Prologue

Pythagorean Tetractys: the capstone of the 9th triangular number.

1. The Tetractys:

   • The Pythagorean Tetractys is a triangular arrangement of 10 points, structured in four rows of musica universalis:

     • 1 point at the top (Monad, the Point, 0-D)
     • 2 points in the second row (Dyad, Duality, 1-D)
     • 3 points in the third row (Triad, Harmony, 2-D)
     • 4 points in the fourth row (Tetrad, Kosmos, 3-D)

   • The total number of points is $1 + 2 + 3 + 4 = 10$, which is the 4th triangular number.

2. The 9th Triangular Number:

   • The 9th triangular number is $T_9 = 45$, which represents a triangle with 9 rows of points.

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3. The Capstone:

   • Visualize the 9th triangular number as a large triangle with 45 points. The Tetractys (10 points) could be seen as the capstone or symbolic apex of this larger structure.

   • The Tetractys, as a sacred symbol in Pythagorean philosophy, represents the unity and harmony of the cosmos. Placing it as the “capstone” of the 9th triangular number is a symbol of the culmination of numerical and geometric perfection.

Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all.