.4 V.4.6.12 V34.6 V36

Symmetry mutationsEdit

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
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paracomp.
*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. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4

Deltoidal trihexagonal tilingEdit

Deltoidal trihexagonal tiling
 
TypeDual semiregular tiling
Faceskite
Coxeter diagram     
Symmetry groupp6m, [6,3], (*632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronRhombitrihexagonal tiling
Face configurationV3.4.6.4
Propertiesface-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 tilingsEdit

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 quadrilaterals. 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 mutationsEdit

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.

*n32 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) tilingEdit

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

See alsoEdit

NotesEdit

  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. ^ "Archived copy". Archived from the original on 2006-09-09. Retrieved 2006-09-09.CS1 maint: archived copy as title (link)
  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:0908.3257, doi:10.4169/math.mag.84.4.283, MR 2843659.
  7. ^ Weisstein, Eric W. "Dual tessellation". MathWorld. (See comparative overlay of this tiling and its dual)
  8. ^ Tilings and Patterns

ReferencesEdit