Snub trihexagonal tiling
Template:Uniform tiling stat table In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.
Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille).
There are three regular and eight semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry.
There is only one uniform coloring of a snub trihexagonal tiling. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".)
Circle packing
The snub trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).[1] The lattice domain (red rhombus) repeats 6 distinct circles. The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from the triangular tiling.
Related polyhedra and tilings
Template:Hexagonal tiling small table
Symmetry mutations
This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram Template:CDD. These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons. Template:Snub table
6-fold pentille tiling
Template:Infobox face-uniform tiling In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.[2] It is one of the 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on a flower.[3] Each of its pentagonal faces has four 120° and one 60° angle.
It is the dual of the uniform snub trihexagonal tiling,[4] and has rotational symmetries of orders 6-3-2 symmetry.
Variations
The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedral pentagonal tiling type 5. In one limit, an edge-length goes to zero and it becomes a deltoidal trihexagonal tiling.
| General | Zero length degenerate |
Special cases | |||
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| Error creating thumbnail: (See animation) |
File:1-uniform 6 dual.svg Deltoidal trihexagonal tiling |
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 a=b, d=e A=60°, D=120° |
 a=b, d=e, c=0 A=60°, 90°, 90°, D=120° |
 a=b=2c=2d=2e A=60°, B=C=D=E=120° |
 a=b=d=e A=60°, D=120°, E=150° |
 2a=2b=c=2d=2e 0°, A=60°, D=120° |
Error creating thumbnail: a=b=c=d=e 0°, A=60°, D=120° |
Related k-uniform and dual k-uniform tilings
There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles; for example, labeling F for V34.6, C for V32.4.3.4, B for V33.42, H for V36:
| uniform (snub trihexagonal) | 2-uniform | 3-uniform | |||
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| F, p6 (t=3, e=3) | FH, p6 (t=5, e=7) | FH, p6m (t=3, e=3) | FCB, p6m (t=5, e=6) | FH2, p6m (t=3, e=4) | FH2, p6m (t=5, e=5) |
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| dual uniform (floret pentagonal) | dual 2-uniform | dual 3-uniform | |||
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File:Floret Pentagonal Variation 1.svg | File:Floret Pentagonal Variation 2.svg | File:Floret Pentagonal Variation 3.svg | File:Floret Pentagonal Variation 4.svg | 
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| 3-uniform | 4-uniform | ||||
| FH2, p6 (t=7, e=9) | F2H, cmm (t=4, e=6) | F2H2, p6 (t=6, e=9) | F3H, p2 (t=7, e=12) | FH3, p6 (t=7, e=10) | FH3, p6m (t=7, e=8) |
| File:Snub Trihexagonal Variation 6.svg | File:Snub Trihexagonal Variation 8.svg | File:Snub Trihexagonal Variation 9.svg | File:Snub Trihexagonal Variation 10.svg | File:Snub Trihexagonal Variation 11.svg | Error creating thumbnail: |
| dual 3-uniform | dual 4-uniform | ||||
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Error creating thumbnail: | File:Floret Pentagonal Variation 9.svg | Error creating thumbnail: | File:Floret Pentagonal Variation 11.svg | Error creating thumbnail: |
Fractalization
Replacing every V36 hexagon by a rhombitrihexagon furnishes a 6-uniform tiling, two vertices of 4.6.12 and two vertices of 3.4.6.4.
Replacing every V36 hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 32.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V36 hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.42.6, and one vertex of 3.4.6.4.
In each fractal tiling, every vertex in a floret pentagonal domain is in a different orbit since there is no chiral symmetry (the domains have 3:2 side lengths of in the rhombitrihexagonal; in the truncated hexagonal; and in the truncated trihexagonal).
| Rhombitrihexagonal | Truncated Hexagonal | Truncated Trihexagonal |
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File:Snub Trihexagonal Fractalization 2.svg | 
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Error creating thumbnail: |
Related tilings
Template:Dual hexagonal tiling table
See also
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn [1]
- Template:Cite book (Chapter 2.1: Regular and uniform tilings, p. 58-65)
- Template:The Geometrical Foundation of Natural Structure (book) p. 39
- Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern R, Dual p. 77-76, pattern 5
- Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, Template:Isbn, pp. 50–56, dual rosette tiling p. 96, p. 114
External links
- ↑ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E
- ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things, 2008, Template:Isbn, Template:Cite web (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p. 288, table)
- ↑ Five space-filling polyhedra by Guy Inchbald
- ↑ Template:MathWorld