Snub trihexagonal tiling

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

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.)

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#circle packing.

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.

Floret pentagonal tiling

Floret pentagonal tiling
TypeDual semiregular tiling
Coxeter diagram
Facesirregular pentagons
Face configurationV3.3.3.3.6
Symmetry groupp6, [6,3]+, (632)
Rotation groupp6, [6,3]+, (632)
DualSnub trihexagonal tiling
Propertiesface-transitive, chiral

In geometry, the floret pentagonal tiling or rosette pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of 15 known isohedral pentagon tilings. It is given its name because its six pentagonal tiles radiate out from a central point, like petals on a flower.[2] Conway calls it a 6-fold pentille.[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.


(See animation)
a=b, d=e
A=60°, D=120°
Deltoidal trihexagonal tiling
a=b, d=e, c=0
60°, 90°, 90°, 120°
Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
V63 V3.122 V(3.6)2 V36 V3.4.12.4 V.4.6.12 V34.6

See also

Wikimedia Commons has media related to Uniform tiling 3-3-3-3-6.

References

  1. ↑ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E
  2. ↑ Five space-filling polyhedra by Guy Inchbald
  3. ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  4. ↑ Weisstein, Eric W. "Dual tessellation". MathWorld.

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