Snub trihexagonal tiling
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
![](/wiki/images/thumb/d/da/2-Uniform_Tiling_20_Colored_by_Regular_Polygon_Orbits.svg/250px-2-Uniform_Tiling_20_Colored_by_Regular_Polygon_Orbits.svg.png)
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 . 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.
6-fold pentille tiling
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 Face figure: |
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 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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![]() 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° |
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:
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 [math]\displaystyle{ 1+\frac{1}{\sqrt{3}}:2+\frac{2}{\sqrt{3}} }[/math] in the rhombitrihexagonal; [math]\displaystyle{ 1+\frac{2}{\sqrt{3}}:2+\frac{4}{\sqrt{3}} }[/math] in the truncated hexagonal; and [math]\displaystyle{ 1+\sqrt{3}:2+2\sqrt{3} }[/math] in the truncated trihexagonal).
Rhombitrihexagonal | Truncated Hexagonal | Truncated Trihexagonal |
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Related tilings
See also
- Tilings of regular polygons
- List of uniform tilings
References
- ↑ 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, ISBN 978-1-56881-220-5, "A K Peters, LTD. - The Symmetries of Things". Archived from the original on 2010-09-19. https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205. Retrieved 2012-01-20. (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p. 288, table)
- ↑ Five space-filling polyhedra by Guy Inchbald
- ↑ Weisstein, Eric W.. "Dual tessellation". http://mathworld.wolfram.com/DualTessellation.html.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. https://archive.org/details/isbn_0716711931. (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. 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, ISBN 978-0866514613, pp. 50–56, dual rosette tiling p. 96, p. 114
External links
- Weisstein, Eric W.. "Uniform tessellation". http://mathworld.wolfram.com/UniformTessellation.html.
- Weisstein, Eric W.. "Semiregular tessellation". http://mathworld.wolfram.com/SemiregularTessellation.html.
- Klitzing, Richard. "2D Euclidean tilings s3s6s - snathat - O11". https://bendwavy.org/klitzing/dimensions/flat.htm#2D.
![]() | Original source: https://en.wikipedia.org/wiki/Snub trihexagonal tiling.
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