Tetraktys

The tetraktys is the figure of ten points arranged in a triangle of four rows — one, two, three, four — and it was the most sacred symbol of the school of Pythagoras. By it the Pythagoreans swore their most binding oath: 'by him who handed down to our soul the tetraktys, fount and root of ever-flowing nature.' Its rows compress the whole of their cosmology: the monad as principle, the dyad as the first going-forth, the triad as the first completion of plane figure, and the tetrad as the first body — point, line, surface, solid.

The sum of its rows is the decad, which the Pythagoreans held to be the perfect number, the turning-point at which counting returns upon itself; and hidden within its rows lie the ratios of the musical consonances — 2:1 the octave, 3:2 the fifth, 4:3 the fourth — so that the same figure is at once arithmetic, geometry, and harmony. Philolaus made the decad the measure that gives all things knowability.

For the later Platonists the tetraktys was a synthema of the intelligible order itself: a visible token that number is not an abstraction from things but the paradigm according to which the Demiurge disposes them. Thomas Taylor's Theoretic Arithmetic gathers the ancient testimonies on this figure and its theology.