Bisected Hexagonal Tiling on:
[Wikipedia]
[Google]
[Amazon]
In 
File:Kisrhombille in deltoidal.svg, 3-6 deltoidal
File:Kisrhombille in rhombille (blue).svg,
* 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, ''Introduction to Tessellations'', 1989, , pp. 50–56
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 truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, 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'' has ...
, 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 positio ...
. It has Schläfli symbol of ''tr''.
Names
Uniform colorings
There is only oneuniform 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 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 semiregularrhombitrihexagonal 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 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 into a central hexagon and surrounding triangles and square, in two different orientations.
Circle packing
The Truncated trihexagonal tiling can be used as acircle 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 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
:
Kisrhombille tiling
The kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent30-60-90 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:
* unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cyc ...
, 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 s ...
s similar to this tiling.)
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 a ...
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
.svg.html){kind=link}
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 bisectedtriangular 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 ...
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)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 transformati ...
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 eightuniform 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 space, hyperbolic plane. Uniform tilings ar ...
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 equilate ...
). 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 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.
See also
* Tilings of regular polygons *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 ...
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, ''Introduction to Tessellations'', 1989, , pp. 50–56
External links
* * * {{Tessellation Euclidean tilings Isogonal tilings Semiregular tilings Truncated tilings