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July 27th, 2009, 05:46 PM   #1
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Self-intersecting 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 self-intersecting polygon and, ignoring the angles, form a simple (non-self-intersecting) polygon with the same side lengths as the self-intersecting polygon (and with side lengths that remain in the same order)?

For instance, lets say we have a self-intersecting 4-gon (quadrilateral) that looks like an hourglass. Intuitively we might imagine that the lengths are rigid rods and the connections are small ball-bearings. 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
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July 27th, 2009, 10:11 PM   #2
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It's trivial to prove for a quadrilateral; just "unfold" it.
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July 28th, 2009, 09:28 AM   #3
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Re: Self-intersecting Polygons and Simple Polygons

The quadrilateral was an instance, I was curious about any self intersecting n-gon.
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