"All Things are Number"
a² + b² = c². This simple formula is one of the most famous theorems in history. It states that for a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. While known to Babylonians and Egyptians, the Pythagorean school is credited with the first proof.
The theorem can be proven visually. If you draw squares on each of the three sides, the area of the largest square is exactly equal to the sum of the areas of the two smaller ones. This is Proposition 47 in Euclid's Elements.
Pythagoras also discovered that musical harmony is based on simple integer ratios of frequencies (like 1:2 for an octave), supposedly after hearing hammers in a blacksmith shop. He believed the entire universe was built upon the harmony of numbers.