From one, two, three, four, a world.
The Pythagorean tetractys
Ten points in four rows. To the Pythagoreans this small triangle was a key to the order of the world. This site shows what it contains in geometry, in number theory and in music.
Start with the introduction →Use the slider to grow the triangle. With four rows it becomes the tetractys.
Introduction
What the tetractys is, how we know about it, and why a triangle of ten points has occupied thinkers for two and a half thousand years.
Geometry
Simplexes, star polygons and the Platonic solids: the forms that unfold from the triangle of points.
Number theory
Figurate numbers, divisibility and primes, seen as arrangements of points rather than strings of digits.
Harmonics
The bond between number and sound: intervals as ratios, and the lambdoma as the table of all ratios.
Mysticism
How the tetractys was read in religious and esoteric traditions, and how much of it can be demonstrated.
Philosophy
Number as the principle of the world: from the Pythagoreans and Plato to the question of why mathematics can describe nature at all.
Number theory and geometry are two views of the same thing: one counts, the other sees.