Soddy's hexlet

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Soddy's hexlet is a theorem that establishes a fact about mutually tangent spheres. It was discovered by Frederick Soddy in 1937.

1 The theorem
2 Proof
3 See also
4 External links

[edit] The theorem

Consider two externally tangent spheres, $A\,$ and $B\,$ , and a third sphere $C\,$ that contains $A\,$ and $B\,$ and is tangent to them. One can construct a chain of spheres $S_1, S_2, S_3, \dots, S_n\,$ , with each $S_n\,$ tangent to $A\,$ and $B\,$ externally and $C\,$ internally and with $S_{n+1}\,$ externally tangent to $S_n\,$ . Soddy proved that, surprisingly, every such chain closes after six spheres -- that is, $S_6\,$ is tangent to $S_1\,$ .

[edit] Proof

The theorem can be proved using geometric inversion. Consider inverting spheres $A\,$ , $B\,$ , and $C\,$ in a sphere centered on the point of tangency of spheres $A\,$ and $C\,$ . Then $A\,$ and $C\,$ invert into parallel planes $A^\prime$ and $C^\prime$ , while $B\,$ inverts into another sphere $B^\prime$ . Since $B\,$ is tangent to $A\,$ and $C\,$ , it follows that $B^\prime$ is tangent to $A^\prime$ and $C^\prime$ ; therefore it is between them. We now construct six spheres $S_1^\prime, S_2^\prime, \ldots, S_6^\prime$ of the same diameter as $B^\prime$ , each one tangent to $A^\prime$ , $B^\prime$ , and $C^\prime$ , in a cyclic chain. We know that this is possible because the $S_n^\prime$ have the same diameter as each other and as $B^\prime$ . Now, we undo the inversion, and see that no matter where $S_1^\prime$ was placed, $S_1, S_2, \ldots, S_6$ form a cyclic chain, which was what we wanted.