# Tetrahexagonal tiling

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.

Tetrahexagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.6)2
Schläfli symbol r{6,4} or ${\begin{Bmatrix}6\\4\end{Bmatrix}}$ rr{6,6}
r(4,4,3)
t0,1,2,3(∞,3,∞,3)
Wythoff symbol 2 | 6 4
Coxeter diagram     or        or         Symmetry group [6,4], (*642)
[6,6], (*662)
[(4,4,3)], (*443)
[(∞,3,∞,3)], (*3232)
Dual Order-6-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

## Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [6,4] kaleidoscope. Removing the last mirror, [6,4,1+], gives [6,6], (*662). Removing the first mirror [1+,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1+,6,4,1+], leaving [(3,∞,3,∞)] (*3232).

UniformColoring FundamentalDomains Schläfli Symmetry        r{6,4} r{4,6}​1⁄2 r{6,4}​1⁄2 r{6,4}​1⁄4 [6,4](*642)     [6,6] = [6,4,1+](*662)   [(4,4,3)] = [1+,6,4](*443)   [(∞,3,∞,3)] = [1+,6,4,1+](*3232)     or   r{6,4} rr{6,6} r(4,3,4) t0,1,2,3(∞,3,∞,3)          =        =        =     or   ## Symmetry

The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold.