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How did Pythagoreans use dots to represent square numbers?
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Pythagoreanism
Greek philosopher and mathematician Pythagoras, portrait bust.
Pythagoreans represented numbers using arrangements of dots or pebbles to visualize their mathematical properties. To represent square numbers (such as 1, 4, 9, 16, ...), they arranged these dots into geometric square shapes.12
This process was closely linked to the use of gnōmones, or "carpenter’s squares." By placing a series of odd numbers around a unit as gnomons, the Pythagoreans could systematically build larger squares. For example, adding a gnomon of 3 dots to a single dot creates a 2x2 square (4), and adding a gnomon of 5 dots to that square creates a 3x3 square (9). This visual method allowed them to demonstrate that the sum of consecutive odd numbers always results in a square number.1