Recall the chambered nautilus introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1 below shows a spiral in rectangular coordinates. How can we describe this curve mathematically?

Figure 1. How can we describe a spiral graph mathematically?

As the point P travels around the spiral in a counterclockwise direction, its distance $d$ from the origin increases. Assume that the distance $d$ is a constant multiple $k$ of the angle $\theta$ that the line segment OP makes with the positive x-axis. Therefore $d\left(P,O\right)=k\theta$, where $O$ is the origin. Now use the distance formula and some trigonometry:

Although this equation describes the spiral, it is not possible to solve it directly for either x or y. However, if we use polar coordinates, the equation becomes much simpler. In particular, $d\left(P,O\right)=r$, and $\theta$ is the second coordinate. Therefore the equation for the spiral becomes $r=k\theta$. Note that when $\theta =0$ we also have $r=0$, so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes $r=a+k\theta$ for arbitrary constants $a$ and $k$. This is referred to as an Archimedean spiral, after the Greek mathematician Archimedes.

Another type of spiral is the logarithmic spiral, described by the function $r=a\cdot {b}^{\theta }$. A graph of the function $r=1.2\left({1.25}^{\theta }\right)$ is given in Figure 2. This spiral describes the shell shape of the chambered nautilus.

Figure 2. A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)