Soddy's hexlet

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 a central sphere (red), and two spheres (not shown) above and below the plane the centers of the hexlet spheres lie on. In addition, the hexlet spheres are tangent to a fourth sphere (blue in Figure 1), which is not tangent to the three others.

According to a theorem published by Frederick Soddy in 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. 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.

Soddy's hexlet is a chain of six spheres, labeled S₁–S₆, 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 S₄ is tangent to S₃ and S₅. The chain is closed, meaning that every sphere in the chain has two tangent neighbors; in particular, the initial and final spheres, S₁ and S₆, are tangent to one another.

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 2r. 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.

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