The Wheel Of Theodorus and Complex Numbers

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.

Consider the vertices of the triangles, the ones that lie on the spiral. We are going to consider these points complex numbers existing in the complex plane. Complex numbers in the complex plane can be conveniently thought of both as points in the plane, and as vectors emanating from the origin to those points. The arithemetic involving complex numbers becomes, in effect, some very slick geometric manipulation of these points and vectors.

The key and driving point here is that multiplying a complex number by i in effect rotates it counter clockwise around the origin by 90 degrees. For example, consider the number 1 + 0*i. This point is located at (1, 0) in the plane. Multiplying by i yields 0 + 1*i, or the point (0, 1), a 90 degree rotation. No matter the number, where or what it is, multiplying by i always rotates 90 degrees.

So consider a point on the spiral, we’ll call it R. To construct the next point on the spiral, we travel perpendicularly, counter clockwise, along a segment of length 1, starting from R, and that is the next point on the spiral Rnew. We can mimic this construction with complex arithmetic.

Let R(n) by the n’th point along the spiral, represented by a complex number … which we will also call R(n). We’ll say that R(1) = 1. First, we want to construct the segment perpendicular to R(n). As I said, this is simply R(n)*i. Next, we want to scale this segment, so as to give it length 1. This is accomplished by dividing by the length of R(n), or distance from R(n) to the origin. This is denoted |R(n)|. Our segment is now i*R(n)/|R(n)|. Displacement from the original point is acomplished through simple addition, yielding the next point as R(n+1) = R(n) + i*R(n)/|R(n)|.

We can improve upon this, however. Because from the construction of the spiral itself, we know that the nth point is a distance of root n away from the origin. Thus the formula becomes, factoring conveniently,

Notice the recursive product structure of this formula. We can unwind the recursion to yield

or

And the above is a very tidy formula indeed, allowing us to calculate any point along the spiral. To demonstrate,

Thus R(2) corresponds on the plane to the point (1, 1), which is clearly correct, one unit over on the x axis, one unit up on the y.

Of course, it’s not as nice as a smooth parameterization, but it’s a very neat encoding of some nice geometry. Complex numbers – not just for taking the square roots of negative numbers any more.