A Wrapped set of Four Tangent Spheres

Geometry Level 5

Four spheres each of radius \( R \) are placed together, in a symmetrical configuration, such that each sphere is tangent to the other three spheres. The centers of the spheres form a regular tetrahedron.

Next, the set of four spheres is tightly wrapped by a sheet of some material of negligible thickness, such that the wrapper encloses all the four spheres and the space between them.

The volume enclosed by the wrapper can be written as \( a R^3 \). Find the factor \( a \), correct to 4 decimal places.

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