An assortment of mathematical marvels Source: http://bit.ly/2LS1usH Gödel proved that if arithmetic is consistent, it must be incomplete, i.e., it has true propositions that can never be proved. Cantor's proof that the infinity of real numbers is greater than the infinity of integers. Fractals of Mandelbrot, Koch, and Sierpinski have infinite levels of detail. The Möbius strip has only one side. The Klein bottle's inside is its outside. The hypercube. Schläfli's formula for vertices, edges, faces, and cells of any 4-dimensional polytope. The five regular polyhedra. Euler's formula for the number of vertices, edges, and faces of any polyhedron. Gibbs's vector cross product. Del operates on scalar and vector fields in 3D, box in 4D. The Gaussian or normal probability distribution is a bell-shaped curve. Euler's formula relating exponentials to sine waves. A special case relating the numbers pi, e, and the imaginary square root of -1. e, expressed as a limit and an infinite series. Calculus, developed by Newton and Leibniz, is based on derivatives (slopes) and integrals (areas) of curves. The derivative of ex is ex. The integral of ex is ex. Napier's constant, e, is the base of natural logarithms and exponentials. e is transcendental. The ratio of successive Fibonacci numbers approaches the golden ratio. An exact formula for the nth Fibonacci number. Each Fibonacci number is the sum of the previous two. The number of spirals in a sunflower or a pinecone is a Fibonacci number. The golden ratio, expressed as a continued fraction. The pentagram contains many pairs of line segments that have the golden ratio. The golden rectangle, a classical aesthetic ideal. Cutting off a square leaves another golden rectangle. A logarithmic spiral is inscribed. The golden ratio, phi. The ratio of a whole to its larger part equals the ratio of the larger part to the smaller. phi is irrational and algebraic. The trigonometric functions. Another form of the Pythagorean theorem. The Pythagorean theorem. A proof by rearrangement. The quadratic equation defines a parabola. Proof that the square root of two is irrational. Pascal's triangle shows the binomial coefficients. The binomial theorem expands powers of sums. The binomial coefficient is the number of ways to choose k objects from a set of n objects, regardless of order. The zeta function of Euler and Riemann, expressed as an infinite series and a curious product over all primes. The prime number theorem of Gauss and Legendre approximates the number of primes less than x. A prime number is divisible only by one and itself. The sieve of Eratosthenes finds primes. Stirling's approximation of n factorial. Euler's gamma function gives factorials for integers but has surprising values for fractions. The product of the numbers from 1 to n is called n factorial. The sum of the numbers from 1 to n. Pi, expressed as an infinite series and an infinite product. Area and volume formulas. Archimedes solved the sphere. The ratio of the circumference of a circle to its diameter is pi. Pi is transcendental, i.e., irrational and non-algebraic. A magic square. All rows, columns, and diagonals have the same sum. Fun arithmetic with the number seven. Fun arithmetic with the number nine.