Snub trihexagonal tiling | |
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Type | Semiregular tiling |
Vertex configuration | 3.3.3.3.6 |
Schläfli symbol | sr{6,3} or |
Wythoff symbol | | 6 3 2 |
Coxeter diagram | |
Symmetry | p6, [6,3]^{+}, (632) |
Rotation symmetry | p6, [6,3]^{+}, (632) |
Bowers acronym | Snathat |
Dual | Floret pentagonal tiling |
Properties | Vertex-transitive chiral |
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 of 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 3 regular and 8 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. (Naming the colors by indices (3.3.3.3.6): 11213.)
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.
Uniform hexagonal/triangular tilings | |||||||||||||||
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Fundamental domains |
Symmetry: [6,3], (*632) | [6,3]^{+}, (632) | |||||||||||||
{6,3} | t{6,3} | r{6,3} | t{3,6} | {3,6} | rr{6,3} | tr{6,3} | sr{6,3} | ||||||||
Config. | 6^{3} | 3.12.12 | (6.3)^{2} | 6.6.6 | 3^{6} | 3.4.6.4 | 4.6.12 | 3.3.3.3.6 |
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 . 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.
n32 symmetry mutations of snub tilings: 3.3.3.3.n | ||||||||
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Symmetry n32 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
232 | 332 | 432 | 532 | 632 | 732 | 832 | ∞32 | |
Snub figures |
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Config. | 3.3.3.3.2 | 3.3.3.3.3 | 3.3.3.3.4 | 3.3.3.3.5 | 3.3.3.3.6 | 3.3.3.3.7 | 3.3.3.3.8 | 3.3.3.3.∞ |
Gyro figures |
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Config. | V3.3.3.3.2 | V3.3.3.3.3 | V3.3.3.3.4 | V3.3.3.3.5 | V3.3.3.3.6 | V3.3.3.3.7 | V3.3.3.3.8 | V3.3.3.3.∞ |
Floret pentagonal tiling | |
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Type | Dual semiregular tiling |
Faces | irregular pentagons |
Coxeter diagram | |
Symmetry group | p6, [6,3]^{+}, (632) |
Rotation group | p6, [6,3]^{+}, (632) |
Dual polyhedron | Snub trihexagonal tiling |
Face configuration | V3.3.3.3.6 |
Properties | face-transitive, chiral |
In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.^{[2]} It is one of 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 tiling, snub trihexagonal tiling,^{[4]} and has rotational symmetry of orders 6-3-2 symmetry.
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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(See animation) |
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° |
a=b=c=d=e 0°, A=60°, D=120° |
There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles, for example (F for V3^{4}.6, C for V3^{2}.4.3.4, B for V3^{3}.4^{2}, H for V3^{6}):
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) | FH^{2}, p6m (t=3, e=4) | FH^{2}, p6m (t=5, e=5) |
3-uniform | 4-uniform | ||||
FH^{2}, p6 (t=7, e=9) | F^{2}H, cmm (t=4, e=6) | F^{2}H^{2}, p6 (t=6, e=9) | F^{3}H, p2 (t=7, e=12) | FH^{3}, p6 (t=7, e=10) | FH^{3}, p6m (t=7, e=8) |
Replacing every V3^{6} 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 V3^{6} hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 3^{2}.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V3^{6} hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.4^{2}.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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Symmetry: [6,3], (*632) | [6,3]^{+}, (632) | |||||
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V6^{3} | V3.12^{2} | V(3.6)^{2} | V3^{6} | V3.4.6.4 | V.4.6.12 | V3^{4}.6 |
Wikimedia Commons has media related to Uniform tiling 3-3-3-3-6 (snub trihexagonal tiling). |
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: CS1 maint: archived copy as title (link) (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)