What is the center of mass of a regular icosahedron with edge length $2$ centimeters? The object is one mass, but is divided exactly in half and each half is a different material. One material is twice as dense as the other. Densities can be taken to be $1$ g/cm$^3$ and $2$ g/cm$^3$, respectively.
I want to use a circumscribed sphere, which touches all vertices, as a model for this. If that is a valid strategy, how can I calculate center of mass for a sphere which is made equally from two materials? I have learned how to calculate center of mass for a uniform sphere, but I have not been taught how to for a sphere like this.