There is a continuum if you vary the starting point, but I was trying to say something about how many there are with a fixed starting point. The number varies according to the starting point, but I think my argument is a good sketch of a proof that for any fixed ellipse that is not a circle and any fixed starting point p on the ellipse, the number of n-gons through p grows exponentially with n. Bill Thurston On Nov 20, 2010, at 3:23 PM, Veit Elser wrote:
I seem to be missing something: isn't it pretty obvious that for large enough n (the number of chords) there will be a continuum of n-gons?
Scale the ellipse so that its arc length is 1 and consider only n-gons with chord length d<D so that a circle of radius d about any point of the ellipse cuts the ellipse in only two points. Call an arbitrary point of the ellipse the origin and let x be the signed arc length around the ellipse with respect to this origin, and clockwise corresponding to increasing x.
Now consider a point at arc length x and a circle of radius d about x. The circle cuts the ellipse at two points, at arc lengths x' and x", where x'>x>x". Define f(x,d) = x'. For 0<d<D, the function f(x,d) is smooth and monotonic increasing in d.
Next define the sequence of functions
f_1(x,d) = f(x,d) f_2(x,d) = f(f(x,d),d) f_3(x,d) = f(f(f(x,d),d),d) etc.
These are also smooth and monotonic increasing in d.
Finally, for any d<D define n by the property
|1+x - f_(n+1)(x,d)| >= |1+x - f_n(x,d)| and |1+x - f_(n-1)(x,d)| > |1+x - f_n(x,d)|
So n is the best approximation to closure of the n-gon.
Well now we have a smooth function f_n(x,d) monotonic in d and it has 1+x in its range. There is a function like this for any x and we can always find a d that gives closure, i.e. f_n(x,d) = 1+x. The arc lengths x thus parameterize a continuous family of n-gons.
Veit
On Nov 20, 2010, at 1:06 PM, Bill Thurston wrote:
Michael Kleber is correct. When I scanned some of this thread, at first I thought that maybe Poncelet's porism would generalize (as Victor Miller suggested:)
Is what you want related to Poncelet's Porism? http://mathworld.wolfram.com/PonceletsPorism.html
There's an argument that would have imply that if there were exactly two solutions to arcs of length L inscribed in an ellipse, for every L, the same phenomenon as in Poncelet's porism would hold: the map of the ellipse to itself gotten by stepping off with steps of length L would be diffeomorphically conjugate to a rotation because the algebraic curve consisting of secants of length L would be an elliptic curve. But feeding the algebra to Mathematica showed this is not true: for each point on the ellipse, there's a degree 4 equation for the other endpoints of the secant, and the curve parametrizing secants of length L has high genus. (approximately genus 16, but I'm not clear enough on algebraic geometry to do it exactly without more work than I'm up for).
The complexification of an ellipse is topologically an annulus, and given a length, L, and a starting point, x, you get an image of a 4-way-branching tree mapped to the annulus. As you vary L and x, the branches sometimes intersect and form loops. Sometimes these are on the real ellipse, sometimes just on the complexification. There are typically 32 critical points for the process, where two of the 4 branches coincide. When L is just a little longer than the minor axis of the ellipse so that there are "less obvious" secants of length L, there are 4 real critical points where the secant is "jammed" --- one end is perpendicular to the ellipse, the other makes an angle, and the perpendicular end can be perturbed in either direction while moving the non-jammed end slightly forward. There are lots of other non-obvious complex critical points
Anyway, for each n, there is an exponential-in-n number of complex values of L with complex inscribed polygons. Even for the case of real polygons where each secant is chosen to advance the obvious way in the smallest possible step, the polygon can be any of the (n-1) n-pointed stars (where retracing is allowed, e.g. going around a triangle 3 times would count as a 9-pointed star). If Kleber's non-obvious secants are allowed, I think the number of n-gons is exponential-in-n even in the real case. That's because there's always an open set of {(starting point p, length L)} where there are four secants, when p is near an end of the minor axis and L is just longer than the minor axis. For any L < major axis, there's a rotation number associated with the process of advancing counterclockwise to the nearest point at distance L. Choose L so that the rotation number is irrational, and L is just greater than the minor axis. Every orbit for L' near L enters the open set wi! th! 4 solutions at regular intervals. By making branching choices at those times, then proceeding "obviously" for a while, and eventually adjusting L slightly to make the polygon close, I think you get exponentially many solutions.
Bill Thurston On Nov 20, 2010, at 9:21 AM, Michael Kleber wrote:
On Sat, Nov 20, 2010 at 12:44 AM, David Wilson <davidwwilson@comcast.net>wrote:
I read your notes and did some imagining, and here are my unjustified beliefs:
For every n >= 3, and for any ellipse, there is a single continuous family of inscribed equilateral n-gons in the ellipse.
I disagree -- I think there are "less obvious" inscribed equilateral n-gons in ellipsis with high eccentricity, in which one or two edges take a short-cut, circumventing a large piece of the ellipse. (Given a point P on the ellipse and a distance d, there may be *four* points on the ellipse at distance d from P, not just two.)
This objection carries through to all the following statements also, I think...
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