A bug would like to walk from the center of one face of an icosahedron to the center of the opposite face. What is the quickest way possible, assuming he must stay on the surface, and at no time can he delve into the icosahedron's interior?

The icosahedron has edges of unit length.

If this minimum distance is \(d\), give your answer as \(d^2\) and to two decimal places.

**Clarification:** The center of a face is defined as the centroid of a triangle.

**Photo credit:** http://home.btconnect.com/

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