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 of sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.

Snub trihexagonal tiling

TypeSemiregular tiling
Vertex configuration
3.3.3.3.6
Schläfli symbolsr{6,3} or
Wythoff symbol| 6 3 2
Coxeter diagram
Symmetryp6, [6,3]+, (632)
Rotation symmetryp6, [6,3]+, (632)
Bowers acronymSnathat
DualFloret pentagonal tiling
PropertiesVertex-transitive chiral

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.)

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.

There is one related 2-uniform tiling, which mixes the vertex configurations of the snub trihexagonal tiling, 3.3.3.3.6 and the triangular tiling, 3.3.3.3.3.3.

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
TypeDual semiregular tiling
Facesirregular pentagons
Coxeter diagram
Symmetry groupp6, [6,3]+, (632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronSnub trihexagonal tiling
Face configurationV3.3.3.3.6
Propertiesface-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.

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

(See animation)
Deltoidal trihexagonal tiling

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 duals to k-uniform tiling, which mixes the 6-fold florets with other tiles, for example:

2-uniform dual 3-uniform dual 4-uniform dual

Fractalization

Replacing every hexagon by a truncated hexagon furnishes a uniform 8 tiling, 5 vertices of configuration 32.12, 2 vertices of configuration 3.4.3.12, and 1 vertex of configuration 3.4.6.4.

Replacing every hexagon by a truncated trihexagon furnishes a uniform 15 tiling, 12 vertices of configuration 4.6.12 and 3 vertices of configuration 3.4.6.4.

In both tilings, every vertex is in a different orbit since there is no chiral symmetry; and the uniform count was from the Floret pentagon region of each fractal tiling (3 side lengths of and 2 side lengths of in the truncated hexagonal; and 3 side lengths of and 2 side lengths of in the truncated trihexagonal).

Fractalizing the Snub Trihexagonal Tiling using the Truncated Hexagonal and Truncated Trihexagonal Tilings
Truncated Hexagonal Truncated Trihexagonal
Dual Fractalization Dual Fractalization
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

See also

References
  1. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E
  2. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 "Archived copy". Archived from the original on 2010-09-19. Retrieved 2012-01-20.CS1 maint: archived copy as title (link) (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  3. Five space-filling polyhedra by Guy Inchbald
  4. Weisstein, Eric W. "Dual tessellation". MathWorld.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
  • Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (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
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