A seashell spiral

Previously, we looked at how to discover a function that describes Humphrey's height. A photograph of our seashell, Humphrey is shown below.

In that previous article, we ended up with this function:

$$\begin{array}{r}f(\omega )=c_0(1+c_1{)}^{\omega }\end{array}\text{\hspace{1em}(Eq. 1)}$$

Fig 1: Humphrey
Copyright: Steve Wilkinson
Conchological Society

Where

• $\omega$ is the worl number,
• $c_0$ is the initial height and
• $c_1$ is the growth factor.

We can imagine that there is some kind of spiral at work here. But what is the mathematics behind this spiral? Our new question is this:

Find a parametric equation [1], $g: \reals \mapsto \reals^2$ that takes a parameter, $\theta \in \reals$ and determines the Cartesian coordinates[2] of the spiral that describes Humphreys height.

From circle to spiral #

The parametric equation for a circle is:
$$\begin{array}{r}c(\theta ,r)=r[\cos (\theta ),\sin (\theta )]\end{array}$$
Where

• $\theta \in [0,2\pi]$ is the parameter
• $r$ is the radius of the circle

We use the notation $[x,y]$ as a row vector to describe a Cartesian coordinate.

A spiral is an extrapolation of this circle equation where the radius of the circle increases as $\theta$ increases. We can generalise this concept by imagining some function that describes a varying radius of the circle.

Then we get this generic function, $s: \reals \mapsto \reals^2$
$$\begin{array}{r}s(\theta ,\mathbf{h})=\mathbf{h}(\theta )[\cos (\theta ),\sin (\theta )]\end{array}\text{\hspace{1em}(Eq. 2)}$$
Where

• $\theta \in \reals$ is the angle of rotation
• $\mathbf h: \reals \mapsto \reals$ determines the distance of the point from the origin at a given angle.

As an example, consider the simplest spiral where the radius is equal to an ever increasing angle:

$$\begin{array}{r}s(\theta ,\lambda (\theta ):=\theta )=\theta [\cos (\theta ),\sin (\theta )]\end{array}$$

Returning to the shell #

In order to prepare Eq. 1 so than it can be used in Eq. 2, we have to choose how $\omega$ can be transformed into a rotation angle. It is clear from the photo that each worl rotates through an angle of $2\pi$. If we say $\theta = 2\omega\pi$ then $\theta$ is the rotation angle required to describe $\omega$ worls. From this we get:
$$\begin{array}{r}\omega =\theta /2\pi \end{array}$$

And from Eq. 1, we restate $f$:
$$\begin{array}{r}f(\theta )=c_0(1+c_1{)}^{\theta /2\pi }\end{array}$$
Where

• $\theta$ is rotation angle,
• $c_0$ is the initial height and
• $c_1$ is the growth factor.

For a seashell, the rotation angle is around the coiling axis[3]. This angle is $0$ when the shell is at its initial height $c_0$.

Now we use Eq. 2 and inject the restated version of $f$:

$$\begin{array}{rl}g(\theta )& =s(\theta ,f)\\ & =f(\theta )[\cos (\theta ),\sin (\theta )]\\ & =c_0(1+c_1{)}^{\theta /2\pi }[\cos (\theta ),\sin (\theta )]\end{array}\text{\hspace{1em}(Eq. 3)}$$

The spiral of Humphrey #

Recall from previous article that we have for Humphrey these approximate values:
$$\begin{array}{rl}{c}_0& =0.347\\ c_1& =0.956\end{array}$$

So, substituting into Eq. 3:
$$\begin{array}{rl}g(\theta )& =0.347(1.956{)}^{\theta /2\pi }[\cos (\theta ),\sin (\theta )]\end{array}$$

Where
$$\begin{array}{r}\theta \in [0;14\pi ]\end{array}$$

In the image below we see a plot of this spiral on the Cartesian plane. It does not look like Humphrey, but it does describe the height of our friend. Note that the graph ends up at $38$mm after the worl $7$. Just as we expect it to do.

Fig. 2: Plot of the Humphrey height spiral

Conclusion #

We have shown here how to obtain an equation for a spiral, using our seashell Humphrey to remind us of the real world. We started with the definition of a circle, generalised a function that transforms a circle and used our knowledge of the seashell to get to a spiral that describes the height.

References #

1. Parametric equation - Wikipedia
2. Cartesian coordinates - Wikipedia
3. Shell Morphology - Wikipedia

• Next: A 3-D seashell spiral
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