This is an image of four disjoint spheres in R3 with 12 common tangents.
This settles a question which has been open since at least 1998 in the affirmative.
These 12 tangents have 6 geometric permutations between them.
The three large spheres each have radius 4/5 and are centred at the vertices of an
equilateral triangle of side length (3)1/2.
The small sphere has radius 1/4 and is centred on the axis of symmetry of the triangle
and is located a distance 35/100 from the plane of the triangle.
Here is a maple script which draws this picture and checks these assertions.
.
Work of Sottile supported by the National Science Foundation
under CAREER Grant DMS-0538734 and Peter Gritzmann of the Technische
Universität München. Modified since: 17 April 2007 by Frank Sottile
.