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Hexlet


Hexlet

Consider two mutually tangent (externally) spheres A and B together with a larger sphere C inside which A and B are internally tangent. Then construct a chain of spheres each tangent externally to A, B and internally to C (so that C encloses the chain as well as the two original spheres). Surprisingly, every such chain closes into a "necklace" after six spheres, regardless of where the first sphere is placed.

This beautiful and amazing result due to Soddy (1937) is a special case of Kollros' theorem. It can be demonstrated using inversion of six identical spheres around an equal center sphere, all of which are sandwiched between two planes (Wells 1991, pp. 120 and 232). This result was given in a Sangaku problem from Kanagawa Prefecture in 1822, more than a century before it was published by Soddy (Rothman 1998).

Moreover, the centers of the six spheres in the necklace and their six points of contact all lie in a plane. Furthermore, there are two planes which touch each of the six spheres, one on either side of the necklace. Finally, the radii r_i of the spheres are related by

 1/(r_1)+1/(r_4)=1/(r_2)+1/(r_5)=1/(r_3)+1/(r_6)

(Rothman 1998).

Soddy's bowl of integers contains an infinite number of nested hexlets. The centers of a Soddy hexlet always lie on an ellipse (Ogilvy 1990, p. 63).


See also

Bowl of Integers, Coxeter's Loxodromic Sequence of Tangent Circles, Daisy, Kollros' Theorem, Seven Circles Theorem, Steiner Chain, Tangent Spheres

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References

Allanson, B. "Soddy's Hexlet." http://members.ozemail.com.au/~llan/soddy.html.Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta Math. 18, 113-121, 1952.Crane, E. "Soddy's Hexlet." http://www.dpmms.cam.ac.uk/~etc21/hexlet/hexlet3.html.Gosset, T. "The Hexlet." Nature 139, 251-252, 1937.Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 49-50, 1976.Morley, F. "The Hexlet." Nature 139, 72-73, 1937.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 60-72, 1990.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.Soddy, F. "The Bowl of Integers and the Hexlet." Nature 139, 77-79, 1937.Soddy, F. "The Hexlet." Nature 139, 154 and 252, 1937.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 120 and 231-232, 1991.

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Hexlet

Cite this as:

Weisstein, Eric W. "Hexlet." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hexlet.html

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