Rhombitrihexagonal tiling

From Wikipedia, the free encyclopedia

Template:Short description

Rhombitrihexagonal tiling
Rhombitrihexagonal tiling
Type Semiregular tiling
Vertex configuration File:Tiling small rhombi 3-6 vertfig.svg
3.4.6.4
Schläfli symbol rr{6,3} or r{63}
Wythoff symbol 3 | 6 2
Coxeter diagram Template:CDD
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 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 Template:CDD, 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, Template:CDD.

Symmetry [6,3], (*632) [6,3+], (3*3)
Name Rhombitrihexagonal Cantic snub triangular Snub triangular
Image File:Rhombitrihexagonal tiling uniform coloring.svg
Uniform face coloring 		File:Rhombitrihexagonal tiling snub edge coloring.svg
Uniform edge coloring 		File:Rhombitrihexagonal tiling snub edge coloring nonuniform.svg
Nonuniform geometry 		File:Snub triangular tiling with rhombitrihexagonal coloring.svg
Limit
Schläfli
symbol 		rr{3,6} s2{3,6} s{3,6}
Coxeter
diagram 		Template:CDD Template:CDD Template:CDD

Examples

File:Wallpaper group-p6m-4.jpg
From The Grammar of Ornament (1856) 		File:Kensington board.svg
The game Kensington 		File:Semi-regular-floor-3464.JPG
Floor tiling, Archeological Museum of Seville, Sevilla, Spain 		File:Nîmes-Temple de Diane-6.jpg
The Temple of Diana in Nîmes, France 		File:0 Mosaïque de Castel Guido - Pal. Massimo 1.JPG
Roman floor mosaic in Castel di Guido
File:Circular rhombitrihexagonal tilng.svg
The tiling can be replaced by circular edges, centered on the hexagons as an overlapping circles grid. In quilting it is called Jacks chain.

There is one related 2-uniform tiling, having hexagons dissected into six triangles.[2][3] 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
File:1-uniform n6.svg
3.4.6.4 		File:Regular hexagon.svg
File:Triangular tiling vertfig.png 		File:2-uniform 18.png
3.3.4.3.4 & 36 		File:D Inset to CH.gif
to CH
Dual Tilings
File:1-uniform 6b.png
3.4.6.4 		File:Regular dodecagon.svg
File:Hexagonal cupola flat.svg 		File:1-uniform 3.png
4.6.12 		File:D Outset to 3.gif
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).[4] The translational lattice domain (red rhombus) contains six distinct circles.

File:1-uniform-6-circlepack.svg

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.)

Uniform hexagonal/triangular tilings Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
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}
Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
File:Uniform tiling 63-t0.svg File:Uniform tiling 63-t01.svg File:Uniform tiling 63-t1.svg File:Uniform tiling 63-t12.svg File:Uniform tiling 63-t2.svg File:Uniform tiling 63-t02.svg File:Uniform tiling 63-t012.svg File:Uniform tiling 63-snub.svg File:Uniform tiling 63-h12.svg
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
File:1-uniform 1 dual.svg File:1-uniform 4 dual1.svg File:1-uniform 7 dual.svg File:1-uniform 1 dual.svg File:1-uniform 11 dual.svg File:1-uniform 6 dual.svg File:1-uniform 3 dual.svg File:1-uniform 10 dual.svg File:1-uniform 11 dual.svg
V63 V3.122 V(3.6)2 V63 V36 V3.4.6.4 V.4.6.12 V34.6 V36

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.

*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 File:Spherical triangular prism.svg File:Uniform tiling 332-t02.svg File:Uniform tiling 432-t02.svg File:Uniform tiling 532-t02.png File:Uniform polyhedron-63-t02.png File:Rhombitriheptagonal tiling.svg File:H2-8-3-cantellated.svg File:H2 tiling 23i-5.png
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 tiling

Page Module:Infobox/styles.css has no content.

Deltoidal trihexagonal tiling
Lua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found.
TypeDual semiregular tiling
Coxeter diagramTemplate:CDD
Wallpaper groupp6m, [6,3], (*632)
Rotation groupp6, [6,3]+, (632)
DualRhombitrihexagonal tiling
Face configurationV3.4.6.4
File:Tiling small rhombi 3-6 dual face.svg
Propertiesface-transitive
File:Aperiodic monotile smith 2023.svg
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 called 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.[5]

The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling.[6] Its faces are deltoids or kites.

File:P5 dual.svg

It is one of seven dual uniform tilings in hexagonal symmetry, including the regular duals.

Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
File:Uniform tiling 63-t2.svg File:Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg File:Rhombic star tiling.svg File:Uniform tiling 63-t0.svg File:Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg File:Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg File:Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
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 three mirrors meeting at a point, and threefold rotation points.[7]

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

File: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.

*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. 		File:Spherical trigonal bipyramid.svg
V3.4.2.4 		File:Spherical rhombic dodecahedron.svg
V3.4.3.4 		File:Spherical deltoidal icositetrahedron.svg
V3.4.4.4 		File:Spherical deltoidal hexecontahedron.svg
V3.4.5.4 		File:Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4 		File:Deltoidal triheptagonal tiling.svg
V3.4.7.4 		File:H2-8-3-deltoidal.svg
V3.4.8.4 		File:Deltoidal triapeirogonal til.png
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 File:Inscribedstar.svg File:Isohedral tiling p4-53.svg File:Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
Configuration V4.4.4.4 V6.4.3.4

See also

Page Module:Side box/styles.css has no content.Page Template:Sister project/styles.css has no content.

Page Template:Plainlist/styles.css has no content.
Script error: No such module "Sister project logo".
Wikimedia Commons has media related to Uniform tiling 3-4-6-4 (rhombitrihexagonal tiling).Template:Preview warning

Notes

Page Template:Reflist/styles.css has no content.

  1. ^ a b Conway, 2008, p288 table
  2. ^ Page Module:Citation/CS1/styles.css has no content.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.
  3. ^ Page Module:Citation/CS1/styles.css has no content."Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
  4. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern B
  5. ^ Page Module:Citation/CS1/styles.css has no content.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.
  6. ^ Script error: No such module "Template wrapper". (See comparative overlay of this tiling and its dual)
  7. ^ Tilings and patterns

References

p40
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn [1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings.
  • Script error: No such module "Template wrapper".
  • Script error: No such module "Template wrapper".
  • Page Module:Citation/CS1/styles.css has no content.Klitzing, Richard. "2D Euclidean tilings x3o6x - rothat - O8".
  • Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern N, Dual p. 77-76, pattern 2
  • Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, Template:Isbn, pp. 50–56, dual p. 116

Lua error in package.lua at line 80: module 'Module:Navbox/configuration' not found.

Retrieved from "https://cosmopedia.co/index.php?title=Rhombitrihexagonal_tiling&oldid=1457401#Deltoidal_trihexagonal_tiling"