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Geometry
- The rectangle that eats itself
- Why the list of perfect solids stops at five
- A circle unrolled into a triangle
- Round is not the only way to be the same width
- The oldest algorithm, drawn as a tiling
- Every square is a stack of odd numbers
- One cone, four curves
- Two squares, four triangles, and no algebra
- An angle that does not care where it stands
- The plane, divided by whoever is nearest
- Euclid proves it without moving anything
Discrete
- One sequence, counting everything
- Seven bridges, and the invention of throwing things away
- More things than boxes
- The primes on a spiral, and a pattern nobody ordered
- Pascal's triangle, in two colours
- Four colours, and a proof nobody can read
- Numbers that wrap
- Six people at a party
Topology
- Every corner pays for itself
- A sphere is a plane plus one point
- Something always stays put
- The surface with one side, and what happens when it is cut
- Three moves, and what they cannot undo
- Nothing on a sphere can be combed flat
Algebra
- The dot product is a shadow
- The directions a map leaves alone
- A matrix is a picture of what happens to the grid
- Multiplying is turning
- Completing the square, by completing a square
Probability
- Getting pi by dropping needles on the floor
- A bell curve assembled out of coin flips
- Twenty-three people
- Bayes' theorem is a picture of a square
- A walk that always comes home, until it does not
- The door that was not opened
Analysis
- A sine wave is a circle seen from the side
- The curve that is its own slope
- The slope of a single point
- One point's worth of information
- A sum whose terms vanish and whose total does not
- Adding up rectangles until they stop being rectangles
- A square wave built entirely out of round ones
- The same terms, in a different order, adding to whatever is asked
- Area is the undoing of slope