Snub trihexagonal tiling

Summary

Snub trihexagonal tiling
Snub trihexagonal tiling
Type Semiregular tiling
Vertex configuration Tiling snub 3-6 left vertfig.svg
3.3.3.3.6
Schläfli symbol sr{6,3} or
Wythoff symbol | 6 3 2
Coxeter diagram CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
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.)

Circle packingEdit

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 tilingsEdit

 
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 mutationsEdit

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.∞
Gyro
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.∞

6-fold pentille tilingEdit

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.

 

VariationsEdit

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°

Related k-uniform and dual k-uniform tilingsEdit

There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles, for example (F for V34.6, C for V32.4.3.4, B for V33.42, H for V36):

Snub Trihexagonal 2-uniform 3-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)
           
           
3-uniform 4-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)
           
           

FractalizationEdit

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   in the rhombitrihexagonal;   in the truncated hexagonal; and   in the truncated trihexagonal).

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

Related tilingsEdit

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 alsoEdit

ReferencesEdit

  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.{{cite web}}: 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 [1]
  • 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

External linksEdit