On the snub cube's $T^3-T^2-T=1$ and snub dodecahedron's $x^3-x^2-x=\phi$

Question

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).

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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 $(4)$?

Answer 1

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,

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