Any four mutually tangent spheres determine six points of tangency. A pair of tangencies 
is said to be opposite if the
two spheres determining 
are distinct from the two spheres determining 
. The six tangencies are therefore grouped into three opposite
pairs corresponding to the three ways of partitioning four spheres into two pairs.
These three pairs of opposite tangencies are coincident (Altshiller-Court 1979, p. 231;
Eppstein 2001).
A special case of tangent spheres is given by Soddy's hexlet, which consists of a chain of six spheres externally tangent to two mutually tangent spheres and internally tangent to a circumsphere. The bends of the circles in the chain obey the relationship
 |
(1)
|
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A Sangaku problem from 1798 asks to distribute 30 identical spheres of radius 
such that they are tangent to a single central sphere of radius
and to four other small spheres. This
can be accomplished (left figure) by placing the spheres at the vertices of an icosidodecahedron (right figure) of side length
, where the radii 
and 
are given by
 |  |  |
(2)
|
 |  |  |
(3)
|
(Rothman 1998).
In general, the bends of five mutually tangent spheres are related by
 |
(4)
|
Solving for 
gives
![kappa_5^+/-=1/2{kappa_1+kappa_2+kappa_3+kappa_4+/-[6(kappa_1kappa_2+kappa_1kappa_3+kappa_1kappa_4+kappa_2kappa_3+kappa_2kappa_4+kappa_3kappa_4)-3(kappa_1^2+kappa_2^2+kappa_3^2+kappa_4^2)]^(1/2)}.](/images/equations/TangentSpheres/NumberedEquation3.svg) |
(5)
|
(Soddy 1937a). Gosset (1937) pointed out that the expression under the square root sign is given by
 |
(6)
|
where 
is the volume of the tetrahedron
having vertices at the centers of the corresponding four spheres. Therefore, the
equation for 
can be written simplify as
 |
(7)
|
where
 |  |  |
(8)
|
 |  |  |
(9)
|
(Soddy 1937b).
In addition, the tetrahedra formed by joining the four points of contact of any one sphere with the other four (when all five are in mutual contact) have opposite edges whose product is the constant
 |
(10)
|
and the volume of these tetrahedra is
 |
(11)
|
(Soddy 1937b). Gosper has further extended this result to 
mutually tangent 
-dimensional hyperspheres,
whose curvatures satisfy
 |
(12)
|
Solving for 
gives
 |
(13)
|
For (at least) 
and 3, the radical equals
 |
(14)
|
where 
is the content of the simplex
whose vertices are the centers of the 
independent hyperspheres.
The radicand can also become negative,
yielding an imaginary 
. For 
, this corresponds to a sphere touching three large bowling
balls and a small BB, all mutually tangent, which is an impossibility.
