Snub Disphenoid

One of the convex Deltahedra also known as the Siamese Dodecahedron. It is Johnson Solid $J_{84}$.

The coordinates of the Vertices may be found by solving the set of four equations

$$1+{x_2}^2+{z_1}^2 = 4$$

$$(x_2-1)^2+(z_3-z_1)^2 = 4$$

$$1+{x_2}^2+(z_3-z_2)^2 = 4$$

$${x_2}^2+{x_2}^2+(z_2-z_1)^2 = 4$$

for the four unknowns $x_2$, $z_1$, $z_2$, and $z_3$. Numerically,

$$x_2 = 1.28917$$

$$z_1 = 1.15674$$

$$z_2 = 1.97898$$

$$z_3 = 3.13572.$$

The analytic solution requires solving the Cubic Equation and gives

$$x_2 = 1-7\cdot 2^{-2/3}(1-i\sqrt{3})\alpha^{-1}-{\textstyle{1\over 6}}\cdot 2^{-1/3}(1+i\sqrt{3})\alpha$$

$$z_1 = {\textstyle{1\over 3}}\cdot 2^{-1/2} [-48+6\beta(1+i\sqrt{3})+\beta^2(1-i\sqrt{3})$$

$$\phantom{=} +147\beta\gamma(\sqrt{3}-i)+42\beta^2\gamma(\sqrt{3}+i)]^{1/2},$$

where

$$\alpha \equiv (12i\sqrt{237}-54)^{1/3}$$

$$\beta \equiv 3^{1/3}(2i\sqrt{237}-9)^{1/3}$$

$$\gamma \equiv (9i+2\sqrt{237}\,)^{-1}.$$