Soddy's hexlet

[2] 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 Kanagawa prefecture.

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.

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.

Thus, if the inversion transformation is chosen judiciously, the problem can be reduced to a simpler case, such as the annular Soddy's hexlet.

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.

[4] This result was probably known to Charles Dupin, who discovered the cyclides that bear his name in his 1803 dissertation under Gaspard Monge.

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

They analysed the packing problems in which circles and polygons, balls and polyhedrons come into contact and often found the relevant theorems independently before their discovery by Western mathematicians.

The sangaku about the hexlet was made by Irisawa Shintarō Hiroatsu in the school of Uchida Itsumi, and dedicated to the Samukawa Shrine in May 1822.

A replica of the sangaku was made from the record and dedicated to the Hōtoku museum in the Samukawa Shrine in August, 2009.

The third problem relates to Soddy's hexlet: "the diameter of the outer circumscribing sphere is 30 sun.

"[7] In his answer, the method for calculating the diameters of the balls is written down and, when converted into mathematical notation, gives the following solution.

Figure 1. A family of hexlets related by a rotation and scaling. The centers of the spheres fall on an ellipse , making it an elliptic hexlet.
Figure 2: An annular hexlet.
Figure 3: A Dupin cyclide, through which the hexlet spheres rotate, always touching. The cyclide is tangent to an inner sphere, an outer sphere and two spheres above and below the "hole" in the "doughnut".
Figure 4: Steiner chain of six circles corresponding to a Soddy's hexlet.
Soddy's hexlet problem in japanese mathematics book Kokonsankan (1832).
Replica of Sangaku at Hōtoku museum in Samukawa Shrine .