Snub trihexagonal tiling

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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 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
Snub hexagonal tiling vertfig.png
Vertex figure: 3.3.3.3.6

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 hexatille, constructed as a snub operation applied to a hexagonal tiling (hexatille).

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[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). 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]

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 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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3
332
[3,3]+
T
432
[4,3]+
O
532
[5,3]+
I
632
[6,3]+
P6
732
[7,3]+
832
[8,3]+...
∞32
[∞,3]+
Snub
figure
Spherical trigonal antiprism.png
3.3.3.3.2
Spherical snub tetrahedron.png
3.3.3.3.3
Spherical snub cube.png
3.3.3.3.4
Spherical snub dodecahedron.png
3.3.3.3.5
Uniform tiling 63-snub.png
3.3.3.3.6
Uniform tiling 73-snub.png
3.3.3.3.7
Uniform tiling 83-snub.png
3.3.3.3.8
Uniform tiling i32-snub.png
3.3.3.3.∞
Coxeter
Schläfli
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{2,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{4,3}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{5,3}
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{6,3}
CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{7,3}
CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{8,3}
CDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{∞,3}
Snub
dual
figure
Uniform tiling 432-t0.png
V3.3.3.3.2
Uniform tiling 532-t0.png
V3.3.3.3.3
Spherical pentagonal icositetrahedron.png
V3.3.3.3.4
Spherical pentagonal hexecontahedron.png
V3.3.3.3.5
Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
V3.3.3.3.6
Ord7 3 floret penta til.png
V3.3.3.3.7
V3.3.3.3.8 Order-3-infinite floret pentagonal tiling.png
V3.3.3.3.∞
Coxeter CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+
(632)
[1+,6,3]
(*333)
[6,3+]
(3*3)
{6,3} t{6,3} r{6,3}
r{3[3]}
t{3,6}
t{3[3]}
{3,6}
{3[3]}
rr{6,3}
s2{6,3}
tr{6,3} sr{6,3} h{6,3}
{3[3]}
h2{6,3}
r{3[3]}
s{3,6}
s{3[3]}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png =
CDel branch 10ru.pngCDel split2.pngCDel node.png or CDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 63-t0.png Uniform tiling 63-t01.png Uniform tiling 63-t1.png
Uniform tiling 333-t01.png
Uniform tiling 63-t12.png
Uniform tiling 333-t012.png
Uniform tiling 63-t2.png
Uniform tiling 333-t2.png
Uniform tiling 63-t02.png
Rhombitrihexagonal tiling snub edge coloring.png
Uniform tiling 63-t012.png Uniform tiling 63-snub.png Uniform tiling 333-t0.pngUniform tiling 333-t1.png Uniform tiling 333-t02.pngUniform tiling 333-t12.png Uniform tiling 63-h12.png
Uniform tiling 333-snub.png
Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36
CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 63-t2.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t2.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 Uniform tiling 63-t0.png Rhombic star tiling.png Uniform tiling 63-t0.png

Floret pentagonal tiling[edit]

Floret pentagonal tiling
Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
Type Dual semiregular tiling
Coxeter diagram CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 6.pngCDel node fh.png
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 is a dual semiregular tiling of the Euclidean plane. It is one of 14 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.[1] Conway calls it a 6-fold pentille.[2] Each of its pentagonal faces has four 120° and one 60° angle.

It is the dual of the uniform tiling, snub trihexagonal tiling,[3] and has rotational symmetry of orders 6-3-2 symmetry.

P7 dual.png

See also[edit]

References[edit]

  1. ^ Five space-filling polyhedra by Guy Inchbald
  2. ^ 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)
  3. ^ Weisstein, Eric W., "Dual tessellation", MathWorld.

External links[edit]