As we know only few math discoveries are considered as shocking and unexpected results, in which open great gates instead of only windows, to the mathematician views. . Dave Richeson at “Division by Zero” has listed some of this kind of discoveries:

### Mathematical surprises

I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental way—Cantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surprise—Andrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.

I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below!

- 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