Thick and thin

• Representing permutations relatively compactly

  Here is a more relative way to represent a permutation with the same economical number of bits.

• Representing permutations compactly

  Here is a way to represent a permutation with a fairly economical number of bits.

• Jordan curve illustrations: drawing the curve

  There is a Jordan curve such that every piecewise-linear path from the inside to the outside intersects the curve infinitely many times. I describe how to draw such a curve here, and do so.

• Jordan curve illustrations: wiggling a course

  There is a Jordan curve such that every piecewise-linear path from the inside to the outside intersects the curve infinitely many times. I define the wiggling precisely here in terms of courses (turn-type sequences).

• Jordan curve illustrations: a triangular example

  There is a Jordan curve such that every piecewise-linear path from the inside to the outside intersects the curve infinitely many times. I sketch the construction of such a curve here.

• Jordan curve illustrations: better pictures

  This post has pictures suggesting a better example of a Jordan curve, using the isotopies suggested last time.

• Jordan curve illustrations: idea for a better example

  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.

• Jordan curve illustrations: attempts

  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.

• Whence topology: main concepts

  Having introduced you to the swamp and the monsters, let me finish by sketching the structures used to make some sense out of it all.

• Whence topology: monsters

  The swamp of topology has monsters lurking inside.

• Whence topology: the swamp

  One is lured into the field of topology by the beauty of the previous problems. But serious attention to the subject reveals a swamp of difficulty with definitions.

• Whence topology: problems

  I like to study topology. These first few blog posts will explain what it’s about. This first discusses some motivating problems.