How did the Pythagoreans represent square numbers with dots?

The Pythagoreans represented square numbers by arranging dots or pebbles into square arrays.12

To build these squares, they utilized gnomons—a term referring to a carpenter’s square shape. By placing a series of odd numbers around a unit in the shape of these right-angled gnomons, they could systematically construct larger squares. For example, adding the next odd number to a smaller square creates the next square in the sequence (e.g., $1^2+3=2^2$, $2^2+5=3^2$, and so on).12

This geometric visualization allowed the Pythagoreans to explore deep numerical relationships, such as the sum of consecutive odd numbers equaling a square ($1+3+5+7=4^2$) and the properties underlying the Pythagorean theorem. By contemplating these figurate numbers, they sought to understand the fundamental structure of the universe through the lens of mathematics.123