An assortment of mathematical marvels

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