Snub trihexagonal tiling

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Snub trihexagonal tiling
Snub trihexagonal tiling
Type Semiregular tiling
Vertex configuration Snub hexagonal tiling vertfig.png
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 ( 11213.)

Circle packing[edit]

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 tilings[edit]

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

Symmetry mutations[edit]

This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure ( and Coxeter–Dynkin diagram CDel node h.pngCDel n.pngCDel node h.pngCDel 3.pngCDel node h.png. 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[edit]

Floret pentagonal tiling
1-uniform 10 dual.svg
TypeDual semiregular tiling
Facesirregular pentagons
Coxeter diagramCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 6.pngCDel node fh.png
Symmetry groupp6, [6,3]+, (632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronSnub trihexagonal tiling
Face configurationV3.
V3. Rotated.png
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.

P7 dual.png


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)
Prototile p5-type5.png
a=b, d=e
A=60°, D=120°
1-uniform 6 dual.svg
Deltoidal trihexagonal tiling
Tiling face 3-4-6-4.svg
a=b, d=e, c=0
60°, 90°, 90°, 120°

Related dual k-uniform tilings[edit]

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
3-uniform 58 dual.svg 3-uniform 59 dual.svg 3-uniform 60 dual.svg 3-uniform 61 dual.svg 4-uniform 150 dual.svg 4-uniform 151 dual.svg


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

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

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
Fractalizing the Snub Trihexagonal Tiling (Truncated Hexagonal).png Fractalizing the Snub Trihexagonal Tiling (Truncated Trihexagonal).png
Dual of Fractalizing the Snub Trihexagonal Tiling (Truncated Hexagonal).png Dual of Fractalizing the Snub Trihexagonal Tiling (Truncated Trihexagonal).png
Dual Fractalization Dual Fractalization

Related tilings[edit]

Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
Uniform tiling 63-t2.svg Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t0.svg Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
V63 V3.122 V(3.6)2 V36 V3.4.6.4 V.4.6.12 V34.6

See also[edit]


  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 "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)
  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 links[edit]