Four rows of dots — one, two, three, four — stacked into a triangle. It totals TEN, and the Pythagoreans held it so sacred they swore oaths on it: the tetractys, the ‘source of ever-flowing nature’. Hidden in it are the ratios of music (2:1 the octave, 3:2 the fifth, 4:3 the fourth) and the tenth triangular number. Slide to build the rows and read the sum.
The tetractys arranges 1+2+3+4 = 10 points in a triangle — the 4th triangular number T₄ = n(n+1)/2 = 10. The Pythagoreans revered the number 10 as complete and read the musical concords in its first ratios: 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth). A fail-loud self-check throws unless 1+2+3+4 = 10 and equals the 4th triangular number. ◆ the arithmetic is exact, node-verified.
LIT: the sum 10, the triangular-number identity, and the small-integer musical ratios are exact. AMBER: ‘source of all nature’ and the oath-worthiness are Pythagorean doctrine — the reverence is the belief, the counting is the math.