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Why did Pythagoreans link odd numbers with limit and even numbers with the unlimited?

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Marble bust of Pythagoras, the ancient Greek mathematician and philosopher, showing a bearded man with a head covering and calm expression. — Pythagoreanism Greek philosopher and mathematician Pythagoras, portrait bust.

In Pythagoreanism, the association of odd numbers with the "limit" and even numbers with the "unlimited" stems from their doctrine of cosmic opposites.1

The Pythagoreans viewed the odd numbers (3, 5, 7, ...) as an active, ordering force that imposes structure and harmony upon the unlimited. This conceptual framework was reinforced by their use of gnomons—pebbles or dots arranged to represent numbers geometrically. When odd numbers are arranged as gnomons around a unit, they consistently produce stable, closed square numbers, which the Pythagoreans associated with the concept of the limit.12

Conversely, even numbers were associated with the "unlimited" because, when depicted as gnomons, they produce "oblong" numbers that allow for infinite variations. This dualistic view was part of a broader table of opposites, where the limiting-odd and the unlimited-even were considered fundamental principles used to explain the generation of the cosmos.123