Detailed exposition at http://mathworld.wolfram.com/PonceletsPorism.html gives reference to King (1994). The fact that this porism generalises to ellipses suggests that it is a theorem of projective geometry, rather than inversive or Laguerre, etc. Elliptic functions apparently take up the role played by trig (or hyperbolic) functions in the Steiner porism. Noteworthy that the harder Poncelet is worked out more thoroughly than the Steiner! WFL On 8/22/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The proof of Steiner's porism, whether for n integer or rational, is anyway trivial via inversive / conformal geometry / Moebius group: invert the chain to a concentric pair of frontier circles, rotate that, then invert back to the new chain.
But Poncelet's is rather harder. At first sight it looks as though a similar trick might work using the Laguerre group: but though this conserves both circles and (separately) lines, it fails to conserve points, whence polygonal vertices bloat up into circles.
The proof offered in Wikipedia involves elliptic functions, and is too terse for me to get a handle. Anybody know of a good elementary demonstration?
WFL
On 8/22/14, Dan Asimov <dasimov@earthlink.net> wrote:
Bill's animation is extremely beautiful.
Just in case this hasn't been mentioned already:
It occurs to me that for any p/q circles case (say GCD(p,q) = 1), if one takes the q-fold cover of the annulus this occurs in, one gets a (topological) annulus -- call it A' -- with a well-defined geometry, and which contains a circular chain of p circles.
Now, every geometrical annulus is conformally equivalent to a standard one: { z in C | 1 <= |z| <= R} for a unique R > 1.
Let the standard annulus conformally equivalent to A' be called A.
Since (locally isometric) covering maps, and conformal maps, preserve circles, we can apply Steiner's porism to annulus A, and this implies that it works in the "fractional circles" case in the first place.
--Dan
P.S. Steiner's porism is very similar to Poncelet's porism.
What the hell is a porism, anyway? Aha -- Webster's New World dictionary defines it as follows. Now it's clear why Steiner's and Poncelet's discoveries are called porisms:
----- porism
• a proposition that uncovers the possibility of finding such conditions as to make a specific problem capable of innumerable solutions -----
PORISM / PRIMOS
On Aug 21, 2014, at 4:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
gosper.org/stein.gif , an animation (of the 3.5 circles case)
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