The Square Root Spiral of Theodorus

Spiral of Theodorus [image Wikimedia Commons].

The square-root spiral is attributed to Theodorus, a tutor of Plato. It comprises a sequence of right-angled triangles, placed edge to edge, all having a common point and having hypotenuse lengths equal to the roots of the natural numbers.

The spiral is built from right-angled triangles. At the centre is an isosceles triangle of unit side and hypotenuse ${\sqrt{2}}$ . Another triangle, with sides ${1}$ and ${\sqrt{2}}$ and hypotenuse ${\sqrt{3}}$ is stacked upon the first. This process continues, giving hypotenuse lengths ${\sqrt{n}}$ for all ${n}$ .

Construction of the Spiral

It is convenient to place the spiral in the complex plane. The ${n}$ -th triagle has sides ${1}$ , ${\sqrt{n}}$ and ${\sqrt{n+1}}$ . We denote the vertices as ${z_n}$ and ${z_{n+1}}$ and the angle at the origin by ${\varphi_n}$ . A typical component triangle is shown in the figure below. From the figure it is clear that

$\displaystyle \tan\varphi_n = \frac{1}{\sqrt{n}} \qquad\mbox{or}\qquad \varphi_n = \arctan\left(\frac{1}{\sqrt{n}}\right) \,.$

Triangular component of the Spiral of Theodorus.

The total angle of the first ${N}$ components, or sum of the first ${N}$ angles, is

$\displaystyle \vartheta_k = \sum_{n=1}^N \varphi_n = 2\sqrt{N} + c_2(N)$

where ${c_2(N)}$ tends to a limit ${C \approx -2.15778}$ as ${N\rightarrow \infty}$ . So, for large ${N}$ the total angle grows like ${2\sqrt{N}}$ .

Clearly the growth of the edge lengths is

$\displaystyle \Delta r = r_{n+1}-r_n = \sqrt{n+1}-\sqrt{n} \approx \frac{1}{2\sqrt{n}} \,.$

The vertices are ${z_n = r_n \exp(i\vartheta_n)}$ . From this we can easily show that

$\displaystyle z_{n+1} = \left( 1+\frac{i}{\sqrt{n+1}} \right) z_n \,.$

This is a first order difference equation for ${z_n}$ . Starting with ${z_0=1}$ , the vertices of the spiral can successively be found.

Archimedean Spiral

The spiral of Theodorus is such that each loop is approximately the same distance ${2\pi}$ from the preceding one. We recall that for the Archimedean spiral ${r=\theta}$ , the distance between consecutive windings is always ${2\pi}$ . We can approximate the square-root spiral by ${r=\theta}$ . In the figure below, the left panel shows the first 530 vertices ${(r_n,\vartheta_n)}$ . In the right panel, a spiral of Archimedes is superimposed on these. We see that as ${n}$ increases, there is ever-closer agreement between the two spirals.