Pi turns up uninvited
A square wave built entirely out of round ones
Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.
AlgebraMultiplying is turning
Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.
ProbabilityGetting pi by dropping needles on the floor
Throw a needle at a lined floor enough times, count how often it crosses a line, and pi falls out. There is no circle anywhere in the experiment.
ProbabilityA walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
NumberTwo squares, and a lattice
Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.
ComputationThe circle that will not square
The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.