John Sharp has this nice article, Beyond the Golden Section - the Golden tip of the iceberg where he recalls how certain constants appear in the,
I. Snub cube:
$$T^3-T^2-T=1\tag1$$
with tribonacci constant $T \approx 1.83929$.
$$x^3-x^2-x=\phi\tag2$$
with golden ratio $\phi$ and root $x \approx 1.94315$.
Note: Incidentally, these two solids are the only Archimedean solids that are chiral (with mirror images).
However, in the Wikipedia for the snub dodecahedron, we find instead the equations,
$$y=z-\frac1{z}\tag3$$
$$z^3-2z = \phi\tag4$$
Q: Where did Sharp get the "tribonacci-like" equation $(2)$? And excluding the obvious relation $x^3-x^2-x = z^3-2z$, how is it related to $(3)$ or $(4)$?
From a related discussion in this recent MO answer, it seems Sharp got the tribonacci-like equation for the snub dodecahedron from Appendix A of "Closed-Form Expressions for Uniform Polyhedra and Their Duals" by Peter Messer.
I. It is known that circumradius $R$ for a snub cube of unit edge length is given by
$$R = \frac12\sqrt{\frac{3-T}{2-T}}=1.34371\dots$$
where $T$ is the real root of,
$$T^3-T^2-T=1\tag1$$
II. On a hunch, after some experimentation, I found that for the snub dodecahedron, one just uses the same formula
$$R = \frac12\sqrt{\frac{3-x}{2-x}}=2.15584\dots$$
where $x$ is the real root of,
$$x^3-x^2-x=\phi\tag2$$
And given
$$z^3-2z = \phi\tag3$$
it turns out the relationship between Sharp's $(2)$ and Wikipedia's $(3)$ is simply,
$$x=\frac{\phi+z}z$$