The Wheel Of Theodorus, or Square Root Spiral, is a rather simple geometric construction that allows you to construct line segments with length of the square root of any integer. Observe the figure below.
You’ll notice I’ve stolen this image from the Figures Speak weblog. I’m currently without my aged copy of the Geometer’s Sketchpad, so crudely wrought and stolen images will have to suffice.
The construction is fairly simple. Starting with a 1-1-root 2 right triangle, use the hypotenuse as one leg, and add a perpendicular segment of length 1 to create another right triangle. Pythathagoras says that the resulting triangle has a hypotenuse of root 3. And so it goes. And it creates this very nice spiral effect.
The construction is very simple, but I would like a way to express this spiral figure itself mathematically. For example, a smooth parameterization would be very nice. Unfortunately I’m not sure one exists. A parameterization exists, but it sort of cheats, and will not be discussed here because it distracts from my main point.
And that main point is that complex numbers provide a very convenient and very elegant way to express the geometric relationships in the figure.
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