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#circle packing.

Snub hexagonal tiling circle packing.png

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
Type Dual semiregular tiling
Faces irregular pentagons
Coxeter diagram CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 6.pngCDel node fh.png
Symmetry group p6, [6,3]+, (632)
Rotation group p6, [6,3]+, (632)
Dual polyhedron Snub trihexagonal tiling
Face configuration V3.
Tiling face 3-3-3-3-6.svg
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.

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 tilings[edit]

Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
Uniform tiling 63-t2.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t0.png 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 [1] (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 [2]
  • Grünbaum, Branko ; and 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]