Soddy's hexlet - Printable Version +- MKLab (https://mklab.gr) +-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1) +--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3) +---- Forum: ARTICLES (https://mklab.gr/forumdisplay.php?fid=13) +---- Thread: Soddy's hexlet (/showthread.php?tid=954)

Soddy's hexlet - mklabgr - 07-07-2026 Soddy's hexlet Summary Soddy’s hexlet is a remarkable result in geometry that describes a closed chain of six spheres, each touching its two neighboring spheres while also remaining tangent to three fixed, mutually tangent spheres. First rediscovered by Frederick Soddy in 1937—though it had already appeared on Japanese Sangaku tablets more than a century earlier—the theorem proves that such a six-sphere configuration always exists, regardless of the sizes of the three given spheres.  Even more surprisingly, there is not just one solution but an infinite family of Soddy’s hexlets, generated through rotations and scaling, making the construction the three-dimensional counterpart of the well-known Steiner chain of circles.  The article also explores the elegant mathematics behind the theorem using inversive geometry, showing how a complex arrangement of tangent spheres can be transformed into a much simpler configuration involving parallel planes before being mapped back to its original form. This approach reveals deep connections with geometric objects such as the Dupin cyclide, whose surface forms the envelope of the rotating hexlets, and explains how special cases produce elliptic, parabolic, or hyperbolic patterns depending on the spheres’ relative sizes.  Beyond its visual appeal, Soddy’s hexlet illustrates the power of geometric transformations, sphere packing, and symmetry, while highlighting the fascinating history of independent mathematical discovery across cultures. It remains an enduring example of how elegant geometric ideas can uncover unexpected relationships that continue to inspire both mathematicians and students today. ARTICLE