The golden ratio,

$$\phi =\frac{1+\sqrt{5}}{2}$$

appears (among other polyhedra) in the Platonic solids *icosahedron* and *dodecahedron*, and it's quite easy to see the significance of the discriminant $d_\phi = 5$ in those two solids. Both $\phi$ and the plastic constant,

$$P=\frac{(9+\sqrt{3d})^{1/3}+(9-\sqrt{3d})^{1/3}}{2^{1/3}\cdot9^{1/3}}=1.324717\dots$$

with $d = 23$, can be found in the vertices of the *snub icosidodecadodecahedron*,

$\hskip2.8in$

$\hskip1.3in$(*Image courtesy of wikipedia and Robert Webb's Stella software*)

This may be a silly question, but why does $P$ appear here? And does $d = 23$ have any geometric significance to that solid? Note that $P$ is also,

$$P = \frac{e^{\pi i/24}}{\sqrt{2}}\frac{\eta(\tau)}{\eta(2\tau)}=1.324717\dots$$

$$\tau = \frac{1+\sqrt{-23}}{2}$$

where $\eta(\tau)$ is the *Dedekind eta function*.