The Sierpiński triangle has exactly zero area but manages to fill space with a consistent, measurable thickness at every scale.
Fractals are famous for being infinitely complex, but this specific shape has a paradoxical geometric property. Even though it technically occupies no space, every small circle drawn on the triangle contains a solid core. This core has a constant proportional radius of roughly 0.288 times the size of the circle. This property means the triangle is uniformly non-flat and fills space in a way that area calculations cannot capture. It reveals a new way to measure the fullness of complex, empty-looking structures in nature.
How Thick Is the Sierpiński Triangle?
arXiv · 2605.01476
Although the Sierpiński triangle has planar area $0$, it is uniformly non-flat: at every point and every scale, its nearby points span a two-dimensional region of comparable size. We prove a sharp version of this statement, showing that the Feng--Wu thickness of $E$ is exactly $\sqrt{3}/6$, the inradius of a unit equilateral triangle. More precisely, if $E$ is the standard Sierpiński triangle of side length $1$ and $B(x,r)$ denotes the closed disk of radius $r$ centered at $x$, then for every $x