geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the snub hexagonal tiling (or ''snub trihexagonal tiling'') is a

semiregular tiling Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Euc ...

of the Euclidean plane. There are four triangles and one hexagon on each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

. It has Schläfli symbol ''sr''. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol ''sr''. Conway calls it a snub hextille, constructed as a

snub A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub. In Awards and Lists For awards, the term "snub" ...

operation applied to a hexagonal tiling (hextille). There are three regular and eight 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 In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following differ ...

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 In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated '' packing de ...

, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (

kissing number In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...

).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 In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilate ...

. :

Snub Trihexagonal Circle Packing with Colored Circles

Related polyhedra and tilings

2-Uniform Tiling 20 Colored by Regular Polygon Orbits

Symmetry mutations

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 In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describe ...

. 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

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...

6-fold pentille tiling

, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of the 15 known

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

pentagon tiling In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular ...

s. Its six pentagonal tiles radiate out from a central point, like petals on a

flower A flower, sometimes known as a bloom or blossom, is the reproductive structure found in flowering plants (plants of the division Angiospermae). The biological function of a flower is to facilitate reproduction, usually by providing a mechani ...

. Each of its pentagonal

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

has four 120° and one 60° angle. It is the dual of the uniform snub trihexagonal tiling, and has rotational symmetries of orders 6-3-2 symmetry. : P7 dual

Variations

The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedral

pentagonal tiling In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular ...

type 5. In one limit, an edge-length goes to zero and it becomes a

deltoidal trihexagonal tiling In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille.Conway, 200 ...

Related k-uniform and dual k-uniform tilings

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:

Fractalization

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+\frac:2+\frac

in the rhombitrihexagonal;

1+\frac:2+\frac

in the truncated hexagonal; and

1+\sqrt:2+2\sqrt

in the truncated trihexagonal).

Related tilings

References

* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008,

* (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * 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, , pp. 50–56, dual rosette tiling p. 96, p. 114

External links

* * * {{Tessellation Chiral figures Euclidean tilings Isogonal tilings Semiregular tilings Snub tilings