This post has pictures suggesting a better example of a Jordan curve, using the isotopies suggested last time.
There is a Jordan curve such that every piecewise-linear path from the inside to the outside intersects the curve infinitely many times. We will begin the construction of such a curve here.
The Jordan curve theorem is famously simple to state and tricky to prove. I want to explain why the Jordan curve theorem ought to be difficult to prove, by showing some pictures of very complicated Jordan curves.
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