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

right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...

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 th ...

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 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 ...

which has one square and one heptagon and one tetrakaidecagon at each vertex.

Naming

The name 3-7 kisrhombille is given by Conway, 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 triangle ...

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. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...

. 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 of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...

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

References

John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...

, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)