- Gödel’s incompleteness theorems
- The discovery of irrational numbers by the Pythagoreans
- Cantor’s theorems—nondenumerability of the continuum and the cardinality of the power set of A is greater than the cardinality of A
- The rational numbers are countable
- The continuum hypothesis can neither be proved nor disproved in ZFC
- The existence of a continuous nowhere differentiable function
- Euler’s solution of the Basel problem
- The existence of non-Euclidean geometries
- The insolvability of quintic equations
- The Monty Hall problem
- Fermat’s non-prime (Euler proved that
is composite)
- The shape of a hanging chain is a catenary
- The existence of space filling curves
- The Banach-Tarski theorem
- The relationship between the complex numbers and the primes (E.g., Riemann zeta function)
- The prime number theorem
- Aperiodic tilings
- Arrow’s impossibility theorem
- Ulam’s spiral of primes
- Andrew Wiles’ proof of Fermat’s Last Theorem
- The use of a computer to prove the four color theorem
- Russell’s paradox
- The Cantor set
- Euler’s polyhedron formula
- The five Platonic solids
- The Brachistochrone problem
- Noncircular figures of constant width
- 0.999…=1
- Lorenz’s “butterfly effect”
- Period 3 implies chaos (and Sharkovsky’s theorem)
- The fundamental theorem of calculus
- Descartes’ discovery of analytic geometry
- Discovery of complex numbers (and their real-world applications)
- Hamilton’s discovery of the quaternions
- There exists a flow in 3-space with closed orbits of every knot and link type
- 19-year-old Gauss’ ruler-and-compass construction of a 17-gon (and its relation to Fermat primes)
- Proving the impossibility of squaring a circle, trisecting an angle, and doubling a cube
- The Euler line
- A complex function that is once differentiable on a disk is infinitely differentiable
- Liouville’s theorem—a function that is bounded and differentiable at every point in the complex plane is constant
- Thomae’s function—a function that is continuous at every irrational number, discontinuous at every rational number
- The elementary linear algebra behind Google’s pagerank
- Kuratowski’s closure-complement theorem
- Surprisingly open problem: does every triangular billiard table have a periodic orbit?
- Surprisingly open problem: the Collatz conjecture/3n+1 problem
- Surprisingly open problem: Goldbach conjecture
- Dirac’s belt trick
- Benford’s law on the distribution of leading digits
- The short proof of the solution to the art gallery problem
- Dropping needles on a hardwood floor to approximate π (Buffon’s needle)
- Robert Conelly’s flexible polyhedron
- The many equivalent interpretations of the Catalan numbers
