The fractal star

By adding smaller and smaller triangles to the sides of the figure at every step, we obtain a fractal outline whose total area converges, once again thanks to a geometric series.

Example — Fractal star

Starting from an equilateral triangle of side $\ell$, three equilateral triangles of side $\ell/3$ are built on the midpoints of the sides (pointing outwards), forming a star. The process repeats on every new side.

First step of the construction of the fractal star: on the triangle of side $\ell$ grow three triangles of side $\ell/3$.

At each step $3\cdot 4^{n-1}$ triangles of side $\ell/3^n$ are added. The area added at step $n$ is: $\Delta A_n = 3\cdot 4^{n-1}\cdot\frac{(\ell/3^n)^2\sqrt{3}}{4} = \frac{3\sqrt{3}\,\ell^2}{4}\cdot\frac{4^{n-1}}{9^n}.$ The total area of the star after infinitely many steps converges (geometric series with $r=4/9<1$).

Links

Topics: Quadratic equations
Concepts: Fractal · Geometric series
Skills: Modelling · Using formulae