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.

Left: Vertices of the spiral, {(r_{n},\vartheta_{n})} for {0\le n\le 530}. Right: Spiral of Archimedes {r=k\vartheta} (blue) superimposed on the spiral of Theodorus.}

Sources

{\bullet} Davis, Philip J., 2001: Spirals: from Theodorus to Chaos. A K Peters, Wellesley, Mass.

{\bullet} Wikipedia article Spiral of Theodorus.

 


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