[wvmcbiot] 12 kissing spheres

Place a unit sphere centered on each vertex of a regular icosahedron with edge length 2: the spheres are tangent.  Fit one more sphere in the center of the icosahedron, just large enough to be tangent to the shell of 12.  What is the radius of this central sphere?  I would be surprised if it too is a unit sphere.

Seeking the aesthetic pleasure of spheres nicely fitting together, vaguely analogous to the hexagonal close packing of circles on a plane.  We need some way of making them stick to each other.

Instead of a central sphere, design a shape, roughly a dodecahedron with curved faces, so that the spheres nicely fit against the curved faces.

Take those thirteen points and form the Voronoi partition.  The central region is a regular dodecahedron.  The outer regions are infinite, truncate them with another shape.  Perhaps those outer spheres.

Octahedron.  Other polyhedra?

Spheres on the vertices of a icosahedral geodesic dome don't quite work: the edge lengths are all slightly different, so identical spheres won't be tangent.  The edge lengths can be made all the same by sacrificing "dome"-ness: simply have an icosahedron with each triangular face subdivided into equal triangles.  The inner shape that is tangent to all the outer spheres is a rounded off dodecahedron.