For every ring containing spheres , there exists a ring of
spheres , each
touching each of the spheres , where
(1)
which can also be written
(2)
This was stated without proof by Jakob Steiner and proved by Kollros in 1938.
The hexlet is a special case with . if more than one turn is allowed, then
(3)
where
and
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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