Noble tetragonal tetracontoctahedron

The noble tetragonal tetracontoctahedron is a noble polyhedron. Its 48 congruent faces are convex irregular quadrilaterals meeting at congruent order-8 vertices. It is a faceting of a semi-uniform truncated octahedral convex hull.

The ratio between the longest and shortest edges is 1:a ≈ 1:1.57021, where a is the positive real root of $$a^6-4a^4+5a^2-3$$.

Vertex coordinates
The vertex coordinates are all permutations of $$\left(±a,\,±b,\,0\right)$$, where $$\frac{a}{b} = \frac{1+\sqrt[3]{\frac{29-3\sqrt{93}}{2}}+\sqrt[3]{\frac{29+3\sqrt{93}}{2}}}{3} \approx 1.46557$$ is the real root of $$x^3-x^2-1$$.