geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

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

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

, one

hexagon 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 is de ...

, and one

dodecagon In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon. Regular dodecagon A regular polygon, 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 ...

on each vertex. It has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

of ''tr''. Rhombic_truncated_trihexagonal_tiling

Names

Uniform colorings

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 diff ...

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 tilings, 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, 200 ...

. 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 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 3 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 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- Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: *triangle, quadrilateral, pentagon, hexagon, octagon ...

, dodeca- and triacontahedron, three

Catalan solid The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...

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

File:Kisrhombille in triangular exploded to hexagonal (yellow).svg, hexagonal
(as exploded triangular) File:Kisrhombille in triangular (yellow).svg,

triangular A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...

File:Kisrhombille in hexakis hexagonal.svg, triangular
(as hexakis hexagonal) File:Kisrhombille in triakis triangular.svg, triakis triangular

Construction from rhombille tiling

Conway Conway may refer to: Places United States * Conway, Arkansas * Conway County, Arkansas * Lake Conway, Arkansas * Conway, Florida * Conway, Iowa * Conway, Kansas * Conway, Louisiana * Conway, Massachusetts * Conway, Michigan * Conway Townshi ...

calls it a kisrhombille for his kis vertex bisector operation applied to the

rhombille tiling 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 ...

. 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 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 Truncation (geometry), truncated triangular 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 geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons, the ''cells'' of the arrangement, line segments and rays, the ''edg ...

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 Horton Conway, John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curv ...

)

wallpaper group A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...

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 tiling In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to t ...

s 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 of a convex polytope is a simple polytope of the same dimension, having a vertex for each Flag (geometry), flag of the original polytope and a Facet (geometry), facet for each face of any dimension of the original pol ...

polyhedra (

zonohedra In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

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

Notes

References

* * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''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 education, mathematics educator known for her educational books about mathematics. Career Britton was born on 6 November 1944. She taught for many years, at Dawson ...

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

External links

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