On the snub cube's $T^3-T^2-T=1$ and snub dodecahedron's $x^3-x^2-x=\phi$ 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_ $T \approx 1.83929$. **II.** Snub dodecahedron: $$x^3-x^2-x=\phi\tag2$$ 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,

$$x=\frac{\phi+z}z$$