Snub trihexagonal tiling
TypeSemiregular tiling
Vertex configuration3.3.3.3.6
Schläfli symbolsr{6,3} or s { 6 3 } {\displaystyle s{\begin{Bmatrix}6\\3\end{Bmatrix}}}
Wythoff symbol| 6 3 2
Coxeter diagram
Symmetryp6, [6,3]+, (632)
Rotation symmetryp6, [6,3]+, (632)
Bowers acronymSnathat
DualFloret pentagonal tiling
PropertiesVertex-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 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 leads to a circle packing, each vertex becoming the center of a circle of fixed diameter. Every circle is in contact with 5 other circles in the packing (kissing number). 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

There is one related 2-uniform tiling, which mixes the vertex configurations 3.3.3.3.6 of the snub trihexagonal tiling and 3.3.3.3.3.3 of the triangular tiling.
Uniform hexagonal/triangular tilings
Fundamental domainsSymmetry: [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.633.12.12(6.3)26.6.6363.4.6.44.6.123.3.3.3.6

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.

n32 symmetry mutations of snub tilings: 3.3.3.3.n vte
Symmetry n32SphericalEuclideanCompact hyperbolicParacomp.
232332432532632732832∞32
Snub figures
Config.3.3.3.3.23.3.3.3.33.3.3.3.43.3.3.3.53.3.3.3.63.3.3.3.73.3.3.3.83.3.3.3.∞
Gyro figures
Config.V3.3.3.3.2V3.3.3.3.3V3.3.3.3.4V3.3.3.3.5V3.3.3.3.6V3.3.3.3.7V3.3.3.3.8V3.3.3.3.∞

6-fold pentille tiling

In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. 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. Each of its pentagonal faces has four 120° and one 60° angle.

It is the dual of the uniform snub trihexagonal tiling, 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.

GeneralZero length degenerateSpecial 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°

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-uniform3-uniform
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)
dual uniform (floret pentagonal)dual 2-uniformdual 3-uniform
3-uniform4-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)
dual 3-uniformdual 4-uniform

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 1 + 1 3 : 2 + 2 3 {\displaystyle 1+{\frac {1}{\sqrt {3}}}:2+{\frac {2}{\sqrt {3}}}} in the rhombitrihexagonal; 1 + 2 3 : 2 + 4 3 {\displaystyle 1+{\frac {2}{\sqrt {3}}}:2+{\frac {4}{\sqrt {3}}}} in the truncated hexagonal; and 1 + 3 : 2 + 2 3 {\displaystyle 1+{\sqrt {3}}:2+2{\sqrt {3}}} in the truncated trihexagonal).

Fractalizing the Snub Trihexagonal Tiling using the Rhombitrihexagonal, Truncated Hexagonal and Truncated Trihexagonal Tilings
RhombitrihexagonalTruncated HexagonalTruncated Trihexagonal

Related tilings

Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632)[6,3]+, (632)
V63V3.122V(3.6)2V36V3.4.6.4V.4.6.12V34.6

See also

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN978-1-56881-220-5
  • Grünbaum, Branko; Shephard, G. C. (1987). . New York: W. H. Freeman. ISBN0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p.58-65)
  • Williams, Robert (1979). . Dover Publications, Inc. ISBN0-486-23729-X.
  • 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, ISBN978-0866514613, pp.50–56, dual rosette tiling p.96, p.114

External links