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 hexagonal tiling is a semiregular tiling of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

. There are 2

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 ...

s (12-sides) and one

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

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 ...

. As the name implies this tiling is constructed by a

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

operation applies 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 ...

, leaving dodecagons in place of the original

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 ...

s, and new triangles at the original vertex locations. It is given an extended

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 ''t''. Conway calls it a truncated hextille, constructed as a

operation applied to a

(hextille). There are 3 regular and 8 semiregular tilings in the plane.

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 differ ...

of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.) Uniform polyhedron-63-t01

Topologically identical tilings

The

al faces can be distorted into different geometries, such as:

Related polyhedra and tilings

Wythoff constructions from hexagonal and triangular tilings

Like the

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

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 the f ...

s that can be based from the regular hexagonal tiling (or the dual

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 ...

). 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 is topologically related as a part of sequence of uniform truncated polyhedra with

vertex configuration In geometry, a vertex configurationCrystallography ...

s (3.2n.2n), and ,3

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

symmetry.

Related 2-uniform tilings

Two 2-uniform tilings are related by dissected the

s into a central hexagonal and 6 surrounding triangles and squares.

Circle packing

The truncated hexagonal 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.Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G 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 of ...

). This is the lowest density packing that can be created from a uniform tiling. : 1-uniform-4-circlepack

Triakis triangular tiling

The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral

with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by

face configuration In geometry, a vertex configurationCrystallography ...

V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles. Conway calls it a kisdeltille, constructed as a

kis Kis or KIS may refer to: Places * Kiş, Khojavend, Azerbaijan * Kiş, Shaki, Azerbaijan * Kish (Sumer) (Sumerian: Kiš), an ancient city in Sumer * Kis, Babol Kenar, a village in Mazandaran Province, Iran * Kis, Bandpey-ye Gharbi, a village in ...

operation applied to a

(deltille). In Japan the pattern is called asanoha for ''hemp leaf'', although the name also applies to other triakis shapes like the

triakis icosahedron In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron. Cartesian coordinates Let \phi be the golden ratio. The 12 po ...

and

triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube. It can be seen as an octahedron with triangular ...

. It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex. : P4 dual

It is one of eight

edge tessellation In geometry, an edge tessellation is a partition of the plane into non-overlapping polygons (a tessellation) with the property that the reflection of any of these polygons across any of its edges is another polygon in the tessellation. All of the r ...

s, tessellations generated by reflections across each edge of a prototile..

Related duals to uniform tilings

It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.

References

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

* (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * * Keith Critchlow, ''Order in Space: A design source book'', 1970, p. 69-61, Pattern E, Dual p. 77-76, pattern 1 * Dale Seymour and Jill Britton, ''Introduction to Tessellations'', 1989, , pp. 50–56, dual p. 117

External links

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