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...
For even n = 2k, an inscribed n-gon will have 180-degree symmetry about the center of the ellipse. For any point P on the ellipse, there will be a unique n-gon with vertex P.
For odd n = 2k+1, the behavior will be more complex. In low eccentricity ellipses, for any point P on the ellipse, I think there will be a unique n-gon with vertex P, as in the even n case (clearly true for the circle). However, in high eccentricity ellipses, we will see something interesting:
Envision the ellipse elongated horizontally. The n-gon will be small in area compared to the ellipse. Near the center of the ellipse, the n-gon will look like a slightly fattened trapezoid. There will be k sides on one arc of the ellipse (say the bottom) and k-1 sides on the opposite arc (top). The remaining 2 sides will cut across the ellipse to complete the closed polygon. The polygon will slide toward one end (say right end) of the ellipse. The nearer vertex (on the bottom arc) will eventually slide to the end of the ellipse. At this point, the n-gon will be crammed into the right end of the ellipse, with its left side vertical. I will guess that this is when the polygon sides are shortest. Then the end point continues to slide upward and the process continues in reverse and upside down. The n-gon moves to the left, and as it reaches the middle of the ellipse, it is upside down from its start position, with k sides on the top arc and k-1 on the bottom. We then slide the n-gon to the left end, and run the closest vertex down across the left end of the ellipse, and the trapezoid flips back to right side up. We then slide the polygon back to its original position, with vertices rotated counterclockwise. Performed quickly, it will look like a small trapezoid bouncing back and forth between the ends of the ellipse, flipping upside down on each bounce. I'm not sure if there is a cutoff point or a smooth transition between the simple rotational behavior of the circle and the bouncing behavior of the highly eccentric ellipse.
It is interesting to note that for a point P on ellipse close to the center, this process generates n distinct polygons with vertex P, but if P is an end of the ellipse, only 1 polygon has vertex P.
----- Original Message ----- From: "Henry Baker" <hbaker1@pipeline.com> To: "Andy Latto" <andy.latto@pobox.com> Cc: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, November 18, 2010 11:01 PM Subject: Re: [math-fun] equilateral elliptical n-gons
Andy is right. The recursive subdivision doesn't work correctly -- it produces pairs of equal sides, but those pairs don't necessarily equal pairs futher away.
Back to the drawing board...
At 02:37 PM 11/18/2010, Andy Latto wrote:
On Thu, Nov 18, 2010 at 5:10 PM, Henry Baker <hbaker1@pipeline.com> wrote:
[My statement below about "notchy" is *wrong*. Given an ellipse, you can start my recursive perpendicular bisector subdivision from _any_ line through the center of the ellipse.
But as someone pointed out, the recursive perpendicular bisector subdivsion doesn't work. You replace side AB with sides AC and CB, where |AC| = |CB|, and you replace side XY with sides XZ and ZY, with |XZ| = |ZY|, but there's no reason to expect that just because |AB| = |XY|, that |AC| should equal |XZ|.
Andy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
----- No virus found in this message. Checked by AVG - www.avg.com Version: 10.0.1153 / Virus Database: 424/3266 - Release Date: 11/19/10
----- No virus found in this message. Checked by AVG - www.avg.com Version: 10.0.1153 / Virus Database: 424/3266 - Release Date: 11/19/10
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.