Is there a mapping from an icosahedron to a sphere?

Question

I recently found a mapping from a cube to a sphere (see Can someone please explain the cube to sphere mapping formula to me?), but is there also such a mapping from an icosahedron to a sphere? If so, is there also a way to map the sphere back to an icosahedron?

Answer 1

Contrary to the explanatiin there, I suppose a simple radial scaling $$\begin{pmatrix}x\cr y\cr z\end{pmatrix}\mapsto \begin{pmatrix}x/\sqrt{x^2+y^2+z^2}\cr y/\sqrt{x^2+y^2+z^2}\\z/\sqrt{x^2+y^2+z^2}\end{pmatrix}$$ would be good enough (esp. as the icosahedron is already close to a sphere.

For a map sphere $\to $ icosahedron, I would collect the unit normal vectors for the faces, then determine which of these has the largest scalar product with the given point on the sphere, then divide the point by that scalar product. (This maps to the circumscribed, not inscribed icosahedron, if that matters)