WIKIPEDIA: Mathematical surprises

• 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

Πηγή: divisbyzero.com/2010/08/18/mathematical-surprises/