How did Pythagoreans use dots to represent triangular numbers?

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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.

To the Pythagoreans, numbers were invested with specific characteristics and were often visualized as geometric shapes formed by dots or pebbles.1

Pythagoreans represented triangular numbers—such as 1, 3, 6, 10, 15, and 21—by arranging dots into the shape of a triangle. This process began with a single dot (the unit) and could be extended by adding a series of natural numbers to form larger triangles. A notable example is the tetraktys, a "perfect triangle" formed by ten dots arranged in four rows (1 + 2 + 3 + 4 = 10).213

This method of using dots to visualize numbers allowed the Pythagoreans to explore the relationships between different figurate numbers, such as the observation that the sum of two adjacent triangular numbers always results in a square number.1