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The inscribed square problem (summary here) is currently open:

Does every Jordan curve admit an inscribed square?

(It is not required that the vertices of the square appear along the curve in any particular order)

I failed to come up with a curve that needs the precision about the order of the vertices. Is there a trivial example? Or is it that we simply do not care?

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2 Answers 2

Let me preface this with the fact that I am unfamiliar with this problem or attempts at proving/disproving the conjecture.

I can't include a picture in a comment so I post this here...

Here's an attempt at drawing a curve such that you can't have a square appear along the perimeter with its vertices "in order". I can't see any other squares that match up with my curve, but then again maybe I missed something.

attempt at a counterexample

Thanks for pointing out this interesting conjecture. :)

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Your concave thing failed you :-) –  jmad Mar 22 '12 at 18:16
    
@jmad good catch. Seeing how unrelated those squares are leads me to believe that proving "there is a square with vertices in order given there is any square" may be just as hard as the original conjecture. This makes me lean towards "we simply do not care". Hopefully an expert will appear and clear things up. :) –  Bill Cook Mar 22 '12 at 18:23
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one more

an image

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