How Thick Is the Sierpiński Triangle?

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 $\sqrt{3}\mathrm{/}6$, 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 $x\in E$ and every $0<r\le 1$, the convex hull of $E\cap B(x,r)$ contains an equilateral triangle of side length $r$. Consequently, $\mathrm{conv}(E\cap B(x,r))$ contains a closed disk of radius $(\sqrt{3}\mathrm{/}6)r$; this constant is best possible. The proof is elementary – boundary edges of all construction triangles survive in the limit set, and self-similarity reduces the problem to the normalized range $1\mathrm{/}2\le r\le 1$.