<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >> The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line. WFL On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote:
So an example would be https://flic.kr/p/em4XbX
On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun