November 2010 - mathfun@mailman.xmission.com

[math-fun] equilateral elliptical n-gons
by Henry Baker 18 Nov '10

18 Nov '10

I think that by using the term "regular" instead of "equilateral", I confused everyone. I'm trying to find a simple way to generate n-vertex equilateral ellipses. It is particularly important for the application I have in mind that the vertices are at equal spacing along the perimeter of the ellipse, and most importantly, that the sequence closes after one trip around -- i.e., there can't be any gap. The reason is that I want to use these little vectors as accelerations, and if they don't sum up exactly to zero, then the velocities & positions don't have any hope of closing up.

2 1

Re: [math-fun] regular _elliptical_ n-gons?
by Dan Asimov 18 Nov '10

18 Nov '10

I wrote, regarding a conformal map from the circle and its interior to an ellipse and its interior, taking the center to the center: << The vertices of a regular n-gon inscribed in the circle will be mapped to n vertices with the same angular separation along the ellipse (with respect to the its center). >> This is not true. The radii of the circle ending at vertices of a regular n-gon inscribed in the circle will be taken to some curves from the center of the ellipse to the ellipse, and the curves will meet at equal angles, but that's all that can be said in this vein. * * * Question: Suppose we have an N-gon with all sides equal that's inscribed in an ellipse. Assume the sides rigid and the ellipse to be a track in which the vertices can slide. Under what conditions will this N-gon actually slide around the ellipse? --Dan Those who sleep faster get more rest.

1 0

Re: [math-fun] regular _elliptical_ n-gons?
by Dan Asimov 18 Nov '10

18 Nov '10

<< In playing with the [Siebeck] Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal. Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem. What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way? I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look. >> If you can conformally map the interior of a circle to the interior of an ellipse, then there is a reasonable definition of an elliptical polygon. If such a map is constrained to take the circle's center to the ellipse's center, then what remains is a one-parameter family of such maps, determined by a rotation of the circle. And this circular family of conformal maps extends to such a family of homeomorphisms of the circle as a curve, to the ellipse as a curve. And this conformal map exists, by the Riemann mapping theorem. More information about it is in a paper by Gabor Szego, "Conformal Mapping of the Interior of an Ellipse onto a Circle", in the Aug.-Sept., 1950 American Mathematical Monthly (v. 57, no. 7). But I'm not sure there's a useful formula for the map of curves given there, though there are a number of Gosperesque formulas. The vertices of a regular n-gon inscribed in the circle will be mapped to n vertices with the same angular separation along the ellipse (with respect to the its center). These might make an appropriate definition for the vertices of an elliptical regular n-gon. Admittedly this answer a question different from the one asked. --Dan Those who sleep faster get more rest.

1 0

Re: [math-fun] regular _elliptical_ n-gons?
by Henry Baker 18 Nov '10

18 Nov '10

Bingo! Yes, this is exactly the sort of thing I had in mind. So the positions of the points on these ellipses can be calculated as the roots of polynomials? (Cool!) But it does appear that there are integer constraints governing various ratios about the possible ellipses constructed in this fashion. At 09:05 PM 11/17/2010, James Buddenhagen wrote: >Rita the dog asked a more specific but related question on Y!A a while >back, with links to a couple of pictures in the additional details. >See here: http://answers.yahoo.com/question/index;_ylt=As6rCggttnM1wrVEcxQE_VXty6IX;_… > >On Wed, Nov 17, 2010 at 10:39 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> In playing with the Marden Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal. >> >> Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem. >> >> What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way? >> >> I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look.

1 0

Re: [math-fun] regular _elliptical_ n-gons?
by Henry Baker 18 Nov '10

18 Nov '10

You are correct. Kalman apologizes, but then goes on to call it the Marden theorem anyway! At 08:17 PM 11/17/2010, Dan Asimov wrote: >It seems that Dan Kalman called it the Marden theorem solely because he read about it in a book by Marden, even though Marden attributes the theorem to Siebeck. I would call it the Siebeck theorem. > >--Dan > ><< In playing with the Marden Theorem, . . .

1 0

[math-fun] regular _elliptical_ n-gons?
by Henry Baker 18 Nov '10

18 Nov '10

In playing with the Marden Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal. Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem. What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way? I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look.

3 2

Re: [math-fun] regular _elliptical_ n-gons?
by Henry Baker 18 Nov '10

18 Nov '10

I just looked at the link you referenced. No, Poncelet's Porism (interesting though it may be) appears to involve n-gon sides which _aren't_ of equal length. I'm looking for n-gons with all n sides equal whose vertices all lie on an ellipse (i.e., the ellipse circumscribes the n-gon). At 08:51 PM 11/17/2010, Victor Miller wrote: >Is what you want related to Poncelet's Porism? >http://mathworld.wolfram.com/PonceletsPorism.html > >On Wed, Nov 17, 2010 at 11:39 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> In playing with the Marden Theorem, I started thinking about approximations >> to ellipses, and was wondering about regular elliptical n-gons. I'm not >> talking about a regular n-gon stretched/squashed or squashed in the X or Y >> direction, because for stretched/squashed n-gons the lengths of those sides >> are no longer equal. >> >> Since the perimeter of an ellipse isn't easily calculated, I would imagine >> that this will be a difficult problem. >> >> What I'm really looking for is a kind of inverse to the ellipse perimeter >> problem: given n sides of length 1 and a major/minor radius ratio >> (alternately an eccentricity), how to draw the ellipse itself. Which >> major/minor ratios and/or eccentricities can actually be represented this >> way? >> >> I would assume that someone has looked into this sort of problem, but I >> wouldn't even know where to begin to look.

1 0

Re: [math-fun] regular _elliptical_ n-gons?
by Henry Baker 18 Nov '10

18 Nov '10

I just occured to me after I sent this message that one might conceivably draw a regular elliptical n-gon using a modification of the classical "2 pins and a string" method. Consider a cyclic chain (like a bicycle chain, except imagine that the links are longer) consisting of m links. Place two pins k links apart. Then place the chain taut against the two pins & stretch it straight & choose one of the joints between two links & draw a dot. Then move to the next joint along the chain & keep the chain taut. In order to handle the case where the perimeter point is collinear with the line of the two foci, the total number of links m should be an even number. It would appear that if the ellipse foci are an integer number (k) of link-lengths apart, then the dots drawn should be a pretty close approximation to a regular elliptical n-gon ?? At 08:41 PM 11/17/2010, Henry Baker wrote: >In playing with the Marden Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal. > >Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem. > >What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way? > >I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look.

1 0

Re: [math-fun] regular _elliptical_ n-gons?
by Dan Asimov 18 Nov '10

18 Nov '10

It seems that Dan Kalman called it the Marden theorem solely because he read about it in a book by Marden, even though Marden attributes the theorem to Siebeck. I would call it the Siebeck theorem. --Dan << In playing with the Marden Theorem, . . . Those who sleep faster get more rest.

1 0

[math-fun] New Version of OEIS
by N. J. A. Sloane 16 Nov '10

16 Nov '10

Announcement: New Version of OEIS! After 2 years of struggle, there is a new version of the On-Line Encyclopedia of Integer Sequences (the OEIS). There are two parts: the main OEIS page, oeis.org, which has all the sequence data, and a wiki, oeis.org/wiki, which has discussion pages. To contribute a new sequence or edit a sequence, you must first register with oeis.org/wiki - click Register, enter your real name in 2 places, enter your email address, etc. When you get issued a password, go to the main page, oeis.org, and login with the same user name and password. Now when you look at a sequence, you will see "history" and "edit" buttons next to the "graph" and "listen" buttons. The "edit" button lets you edit the sequence. To submit a new sequence, use the "Submit new seq." link at the bottom of any sequence page. If you don't login, you won't see the edit button. All contributions will be reviewed by the editors (if you would like to help by becoming an editor, let me know). That this works is due to the enormous efforts of Russ Cox, who has completely rewritten all the OEIS software. David Applegate has also been of great help in getting the new system working. Some 18 months ago we discovered that the mediawiki software was incapable of handling the kind of queries that the OEIS generates. I had almost given up hope that these problems would ever be solved. But thanks to Russ, as of Nov 11 2010, oeis.org is finally running. This means that after 45 years, I no longer have to process every single contribution to the database. Thank you, Russ and Dave: you have saved the OEIS! In the Spring of 2009 we set up the OEIS Foundation, a 501(c)(3) public charity, which owns, runs and collects funds to support the OEIS. The web site is oeisf.org. We need donations! Neil Sloane President, OEIS Foundation Inc.

1 0