In
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 c ...
, 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 Eucli ...
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 position ...
. It has
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
''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
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathemat ...
(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 den ...
, 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 of ...
).
[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 equilater ...
.
:
Related polyhedra and tilings
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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
, 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 visua ...
s.
6-fold pentille tiling
In
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 c ...
, 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.
:
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 pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not ...
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 V3
4.6, C for
V32.4.3.4, B for
V33.42, H for V3
6:
Fractalization
Replacing every V3
6 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 V3
6 hexagon by a
truncated hexagon furnishes a 8-uniform tiling, five vertices of 3
2.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V3
6 hexagon by a
truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.4
2.6, 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
in the rhombitrihexagonal;
in the truncated hexagonal; and
in the truncated trihexagonal).
Related tilings
See also
*
Tilings of regular polygons
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 Eucli ...
*
List of uniform tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dual ...
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