Tangent Spheres

Any four mutually tangent spheres determine six points of tangency. A pair of tangencies (t_i,t_j) is said to be opposite if the two spheres determining t_i are distinct from the two spheres determining t_j. 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


A Sangaku problem from 1798 asks to distribute 30 identical spheres of radius r such that they are tangent to a single central sphere of radius R 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 a, where the radii r and R are given by


(Rothman 1998).

In general, the bends of five mutually tangent spheres are related by


Solving for kappa_5 gives


(Soddy 1937a). Gosset (1937) pointed out that the expression under the square root sign is given by


where V is the volume of the tetrahedron having vertices at the centers of the corresponding four spheres. Therefore, the equation for kappa_5 can be written simplify as




(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


and the volume of these tetrahedra is


(Soddy 1937b). Gosper has further extended this result to n+2 mutually tangent n-dimensional hyperspheres, whose curvatures satisfy


Solving for kappa_(n+1) gives


For (at least) n=2 and 3, the radical equals


where V is the content of the simplex whose vertices are the centers of the n+1 independent hyperspheres. The radicand can also become negative, yielding an imaginary kappa_(n+1). For n=3, this corresponds to a sphere touching three large bowling balls and a small BB, all mutually tangent, which is an impossibility.

See also

Apollonian Gasket, Bowl of Integers, Hexlet, Soddy Circles, Sphere, Tangent Circles, Tetrahedron, Wada Basin

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Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, 1979.Eppstein, D. "Tangent Spheres and Triangle Centers." Amer. Math. Monthly 108, 63-66, 2001.Gosset, T. "The Hexlet." Nature 139, 251-252, 1937.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.Soddy, F. "The Kiss Precise." Nature 137, 1021, 1936.Soddy, F. "The Bowl of Integers and the Hexlet." Nature 139, 77-79, 1937a.Soddy, F. Nature 139, 252, 1937b.

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Tangent Spheres

Cite this as:

Weisstein, Eric W. "Tangent Spheres." From MathWorld--A Wolfram Web Resource.

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