I recently purchased tea where the tea bags were oddly tetrahedron-shaped rather than the common square tea bag shape. My girlfriend told me that the tetrahedron shape maximizes surface area, compared to the common type of bag. I responded why not make the tea bags a shape with a lot of convexities and concavities, like a 3D star or something, since such a shape would have even more surface area. She responded that the cost to produce such a complicated shape would likely be a lot higher, and so the tetrahedron must be some optimal polyhedron that maximizes the surface area to cost ratio.
Let's say for a polyhedron:
an edge costs $1 (since an edge implies a worker having to stitch those sides of the tea bag material together)
The polyhedron must be able to fit inside a sphere with radius 1 centimeter. (So we cannot construct a tetrahedron with huge faces and a small volume)
There can only be finite vertices, edges, and faces.
I want to emphasize that volume doesn't really matter. We just need a few chopped tea leaves to make a cup of tea. But we need those chopped leaves maximally distributed to quicken the dissolving process.
These constraints were chosen for simplicity's sake.
My question is, what polyhedron, if any, meets those 3 constraints and maximizes the surface area to cost ratio?