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 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

s with 4, 6, and 14 triangles meeting at each vertex. The image shows a

Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...

projection of the hyperbolic plane. It is labeled V4.6.14 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the dual tessellation of the truncated triheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex.

Naming

The name 3-7 kisrhombille is given by

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 ...

, seeing it as a 3-7 rhombic tiling, divided by a ''kis'' operator, adding a center point to each rhombus, and dividing into four triangles.

Symmetry

There are no mirror removal subgroups of ,3 The only small index subgroup is the alternation, ,3sup>+, (732).

Related polyhedra and tilings

Three isohedral (regular or quasiregular) tilings can be constructed from this tiling by combining triangles: It is topologically related to a polyhedra sequence; see

discussion Conversation is interactive communication between two or more people. The development of conversational skills and etiquette is an important part of socialization. The development of conversational skills in a new language is a frequent focus ...

. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and are the reflection domains for the (2,3,''n'')

triangle group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triang ...

s – for the heptagonal tiling, the important

(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, 84(''g'' − 1), of it ...

. See also the uniform tilings of the hyperbolic plane with (2,3,7) symmetry. The kisrhombille tilings can be seen as from the sequence of rhombille tilings, starting with the cube, with faces divided or kissed at the corners by a face central point. Morphing of modular tiling to 2 3 7 triangle tiling

Morphing of modular tiling to 2 3 7 triangle tiling

Just as the (2,3,7) triangle group is a quotient of the

modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...

(2,3,∞), the associated tiling is the quotient of the modular tiling, as depicted in the video at right.

References

* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)