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Kollros' Theorem


For every ring containing p spheres, there exists a ring of q spheres, each touching each of the p spheres, where

 1/p+1/q=1/2,
(1)

which can also be written

 (p-2)(q-2)=4.
(2)

This was stated without proof by Jakob Steiner and proved by Kollros in 1938.

The hexlet is a special case with p=3. if more than one turn is allowed, then

 (p-2r)(q-2s)=4rs,
(3)

where r and s are the numbers of turns on both necklaces before closing (M. Buffet, pers. comm., Feb. 14, 2003).


See also

Bowl of Integers, Hexlet, Sphere, Tangent Spheres

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References

Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta Math. 18, 113-121, 1952.Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 50, 1976.

Referenced on Wolfram|Alpha

Kollros' Theorem

Cite this as:

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

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