Inscribe a regular (-ish) polyhedron within a sphere, so the vertices lie on the sphere. This represents material that must be added (padding) to the polyhedron to make it a sphere.

Inscribe a sphere within a polyhedron, so that the faces are tangent to the sphere. This represents material that must be cut away from the polyhedron to make it a sphere.

(The inspiration was, by one of these metrics, the icosahedron is the most spherelike polyhedron, whereas by the other, the dodecahedron is.)

Consider a sphere that is tangent to the edges of a wireframe polyhedron. Does this represent anything special? Wildly speculating, suppose you are permitted both to add and cut away material from the polyhedron to make it a sphere. What is the minimum total material that must be involved?

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