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July 27th, 2009, 06:46 PM  #1 
Newbie Joined: May 2009 Posts: 14 Thanks: 0  Selfintersecting Polygons and Simple Polygons
Intuitively this seems true to me, but I am not certain and haven't been able to find any source specifically addressing it. Any help would be appreciated. Can we take every selfintersecting polygon and, ignoring the angles, form a simple (nonselfintersecting) polygon with the same side lengths as the selfintersecting polygon (and with side lengths that remain in the same order)? For instance, lets say we have a selfintersecting 4gon (quadrilateral) that looks like an hourglass. Intuitively we might imagine that the lengths are rigid rods and the connections are small ballbearings. Can we pick it up, twist it and lay it back down so that it no longer looks like an hourglass, but instead a simple quadrilateral that lays flat on the ground? Its strange, I don't even see this addressed in Coxeter's writings that I have, perhaps because it is so elementary (my Dear Watson). Eric 
July 27th, 2009, 11:11 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,310 Thanks: 1980 
It's trivial to prove for a quadrilateral; just "unfold" it.

July 28th, 2009, 10:28 AM  #3 
Newbie Joined: May 2009 Posts: 14 Thanks: 0  Re: Selfintersecting Polygons and Simple Polygons
The quadrilateral was an instance, I was curious about any self intersecting ngon.


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