View source for Snub trihexagonal tiling

{{Uniform tiles db|Uniform tiling stat table|Unh}}
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 (geometry)|vertex]]. It has [[Schläfli symbol]] ''sr{3,6}''. The [[snub tetrahexagonal tiling]] is a related hyperbolic tiling with Schläfli symbol ''sr{4,6}''.

[[John Horton Conway|Conway]] calls it a '''snub hextille''', constructed as a [[Snub (geometry)|snub]] operation applied to a [[hexagonal tiling]] (hextille).

There are three [[List of regular polytopes#Euclidean tilings|regular]] and eight [[List of uniform tilings|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. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".)

== 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]]).<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E</ref> The lattice domain (red rhombus) repeats 6 distinct circles. <!--The packing density is ... % coverage.)--> The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from the [[triangular tiling#circle packing|triangular tiling]].
:[[File:Snub Trihexagonal Circle Packing with Colored Circles.svg|333x333px]]

== Related polyhedra and tilings ==
[[File:2-Uniform Tiling 20 Colored by Regular Polygon Orbits.svg|thumb|250x250px|There is one related [[2-uniform tiling]], which mixes the [[vertex configuration]]s 3.3.3.3.6 of the snub trihexagonal tiling and 3.3.3.3.3.3 of the [[triangular tiling]].]]
{{Hexagonal_tiling_small_table}}

=== Symmetry mutations ===
This semiregular tiling is a member of a sequence of [[Snub (geometry)|snubbed]] polyhedra and tilings with vertex figure (3.3.3.3.''n'') and [[Coxeter–Dynkin diagram]] {{CDD|node_h|n|node_h|3|node_h}}. These figures and their duals have (n32) rotational [[Orbifold notation|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 [[digon]]s.
{{Snub table}}

{{-}}

== <span class="anchor" id="Floret pentagonal tiling"></span> 6-fold pentille tiling ==

{{Infobox face-uniform tiling|
  name=Floret pentagonal tiling|
  Image_File=[[File:Tiling_snub_3-6_left_dual_simple.svg|280px]]|
  Type=[[List of uniform tilings|Dual semiregular tiling]]|
  Cox={{CDD|node_fh|3|node_fh|6|node_fh}}|
  Face_List=[[Pentagon|irregular pentagons]]|
  Symmetry_Group=p6, [6,3]<sup>+</sup>, (632)|
  Rotation_Group=p6, [6,3]<sup>+</sup>, (632)|
  Face_Type=V3.3.3.3.6<br>Face figure: [[File:Tiling_snub_3-6_left_dual_face.svg|113px|center]]|
  Dual=Snub trihexagonal tiling|
  Property_List=[[face-transitive]], [[Chirality (mathematics)|chiral]]|
}}
In [[geometry]], the '''6-fold pentille''' or '''floret pentagonal tiling''' is a dual semiregular tiling of the Euclidean plane.<ref>John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'', 2008, {{isbn|978-1-56881-220-5}}, {{cite web |url=http://www.akpeters.com/product.asp?ProdCode=2205 |title=A K Peters, LTD. - The Symmetries of Things |accessdate=2012-01-20 |url-status=dead |archiveurl=https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205 |archivedate=2010-09-19 }} (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p. 288, table)</ref> It is one of the 15 known [[Isohedral figure|isohedral]] [[pentagon tiling]]s. Its six pentagonal tiles radiate out from a central point, like petals on a [[flower]].<ref>[https://www.steelpillow.com/polyhedra/five_sf/five.html Five space-filling polyhedra] by Guy Inchbald</ref> Each of its pentagonal [[Face (geometry)|faces]] has four 120° and one 60° angle.

It is the dual of the uniform snub trihexagonal tiling,<ref>{{MathWorld | urlname=DualTessellation | title=Dual tessellation}}</ref> and has [[List of planar symmetry groups#Wallpaper groups|rotational symmetries of orders 6-3-2]] symmetry.
:[[File:P7 dual.png|320px]]

=== Variations ===
The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedral [[Pentagonal tiling#Monohedral convex pentagonal tilings|pentagonal tiling]] type 5. In one limit, an edge-length goes to zero and it becomes a [[deltoidal trihexagonal tiling]].
{| class=wikitable
!General
!Zero length<BR>degenerate
!colspan=4|Special cases
|- align=center
|[[File:P5-type5.png|160px]]<BR>[[:File:Snub_trihexagonal_tiling_variations.gif|(See animation)]]
|[[File:1-uniform 6 dual.svg|160px]]<BR>Deltoidal trihexagonal tiling
|[[File:Floret_pentagonal_tiling-v0.svg|190px]]
|[[File:Floret_hexagonal_tiling-v4.svg|190px]]
|[[File:Floret_hexagonal_tiling-v2.svg|200px]]
|[[File:Floret_hexagonal_tiling-v3.svg|160px]]
|- align=center
|[[File:Prototile p5-type5.png|120px]]<BR>a=b, d=e<BR>A=60°, D=120°
|[[File:Tiling face 3-4-6-4.svg|80px]]<BR>a=b, d=e, '''c=0'''<BR>A=60°, 90°, 90°, D=120°
|[[File:Floret_pentagonal_tiling-v0-tile.svg|150px]]<BR>a=b=2c=2d=2e<BR>A=60°, B=C=D=E=120°
|[[File:Floret_hexagonal_tiling-v4-tile.png|180px]]<BR>a=b=d=e<BR>A=60°, D=120°, E=150°
|[[File:hexatile-parallelogram.svg|150px]]<BR>2a=2b=c=2d=2e<BR>0°, A=60°, D=120°
|[[File:hexatile-trapzoid.svg|150px]]<BR>a=b=c=d=e<BR>0°, A=60°, D=120°
|}

=== Related k-uniform and dual k-uniform tilings ===
There are many [[List of k-uniform tilings|''k''-uniform tilings]] whose duals mix the 6-fold florets with other tiles; for example, labeling '''F''' for V3<sup>4</sup>.6, '''C''' for [[Cairo pentagonal tiling|V3<sup>2</sup>.4.3.4]], '''B''' for [[Prismatic pentagonal tiling|V3<sup>3</sup>.4<sup>2</sup>]], '''H''' for V3<sup>6</sup>:
{| class=wikitable
!uniform (snub trihexagonal)
! colspan="2" |2-uniform
! colspan="3" |3-uniform
|- align=center
|'''F''', p6 (t=3, e=3)
|'''FH''', p6 (t=5, e=7)
|'''FH''', p6m (t=3, e=3)
|'''FCB''', p6m (t=5, e=6)
|'''FH<sup>2</sup>''', p6m (t=3, e=4)
|'''FH<sup>2</sup>''', p6m (t=5, e=5)
|-
|[[File:Snub Trihexagonal Original.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 1.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 2.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 3.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 4.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 5.svg|alt=|200x200px]]
|-
!dual uniform (floret pentagonal)
! colspan="2" |dual 2-uniform
! colspan="3" |dual 3-uniform
|-
|[[File:Floret Pentagonal Original.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 1.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 2.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 3.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 4.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 6.svg|alt=|200x200px]]
|-
! colspan="2" |3-uniform
! colspan="4" |4-uniform
|- align=center
|'''FH<sup>2</sup>''', p6 (t=7, e=9)
|'''F<sup>2</sup>H''', cmm (t=4, e=6)
|'''F<sup>2</sup>H<sup>2</sup>''', p6 (t=6, e=9)
|'''F<sup>3</sup>H''', p2 (t=7, e=12)
|'''FH<sup>3</sup>''', p6 (t=7, e=10)
|'''FH<sup>3</sup>''', p6m (t=7, e=8)
|-
|[[File:Snub Trihexagonal Variation 6.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 8.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 9.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 10.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 11.svg|alt=|200x200px]]
|[[File:Snub Trihexagonal Variation 12.svg|alt=|200x200px]]
|-
! colspan="2" |dual 3-uniform
! colspan="4" |dual 4-uniform
|-
|[[File:Floret Pentagonal Variation 7.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 8.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 9.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 10.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 11.svg|alt=|200x200px]]
|[[File:Floret Pentagonal Variation 12.svg|alt=|200x200px]]
|}

=== Fractalization ===
Replacing every V3<sup>6</sup> hexagon by a [[Rhombitrihexagonal tiling|rhombitrihexagon]] furnishes a 6-uniform tiling, two vertices of 4.6.12 and two vertices of 3.4.6.4.

Replacing every V3<sup>6</sup> hexagon by a [[Truncated hexagonal tiling|truncated hexagon]] furnishes a 8-uniform tiling, five vertices of 3<sup>2</sup>.12, two vertices of  3.4.3.12, and one vertex of 3.4.6.4.

Replacing every V3<sup>6</sup> hexagon by a [[Truncated trihexagonal tiling|truncated trihexagon]] furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.4<sup>2</sup>.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 <math>1+\frac{1}{\sqrt{3}}:2+\frac{2}{\sqrt{3}}</math> in the rhombitrihexagonal; <math>1+\frac{2}{\sqrt{3}}:2+\frac{4}{\sqrt{3}}</math> in the truncated hexagonal; and <math>1+\sqrt{3}:2+2\sqrt{3}</math> in the truncated trihexagonal).
{| class="wikitable"
|+Fractalizing the Snub Trihexagonal Tiling using the [[Rhombitrihexagonal tiling|Rhombitrihexagonal]], [[Truncated hexagonal tiling|Truncated Hexagonal]] and [[Truncated trihexagonal tiling|Truncated Trihexagonal]] Tilings
!Rhombitrihexagonal
!Truncated Hexagonal
!Truncated Trihexagonal
|-
|[[File:Snub Trihexagonal Fractalization 1.svg|frameless|333x333px]]
|[[File:Snub Trihexagonal Fractalization 2.svg|frameless|333x333px]]
|[[File:Snub Trihexagonal Fractalization 3.svg|frameless|333x333px]]
|-
|[[File:Snub Trihexagonal Dual Fractalization 1.svg|frameless|333x333px]]
|[[File:Snub Trihexagonal Dual Fractalization 2.svg|frameless|333x333px]]
|[[File:Snub Trihexagonal Dual Fractalization 3.svg|frameless|333x333px]]
|}

==== Related tilings====

{{Dual hexagonal_tiling_table}}

==See also==
{{Commons category|Uniform tiling 3-3-3-3-6 (snub trihexagonal tiling)}}
* [[Tilings of regular polygons]]
* [[List of uniform tilings]]

== References ==
{{Reflist}}
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]
* {{cite book | author1=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (Chapter 2.1: ''Regular and uniform tilings'', p.&nbsp;58-65)
*{{The Geometrical Foundation of Natural Structure (book)}} p.&nbsp;39
* Keith Critchlow, ''Order in Space: A design source book'', 1970, p.&nbsp;69-61, Pattern R, Dual p.&nbsp;77-76, pattern 5
* Dale Seymour and [[Jill Britton]], ''Introduction to Tessellations'', 1989, {{isbn|978-0866514613}}, pp.&nbsp;50–56, dual rosette tiling p.&nbsp;96, p.&nbsp;114

==External links==
* {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
* {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|s3s6s - snathat - O11}}

{{Tessellation}}

[[Category:Chiral figures]]
[[Category:Euclidean tilings]]
[[Category:Isogonal tilings]]
[[Category:Semiregular tilings]]
[[Category:Snub tilings]]