The Soddy page.
Frederick Soddy 
1877-1960
Michael Fox, Adrian Oldknow, John Rigby and Sir Christopher Zeeman have submitted a two-part article
to the Mathematical Gazett giving details on their discoveries and their proofs of the generalisation of results about Soddy circles of a triangle to Soddy spheres of 4-ball tetrahedra. In the meantime the key results are summarised in the note below:
The Soddy spheres of a 4-ball tetrahedron (3pp) (Word
139 Kb)
The Oldknow article giving the constructions
for Cabri 3D:
Exploring 4-ball tetrahedra with Cabri 3D
(5pp) (Word
1293 Kb)
A zip file of the Cabri 3D
constructions (Zip
295 Kb)
(a) 
(b)
(c)
(a) The centres of 4 touching spheres
form a 4-ball tetrahedron
(b) Four spherical fruit can lie inside a
transparent spherical fruit bowl,
and touch a small spherical fruit in their
middle!
(c) Three of the authors hard at work in
the Hertford College SCR.