Mathematics

The Fundamental Theorem of Calculus - In college, I didn't really understand the FTC, particularly the differential form.  I wrote the commentary that I wish I had had then.  The intended audience is students who have seen integration but want a deeper understanding.

The Igloo Curtain - This introduction to line integrals concerns an Eskimo wishing to hang a curtain across his igloo along a circular arc.  The Igloo Curtain is generalized in one direction in a paper called A Tale of Two Eskimos, which extends the problem to surface integrals in the fourth dimension; and in another direction in Boring Spheres, which extends the problem by finding the arc which has associated with it a maximum area.

Infinitesimal Integration - What -- did you say infinitesimals?  That's so 17th century! ... But wait, maybe these tools used regularly by Newton and Euler don't deserve the disfavor they've received since Riemann.  Here is my take on a very practical concept.

The Inner Circle Paradox - A surprising geometric paradox kicks in at the 10th dimension.

Into the Desert - Suppose that we draw a horizontal line above one row of an infinite checkerboard of squares.  Below the line we place an army of checkers, one per square.  Above the line is the desert.  A move consists of jumping one checker over another horizontally or vertically, removing the jumped checker.  The goal is to place a checker as far into the desert as possible.  Surprisingly, it is impossible to advance 5 squares!

The Petersburg Paradox - Paul and Peter agree to play a coin-toss game.  If a head is thrown on the first toss, Paul will pay Peter $1.  If the first toss is a tail but the second toss is a head, Paul will pay Peter $2.  For every toss that Peter can delay getting his first head, Paul will double the payoff.  How much should Peter pay Paul for the privilege of playing this game? 

Slicing An Orange - If you cut an orange of diameter d into d equally-spaced, parallel slices, then which slices have more peel, the large diameter slices in the middle or the small diameter slices towards the ends?

Unbiased Estimators - Statistics students wonder why we divide by n-1 when calculating sample variances and standard deviations and are not satisfied with the "degrees of freedom" answer that textbooks provide.  Here is a more satisfying answer.

Zebra Patterns - There is an intriguing pattern in the decimals of this obscure number.

"The highest form of pure thought is in mathematics."  -- Plato