Section

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.

1. 01What is the tetractys?Ten points in four rows, one of the oldest figures in mathematics, and why it is more than a sum.
2. 02From point to solidThe four rows of the tetractys as the four steps of space, and how stacked triangles become a tetrahedron.
3. 03Simplexes and star polygonsWhen every point is joined to every other, the lines fall apart into stars, and the number of lines is again a triangular number.
4. 04The triangle of points – just 1 + 2 + 3 + 4 = 10?Why ten, of all numbers? Three observations in which one series of numbers catches up with another at exactly this point.
5. 05The key to the tetractysA number can stand to a divisor in only three ways. Together with the number itself they give four basic figures, and ten is the first number that contains all four.
6. 06Number and soundOctave, fifth and fourth are contained in the first four numbers. The lambdoma makes every ratio visible and audible at once.
7. 07Error and conjecture in music theoryHans Kayser saw the tetractys mainly in the numbers 6, 8, 9 and 12, and considered the ten a simplification. The case against that view.
8. 08The School of AthensIn Raphael's famous fresco a boy holds a small tablet up to Pythagoras. On it is the tetractys, together with the music theory of the Pythagoreans.
9. 09The ancient sourcesWhat ancient texts report about the tetractys, from Philolaus to Lucian, and how reliable this tradition is.
10. 10Pythagoras and the Pythagorean brotherhoodMan, teacher, priest, philosopher, politician, natural scientist and healer – what can be said about Pythagoras and what is legend.