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Re: Interesting problem

Cross-posted to 2 groups.

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On Wed, 25 May 2016 16:20:03 -0500, Odd Bodkin wrote:

> 
> Nice to see old 5th-grade problems are still interesting to you.
> 

Odd Ball gives one explanation, but can he attain a full generalization?

In the first place, gears are unnecessary.  We can consider rolling circles
as being the limit of infinite "teeth."

We can also transform a straight "line" into a circle by constructing
polygonal, or n-gon, intermediates.  In other words, from our straight
line we construct a 2-gon*, triangle(3-gon), a square(4-gon), a pentagon, etc.
until we finally arrive at our circle, which is an "infinity-gon."

We roll our circle along each of these n-gons.  How does the motion
of the circle behave in each case?

* Yes, Odd Ball, a 2-gon does exist.  It is a degenerate polygon
with 2 sides and 2 vertices.  We can easily make one from our original
straight line.

Even more illustrative is to consider what happens when a circle is
rolled around a 1-gon, which has 1 side and 1 vertex.  Maybe Odd Ball
will enlighten us.

C'mon, Odd Ball.  Let's see you fully generalize this 5th-grade problem.

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Thread

Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-26 07:05 +1000
  Re: Interesting problem Poutnik <poutnik4nntp@gmail.com> - 2016-05-25 23:17 +0200
    Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-27 03:57 +1000
    Re: Interesting problem David Canzi <dmcanzi@uwaterloo.ca> - 2016-05-30 03:12 -0400
      Re: Interesting problem ClutterFreak <ClutterFreak@FakeAddress.com> - 2016-05-31 20:05 -0500
        Re: Interesting problem David Canzi <dmcanzi@uwaterloo.ca> - 2016-06-02 22:25 -0400
          Re: Interesting problem ClutterFreak <ClutterFreak@FakeAddress.com> - 2016-06-02 23:52 -0500
          Re: Interesting problem Fabian Russell <fb@zen.info> - 2016-06-03 09:52 +0000
            Re: Interesting problem noTthaTguY <abu.kuanysh05@gmail.com> - 2016-06-03 09:56 -0700
  Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-25 16:20 -0500
    Re: Interesting problem Tom Roberts <tjroberts137@sbcglobal.net> - 2016-05-25 18:31 -0500
    Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-27 03:25 +1000
      Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-26 13:32 -0500
        Re: Interesting problem HGW <hw@...> - 2016-05-27 12:15 +1000
          Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-27 04:26 -0500
            Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-27 19:52 +1000
              Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-27 05:46 -0500
                Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-27 21:55 +1000
                Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-27 07:23 -0500
                Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-27 07:26 -0500
                Re: Interesting problem ClutterFreak <ClutterFreak@FakeAddress.com> - 2016-05-27 09:38 -0500
                Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-27 09:47 -0500
                Re: Interesting problem "hanson" <hanson@quick.net> - 2016-05-27 08:00 -0700
                Re: Interesting problem ClutterFreak <ClutterFreak@FakeAddress.com> - 2016-05-27 10:22 -0500
                Re: Interesting problem "hanson" <hanson@quick.net> - 2016-05-27 09:02 -0700
                Re: Interesting problem ClutterFreak <ClutterFreak@FakeAddress.com> - 2016-05-27 18:43 -0500
                Re: Interesting problem ClutterFreak <ClutterFreak@FakeAddress.com> - 2016-05-27 20:16 -0500
                Re: Interesting problem "hanson" <hanson@quick.net> - 2016-05-27 08:14 -0700
                Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-28 19:47 +1000
                Re: Interesting problem Odd Bodkin <bodkinodd@gmail.com> - 2016-05-29 13:52 -0500
                Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-30 07:09 +1000
                Re: Interesting problem Sam Wormley <swormley1@gmail.com> - 2016-05-29 16:26 -0500
                Re: Interesting problem Fabian Russell <fb@zen.info> - 2016-05-30 04:25 +0000
                Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-31 21:00 +1000
                Re: Interesting problem Fabian Russell <fb@zen.info> - 2016-05-31 22:46 +0000
                Re: Interesting problem HGW <hw@...> - 2016-06-02 09:32 +1000
                Re: Interesting problem noTthaTguY <abu.kuanysh05@gmail.com> - 2016-06-01 18:34 -0700
                Re: Interesting problem Mikkel Haaheim <mikkelhaaheim@gmail.com> - 2016-05-27 08:08 -0700
    Re: Interesting problem Fabian Russell <fb@zen.info> - 2016-05-27 22:03 +0000
      Re: Interesting problem "HGWilson, DSc." <hgw@....> - 2016-05-28 20:10 +1000

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