Cosmopoeia
Sections

Section

Foundations

The mathematics behind this website, step by step and for non-specialists: what a proof is, how a few basic assumptions lead to numbers, divisors and figures, and why the theorems about the ten and the primes really hold. With proofs to unfold and checks by computer.

  1. 00The language of mathematicsA reading aid for non-specialists: what symbols such as ∀, ∣, ⇔ and Σ mean, how a mathematical paper is structured, and how to read a proof in Lean line by line.
  2. 01What is a proof?Why examples are not enough, what a proof is built from, and a first proof using the triangle of dots of the tetractys: 1 + 2 + 3 + 4 = 10, and why it continues the same way for every number.
  3. 02CountingThe natural numbers from nothing but a zero and a successor: Peano's axioms, addition by recursion, mathematical induction, and the gnomon of the Pythagoreans, which joins the odd numbers into squares.
  4. 03DividingWhat it means for one number to divide another, why division with remainder always works and is unique, and how the Euclidean algorithm finds the greatest common divisor, pictured as a rectangle from which squares are cut.
  5. 04Prime numbersWhat a prime is and why 1 is not one, why there are infinitely many, why every number has exactly one factorisation into primes, and why the sieve of Eratosthenes only starts striking out at the square of a prime.
  6. 05Arithmetic with remaindersCongruences after Gauss: calculating on a clock. Why all primes from 5 on lie in the rhythm of six, why p² − 1 is always divisible by 24, and why 24 is the largest number with this property.
  7. 06Numbers as figuresWhy the figure n/k falls into exactly as many parts as the greatest common divisor says, why ten is the first number with all four kinds of figure, and why primes are exactly the simplexes without a compound figure.
  8. 07AnglesWhy a triangle has 180 degrees, a polygon (n − 2) · 180 degrees and a star (n − 2k) · 180 degrees, why only triangle, square and hexagon fill the plane, and why ten is unique among the angle sums.
  9. 08RatiosFractions as ratios, the intervals of the tetractys 2 : 1, 3 : 2 and 4 : 3, why twelve fifths never make exactly seven octaves, and why the division table contains every fraction exactly once as a new one.
  10. 09The incommensurableLengths without a common measure: why side and diagonal of the square stand in no ratio of numbers, why the Euclidean algorithm then never ends, and how the pentagram, the emblem of the Pythagoreans, carries the same discovery within it.
  11. 10Approaching πHow Archimedes trapped the circle between two polygons and pinned down π to two decimal places, which doubling formula does it, and why the folded squares of Andreas Weiss rebuild exactly this method.
  12. 11What remains openWhat this section has proven, what is proven but would lead too far here, what nobody knows, and which statements of this website are interpretations. A map of the certain and the open.