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.