Snub trihexagonal tiling

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).Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E 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.

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File:2-Uniform Tiling 20 Colored by Regular Polygon Orbits.svg, which mixes the vertex configurations 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}}

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 {{CDD|node_h|n|node_h|3|node_h}}. 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.

{{Snub table}}

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{{Infobox face-uniform tiling|

name=Floret pentagonal tiling|

Image_File=280px|

Type=Dual semiregular tiling|

Cox={{CDD|node_fh|3|node_fh|6|node_fh}}|

Face_List=irregular pentagons|

Symmetry_Group=p6, [6,3]⁺, (632)|

Rotation_Group=p6, [6,3]⁺, (632)|

Face_Type=V3.3.3.3.6
Face figure: File:Tiling_snub_3-6_left_dual_face.svg|

Dual=Snub trihexagonal tiling|

Property_List=face-transitive, chiral|

}}

In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.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) It is one of the 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on a flower.[https://www.steelpillow.com/polyhedra/five_sf/five.html Five space-filling polyhedra] by Guy Inchbald Each of its pentagonal faces has four 120° and one 60° angle.

It is the dual of the uniform snub trihexagonal tiling,{{MathWorld | urlname=DualTessellation | title=Dual tessellation}} and has rotational symmetries of orders 6-3-2 symmetry.

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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.

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

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 $1+\backslash frac\{1\}\{\backslash sqrt\{3\}\}:2+\backslash frac\{2\}\{\backslash sqrt\{3\}\}$ in the rhombitrihexagonal; $1+\backslash frac\{2\}\{\backslash sqrt\{3\}\}:2+\backslash frac\{4\}\{\backslash sqrt\{3\}\}$ in the truncated hexagonal; and $1+\backslash sqrt\{3\}:2+2\backslash sqrt\{3\}$ in the truncated trihexagonal).

{{Dual hexagonal_tiling_table}}

{{Commons category|Uniform tiling 3-3-3-3-6 (snub trihexagonal tiling)}}

• Tilings of regular polygons
• List of uniform tilings

{{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. 58-65)
• {{The Geometrical Foundation of Natural Structure (book)}} 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

• {{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