Soddy's hexlet

[
ellipse.]

In geometry, Soddy's hexlet is a chain of six spheres (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, these three spheres are shown as an outer circumscribing sphere "C" (blue), and two spheres "A" and "B" (green) above and below the plane of their centers. In addition, the hexlet spheres are tangent to a fourth sphere "D" (red in Figure 1), which is not tangent to the three others.

According to a theorem published by Frederick Soddy in 1937, [Harvnb|Soddy|1937] it is always possible to find a hexlet for any choice of mutually tangent spheres "A", "B" and "C". Indeed, there is an infinite family of hexlets related by rotation and scaling of the hexlet spheres (Figure 1); in this, Soddy's hexlet is the spherical analog of a Steiner chain of six circles. [Harvnb|Ogilvy|1990] Consistent with Steiner chains, the centers of the hexlet spheres lie in a single plane, on an ellipse. Soddy's hexlet was also discovered independently in Japan, as shown by Sangaku tablets from 1822 in the Kanagawa prefecture. [Harvnb|Rothman|1998]

Definition

Soddy's hexlet is a chain of six spheres, labelled "S"₁–"S"₆, each of which is tangent to three given spheres, "A", "B" and "C", that are themselves mutually tangent at three distinct points. (For consistency throughout the article, the hexlet spheres will always be depicted in grey, spheres "A" and "B" in green, and sphere "C" in blue.) The hexlet spheres are also tangent to a fourth fixed sphere "D" (always shown in red) that is not tangent to the three others, "A", "B" and "C".

Each sphere of Soddy's hexlet is also tangent to its neighbors in the chain; for example, sphere "S"₄ is tangent to "S"₃ and "S"₅. The chain is closed, meaning that every sphere in the chain has two tangent neighbors; in particular, the initial and final spheres, "S"₁ and "S"₆, are tangent to one another.

Annular hexlet

The annular Soddy's hexlet is a special case (Figure 2), in which the three mutually tangent spheres consist of a single sphere of radius "r" (blue) sandwiched between two parallel planes (green) separated by a perpendicular distance 2"r". In this case, Soddy's hexlet consists of six spheres of radius "r" packed like ball bearings around the central sphere and likewise sandwiched. The hexlet spheres are also tangent to a fourth sphere (red), which is not tangent to the other three.

The chain of six spheres can be rotated about the central sphere without affecting their tangencies, showing that there is an infinite family of solutions for this case. As they are rotated, the spheres of the hexlet trace out a torus (a doughnut-shaped surface); in other words, a torus is the envelope of this family of hexlets.

olution by inversion

The general problem of finding a hexlet for three given mutually tangent spheres "A", "B" and "C" can be reduced to the annular case using inversion. This geometrical operation always transforms spheres into spheres or into planes, which may be regarded as spheres of infinite radius. A sphere is transformed into a plane if and only if the sphere passes through the center of inversion. An advantage of inversion is that it preserves tangency; if two spheres are tangent before the transformation, they remain so afterwards. Thus, if the inversion transformation is chosen judiciously, the problem can be reduced to a simpler case, such as the annular Soddy's hexlet. Inversion is reversible; repeating an inversion in the same point returns the transformed objects to their original size and position.

Inversion in the point of tangency between spheres "A" and "B" transforms them into parallel planes, which may be denoted as "a" and "b". Since sphere "C" is tangent to both "A" and "B" and does not pass through the center of inversion, "C" is transformed into another sphere "c" that is tangent to both planes; hence, "c" is sandwiched between the two planes "a" and "b". This is the annular Soddy's hexlet (Figure 2). Six spheres "s"₁–"s"₆ may be packed around "c" and likewise sandwiched between the bounding planes "a" and "b". Re-inversion restores the three original spheres, and transforms "s"₁–"s"₆ into a hexlet for the original problem. In general, these hexlet spheres "S"₁–"S"₆ have different radii.

An infinite variety of hexlets may be generated by rotating the six balls "s"₁–"s"₆ in their plane by an arbitrary angle before re-inverting them. The envelope produced by such rotations is the torus that surrounds the sphere "c" and is sandwiched between the two planes "a" and "b"; thus, the torus has an inner radius "r" and outer radius 3"r". After the re-inversion, this torus becomes a Dupin cyclide (Figure 3).

Dupin cyclide

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. [Harvnb|Coxeter|1952] This result was probably known to Charles Dupin, who discovered the cyclides that bear his name in his 1803 dissertation under Gaspard Monge. [Harvnb|O'Connor|Robertson|2000]

Relation to Steiner chains

The intersection of the hexlet with the plane of its spherical centers produces a Steiner chain of six circles.

ee also

*Descartes' theorem

Notes

References

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External links

*mathworld|title=Hexlet|urlname=Hexlet
*cite web|url=http://members.ozemail.com.au/~llan/soddy.html|title=Animation of Soddy's hexlet|author= B. Allanson