The Sierpiński triangle is a self-similar fractal: a geometric structure that reproduces the same pattern at infinitely different scales.

Unlike some fractals, which are devilishly complex mathematically, the triangle is easy to describe. It can be constructed by taking an equilateral triangle, splitting it into four smaller equilateral triangles, and then removing the central one. This process can be repeated arbitrarily many times, creating a more and more detailed shape.

The process of creating a Sierpiński triangle

The triangle is named after the Polish mathematician Wacław Sierpiński (1882–1969), who described it mathematically in 1915. But he wasn’t the first to discover it; perhaps because of its simplicity, it has been used in art and design since at least the 13th century.

Perhaps most pleasingly, if you treat the shape as musical notes, the resulting melody is quite beautiful: