### 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 $\phi = 2 \textrm{cos} \frac{\pi}{5}$ 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.

$S: z \mapsto - \frac{1}{z}$

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

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 $\phi = 2 \textrm{cos} \frac{\pi}{5}$ 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.

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