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Stromquist's Theorem: If the simple closed curve J is "nice enough" then it has an inscribed square.
"Nice enough" includes polygons.
Read more about it here: www.webpages.uidaho.edu/~markn/squares
An "inscribed square" means that the corners of a square overlap with the curve.

I would like to suggest a counter-example:

The curve connected by the points$$ (0.2,0),\ (1,0),\ (1,1),\ (0,1),\ (0,0.2),\ (-0.2, -0.2),\ (0.2,0).$$ Link to plot: http://www.freeimagehosting.net/uploads/5b289e6824.png
Can this curve be incribed by a square?

(An older version of this question had another example: a triangle on top of a square (without their mutual side.) )

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Regarding your edit: (0.2, 0) — (1, 0.2) — (0.8, 1) — (0, 0.8) (and many others)

http://www.imgftw.net/img/326639277.png

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What about the square on the bottom- with it and the triangle's mutual side returned?

Edit: Btw. Don't be disheartened that your counterexample didn't come through- looking for counterexamples to proved theorems will never give a genuine counterexample (unless ZFC turns out to be inconsistent!!;)), but it can provide valuable insight into what the theorem means.

Here you can see from experience that 'most of the square' may be contained within the curve- this is something that may not be obvious looking at the examples of Stormquist in action in books on the topic.

Trying to find counterexamples, IMHO, is a great way to develop as a mathematician- just do not be afraid to be wrong ;)

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  • $\begingroup$ I revised the example. Can you tell what is wrong with the newer one? $\endgroup$ – metame Aug 5 '10 at 13:50

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