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 . Another triangle, with sides and and hypotenuse is stacked upon the first. This process continues, giving hypotenuse lengths for all .

**Construction of the Spiral**

It is convenient to place the spiral in the complex plane. The -th triagle has sides , and . We denote the vertices as and and the angle at the origin by . A typical component triangle is shown in the figure below. From the figure it is clear that

Triangular component of the Spiral of Theodorus.

The total angle of the first components, or sum of the first angles, is

where tends to a limit as . So, for large the total angle grows like .

Clearly the growth of the edge lengths is

The vertices are . From this we can easily show that

This is a first order difference equation for . Starting with , 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 from the preceding one. We recall that for the Archimedean spiral , the distance between consecutive windings is always . We can approximate the square-root spiral by . In the figure below, the left panel shows the first 530 vertices . In the right panel, a spiral of Archimedes is superimposed on these. We see that as increases, there is ever-closer agreement between the two spirals.

** Sources **

Davis, Philip J., 2001: *Spirals: from Theodorus to Chaos*. A K Peters, Wellesley, Mass.

Wikipedia article *Spiral of Theodorus.*

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