The Sierpiński Triangle
An endless pattern that resembles itself
Summary
Split each triangle into 4 smaller triangles and remove the middle one. Do the same with the 3 that remain. With this simple rule, infinite repetition produces a self-similar fractal whose area shrinks to zero.
In depth
When the Polish mathematician Wacław Sierpiński described this structure in 1915, he was presenting one of the early examples that showed the limits of classical geometry. The Sierpiński triangle is one of the classic examples of a fractal: every part is a shrunken copy of the whole (self-similarity). In classical geometry, an object has dimension 1 (a line), 2 (a surface) or 3 (a volume). The dimension of the Sierpiński triangle, however, is log 3 / log 2 ≈ 1.585, which is not a whole number. This is where the idea of "fractal dimension" (Hausdorff dimension) comes in. After n iterations the structure contains 3ⁿ triangles; the total area shrinks as (3/4)ⁿ × the starting area and goes to zero in the limit, while the total perimeter goes to infinity. Endless detail, zero area: the paradox of fractals.
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