The spiral of similar triangles

A configuration of similar figures that shrink with a constant ratio generates a sum of areas that is a geometric series: if the ratio is less than 1, the sum converges to a finite value.

Example — Spiral of similar triangles ("hermit crab")

A spiral of consecutive right-angled triangles, each built on the hypotenuse of the previous one, with similarity ratio $k$.

If the area of the first triangle is $A_1$, the subsequent areas are: $A_1,\; k^2 A_1,\; k^4 A_1,\; k^6 A_1,\; \ldots$ The sum of the infinitely many triangles is a geometric series: $A_{\text{tot}} = A_1\cdot\sum_{n=0}^{\infty}(k^2)^n = \frac{A_1}{1-k^2}, \qquad |k|<1.$

Example — A numerical case

With $k=\dfrac{\sqrt{3}}{2}$ and $A_1 = \dfrac{\sqrt{3}}{2}$: $A_{\text{tot}} = \frac{\sqrt{3}/2}{1-3/4} = \frac{\sqrt{3}/2}{1/4} = 2\sqrt{3}$

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