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
(3)
where
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. Referenced
on Wolfram|Alpha Kollros' Theorem
Cite this as:
Weisstein, Eric W. "Kollros' Theorem."
From MathWorld https://mathworld.wolfram.com/KollrosTheorem.html
Subject classifications
spheres, there exists a ring of
spheres, each
touching each of the 
spheres, where
. if more than one turn is allowed, then
and 
are the numbers of turns on both necklaces before closing (M. Buffet, pers.
comm., Feb. 14, 2003).
