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 truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

, and one

dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational sym ...

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 ...

of ''tr''. Rhombic_truncated_trihexagonal_tiling

Names

Uniform colorings

There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares.

Related 2-uniform tilings

The ''truncated trihexagonal tiling'' has three related

2-uniform tiling A ''k''-uniform tiling is a tiling of tilings of the plane by convex regular polygons, connected edge-to-edge, with ''k'' types of vertices. The 1-uniform tiling include 3 regular tilings, and 8 semiregular tilings. A 1-uniform tiling can be defi ...

s, one being a 2-uniform coloring of the semiregular

rhombitrihexagonal 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, 2 ...

. The first dissects the hexagons into 6 triangles. The other two dissect the

s into a central hexagon and surrounding triangles and square, in two different orientations.

Circle packing

The Truncated 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 3 other circles in the packing ( kissing number).Order in Space: A design source book, Keith Critchlow, p.74-75, pattern D : 1-uniform-3-circlepack

Kisrhombille tiling

The kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent

30-60-90 triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...

s with 4, 6, and 12 triangles meeting at each vertex. Subdividing the faces of these tilings creates the kisrhombille tiling. (Compare the disdyakis hexa-, dodeca- and triacontahedron, three

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...

s similar to this tiling.) File:Kisrhombille in deltoidal.svg, 3-6 deltoidal File:Kisrhombille in rhombille (blue).svg,

rhombille In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape ar ...

File:Kisrhombille in hexagonal (red).svg,

hexagonal In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

triangular

Construction from rhombille tiling

Conway calls it a kisrhombille for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 3-7 kisrhombille. It can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected

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 ...

divided into 6 triangles, or as an infinite arrangement of lines in six parallel families.) It is labeled V4.6.12 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 12 triangles.

Symmetry

The ''kisrhombille tiling'' triangles represent the fundamental domains of p6m, ,3(*632

orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...

)

wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformatio ...

symmetry. There are a number of small index subgroups constructed from ,3by mirror removal and alternation. ⁺,6,3creates *333 symmetry, shown as red mirror lines. ,3⁺creates 3*3 symmetry. ,3sup>+ is the rotational subgroup. The commutator subgroup is ⁺,6,3⁺ which is 333 symmetry. A larger index 6 subgroup constructed as ,3* also becomes (*333), shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12.

Related polyhedra and tilings

There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual

). Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The ''truncated triangular tiling'' is topologically identical to the hexagonal tiling.)

Symmetry mutations

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For ''p'' < 6, the members of the sequence are

omnitruncated In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortc ...

polyhedra (

zonohedra In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...

), shown below as spherical tilings. For ''p'' > 6, they are tilings of the hyperbolic plane, starting with the

truncated triheptagonal tiling In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of Uniform colorings There is only o ...

Notes

References

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

* Keith Critchlow, ''Order in Space: A design source book'', 1970, p. 69-61, Pattern G, Dual p. 77-76, pattern 4 * Dale Seymour and

Jill Britton Jill E. Britton (6 November 1944 – 29 February 2016) was a Canadian mathematics educator known for her educational books about mathematics. Career Britton was born on 6 November 1944. She taught for many years, at Dawson College in Westmount ...

, ''Introduction to Tessellations'', 1989, , pp. 50–56

External links

* * * {{Tessellation Euclidean tilings Isogonal tilings Semiregular tilings Truncated tilings