Soddy's hexlet

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Soddy's hexlet is a theorem about mutually tangent spheres published by Frederick Soddy in 1937,^[1] and is the three-dimensional analog of the problem of Apollonius.^[2] Given three mutually tangent spheres (denoted A, B, and C), it is always possible to find a chain of six other spheres (S₁-S₆, the "hexlet") that are tangent to the three original spheres and successively tangent to each other. For example, sphere S₃ is tangent to A, B, and C, and also to spheres S₂ and S₄. The chain of spheres closes perfectly; for example, sphere S₆ is tangent to spheres S₅ and also to S₁. Furthermore, Soddy showed that there are an infinite number of such hexlets for a given set of three original spheres.

A Dupin cyclide. It is fairly easy to imagine a 1-parameter family of spheres inside this "tube". The tube is also the envelope of a 1-parameter family of spheres touching it (in circles) from the outside. This outer family is harder to visualize and includes two planes, which are "spheres through infinity".

The envelope of Soddy's hexlets is a Dupin cyclide, an inversion of the torus. Thus Soddy's construction shows that a cyclide of Dupin is the envelope of a 1-parameter family of spheres in two different ways, and each sphere in either family is tangent to two spheres in same family and three spheres in the other family.^[3] This result was probably known to Charles Dupin, who discovered the cyclides that bear his name in his 1803 dissertation under Gaspard Monge.^[4]

Soddy's hexlet was also discovered independently in Japan, as shown by Sangaku tablets from 1822 in the Kanagawa prefecture.^[5]

1 Simple proof by inversion
2 See also
3 Notes
4 References
5 External links

[edit] Simple proof by inversion

Soddy's hexlet theorem can be proved most readily by geometric inversion, which transforms spheres into spheres or into planes (which can be regarded as spheres of infinite radius). If the center of inversion lies on the sphere itself, the sphere is transformed into a plane; otherwise, it is transformed into another sphere.

Inversion in the point of tangency between spheres A and C transforms them into parallel planes, which may be denoted as a and c. Since sphere B is tangent to both A and C and since tangency is not affected by inversion, B must be transformed into another sphere b of radius r that is tangent to both planes; hence, b is sandwiched between the two planes a and c. Six balls (s₁-s₆) of the same radius r may be packed around b and also sandwiched between the bounding planes a and c, just as six pennies may be packed around a single penny on a table. Re-inversion restores the three original spheres, and transforms the six surrounding balls into the six spheres of Soddy's hexlet, S₁-S₆, which generally have different radii.

An infinite variety of hexlets may be generated by rotating the six balls s₁-s₆ in their plane by an arbitrary angle before re-inverting them. The envelope produced by such rotations is the torus that surrounds the sphere b and is sandwiched between the two planes a and c; thus, the torus has an inner radius r and outer radius 3r. After the re-inversion, this torus becomes a Dupin cyclide.

[edit] See also

Descartes' theorem

[edit] References

Coxeter, HSM (1952), “Interlocked rings of spheres”, Scripta Mathematica 18: 113–121 .
O'Connor, John J. & Robertson, Edmund F. (2000), “Pierre Charles François Dupin”, MacTutor History of Mathematics archive .
Ogilvy, C.S. (1990), Excursions in Geometry, Dover, ISBN 0-486-26530-7 .
Soddy, Frederick (1937), “The bowl of integers and the hexlet”, Nature (London) 139: 77–79 .
Rothman, T (1998), “Japanese Temple Geometry”, Scientific American 278: 85–91 .