Cosmopoeia
Sections

Section

Introduction

The small figure where everything begins: what the tetractys is, how we know about it and why it 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. 02The tetractys at a glanceNumbers as quantities, cycles in four phases, the first stroke of the compass, the ten in the division table and the pentagram in DNA: a walk through the basic ideas of this site.
  3. 03From point to solidThe four rows of the tetractys as the four steps of space, and how stacked triangles become a tetrahedron.
  4. 04Simplexes 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.
  5. 05The 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.
  6. 06The numbers from one to tenFor the Pythagoreans every number up to ten had a nature of its own: the monad as origin, the dyad as opposition, five as marriage and seven as the motherless virgin. What the old books of numbers say about each one, what of it is mathematics and what interpretation.
  7. 07The 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.
  8. 08Number and soundOctave, fifth and fourth are contained in the first four numbers. The lambdoma makes every ratio visible and audible at once.
  9. 09Error 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.
  10. 10The 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.
  11. 11The ancient sourcesWhat ancient texts report about the tetractys, from Philolaus to Lucian, and how reliable this tradition is.
  12. 12Pythagoras: the manOrigins, years of study, travels and the end in Metapontum: what the ancient accounts say about the life of Pythagoras and how much of it is certain.
  13. 13Pythagoras: the teachingWith the Pythagoreans conceptual thinking entered a world view shaped by myth. What the teaching covered and why master and pupils can hardly be told apart.
  14. 14Pythagoras: the priestA god who never loses sight of humankind, the judgement of souls, the transmigration of souls and sparing all that has a soul: the religious side of Pythagoras.
  15. 15Pythagoras: the philosopherThe eternal essence of number, opposites as a creative force, the table of ten pairs of opposites, the music of the spheres and why Pythagoras called the world cosmos.
  16. 16Pythagoras: the statesmanFriendship of all with all: the speeches to the city fathers, the young and the women of Croton, the ordered state of the soul as the aim of education and music as the first educator.
  17. 17Pythagoras: the natural scientistArithmetician, geometer, astronomer and music theorist: thinking in analogies, the Pythagorean theorem, the forbidden dodecahedron, the central fire and the legend of the smithy.
  18. 18Pythagoras: the healerRegimen rather than the knife, music as medicine, Alcmaeon's teaching of the balance of forces and the daily training of memory.
  19. 19Pythagoras: the Pythagorean brotherhoodAdmission tests and five years of silence, the oath on the sacred four, listeners and learners, the day in the order, Damon and Phintias and the end in persecution.