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Snub trihexagonal tiling

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Title: Snub trihexagonal tiling  
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Subject: Snub trioctagonal tiling, Hexagonal tiling small table, Pentagonal icositetrahedron, Gilbert tessellation, Pinwheel tiling
Collection: Chiral Figures, Euclidean Tilings, Isogonal Tilings, Semiregular Tilings, Snub Tilings
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Snub trihexagonal tiling

Snub trihexagonal tiling
Snub trihexagonal tiling
Type Semiregular tiling
Vertex configuration
3.3.3.3.6
Schläfli symbol sr{6,3}
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.)

Contents

  • Circle packing 1
  • Related polyhedra and tilings 2
    • Symmetry mutations 2.1
    • Floret pentagonal tiling 2.2
      • Variations 2.2.1
      • Related tilings 2.2.2
  • See also 3
  • References 4
  • External links 5

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.

Related polyhedra and tilings

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.
Uniform hexagonal/triangular tilings
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. 63 3.12.12 (6.3)2 6.6.6 36 3.4.6.4 4.6.12 3.3.3.3.6

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
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
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.∞
Gryro
figures
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

Floret pentagonal tiling
Type Dual semiregular tiling
Coxeter diagram
Faces irregular pentagons
Face configuration V3.3.3.3.6
Symmetry group p6, [6,3]+, (632)
Rotation group p6, [6,3]+, (632)
Dual Snub trihexagonal tiling
Properties face-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°

Related tilings

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

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 [3] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  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 [4]
  • , p. 58-65) Regular and uniform tilings (Chapter 2.1:  
  • 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

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