Carbon Beauty II

In the buckyball paper by Singerman and Martin, the genus 70 buckyball curve appears as the $p=11$ analogue of the Klein surface for $p=7$. The construction relies on the Hecke group $H5$, generated by

$S:z\mapsto -\frac{1}{z}$
$T:z\mapsto -\frac{1}{z+\phi }$

where $\phi =\frac{1+\sqrt{5}}{2}$ is the golden ratio. The golden ratio turns up in many places in noncommutative geometry, for example as weights for a quantum groupoid. Note that the modular group is also a Hecke group for $\phi =1$. By a theorem of Hecke, $H5$ is discrete precisely because $\phi =2\text{cos}\frac{\pi }{5}$ where 5 is an ordinal. Note that the special phase $\frac{\pi }{5}$ (or double this) also has nice properties in relation to the Jones polynomial, which is universal for quantum computation at a 5th root of unity.