A seashell's height

Buccinum humphreysianum is a very handy marine snail when it comes to the discovery of a mathematical equation that describes a shell. The 7 to 8 tumid worls[2] of this species are stacked neatly on top of each other. This stack provides a splendid and authoritative image for what an ideal shell should look like (at least to me).

If you do not know what I mean, just have a look below at photograph of Humphrey's old home (the markings is discussed below). Surely you will agree that it is worth a little of our time to get to the bottom of the mathematics that describe this beautifully organised structure.

It is always good to be clear on what you want to achieve. So let us start with a simple idea:

Find a function $f(\omega)$ that predicts the height of Buccinum humphreysianum at the $\omega$ -th worl.

What can we measure? #

The total height of our shell has been measured by the photographer as $38$ mm [1]. And, if we imagine the worl that was still forming to be more complete, a count $7$ worls is available on this photo. These are marked on the photo.

Let's take a closer look at the tumidity of Humphrey's worls. Can you measure the height of each worl directly from the photograph? My measurements are shown below. And yes, they do add up to $38$ , which is just proper.

Worl	Worl (mm)
1	0.7
2	0.9
3	1.4
4	2.8
5	5.2
6	9
7	18

We are interested in the total height grown from the top. If we add up the height of each worl that contribute to the total height, we get data that more closely resembles our problem space:

$\omega$	Height (mm)
1	0.7
2	1.6
3	3
4	5.8
5	11
6	20
7	38

Some intuitions #

The taller the shell is, the larger the bottom worl. We assume that the ratio between the height of a worl and its predecessor is constant. Let $g$ be this constant growth factor.

We also take it as a basic truth that something cannot grow from nothing. So there has to be a value for the smallest initial height. We denote this initial height as $k$ .

We can now write down the sequence of expressions that show the total height at each worl:
$f (0) f (1) f (2) = k = f (0) + f (0) \times g = f (0) (1 + g) = k (1 + g) = f (1) (1 + g) = f (2) = k (1 + g) (1 + g) = k (1 + g)^{2}$

An equation emerges from this sequence
$f (ω) = k (1 + g)^{ω}$

Solving the puzzle #

For Humphrey's home, we now need to get values for $k$ and $g$ . There are two unknowns, $k$ and $g$ , and we have many equations to choose from. So it is a straight forward process to calculate values for the constants:

Let us take the first row in our total height measurements:

$0.7 1 + g = k (1 + g)^{1} = 0.7 / k$

And now the last row:
$38 (1 + g)^{7} = k (1 + g)^{7} = 38 / k$

And now we combine
$(0.7 / k)^{7} (0 . 7^{7} / k^{7}) (0 . 7^{7} / 38) k^{6} = 38 / k = 38 / k = k^{7} / k = 0 . 7^{7} / 38 = 0 . 7^{7 / 6} / 3 8^{1 / 6} = 0.36$

And

$1 + g g = 0.7 / 0.36 = 0.7 / 0.36 - 1 = 0.94$

So we have an equation:
$f (n) = 0.36 (1 + 0.94)^{n} = 0.36 (1.94)^{n}$

How good is the model? #

Working out our predicted values show that our model is not too far off:

$\omega$	Height	$f(\omega)$	Error
1	0.7	0.7	0
2	1.6	1.35	0.25
3	3	2.63	0.37
4	5.8	5.1	0.7
5	11	9.89	1.11
6	20	19.19	0.81
7	38	37.23	0.77

Using the model #

For good values of the parameters $k$ and $g$ this equation is now our model:

$f (ω) = k (1 + g)^{ω}$

To use this model all you need is to take two measurements.

Measure the first worl, $w_1$ and the total height $w_n$ . Also decide on the final worl number $n$ . Note that $n$ does not have to be an integer. Worls are normally counted so that the final worl is a fraction.

From these three numbers, $w_1,w_n$ and $n$ , anyone can estimate the length at any worl, $\omega$ .

Just set $g = (w_1/w_n)^{\frac {-1}{n-1}} - 1$ and $k = w_1/(1 + g)$ and use the model $f(\omega)$ .

Here's how we get the equation for $g$ :
$w_{1} w_{n} w_{1} / w_{n} w_{1} / w_{n} (w_{1} / w_{n})^{\frac{1}{1 - n}} g = k (1 + g)^{1} = k (1 + g)^{n} = k (1 + g) / (k (1 + g)^{n}) = (1 + g)^{1 - n} = 1 + g = (w_{1} / w_{n})^{\frac{- 1}{n - 1}} - 1$

And for $k$ :
$w_{1} k = k (1 + g) = w_{1} / (1 + g)$

For our photographed shell we have:
$g k = (0.7 / 38)^{- 1 / 6} - 1 = 0.94 = 0.7 / (1 + 0.94) = 0.36$

In conclusion #

We have shown how an observation from nature can lead us down a path to discover an interesting mathematical formula. Defining such a model is not a complicated process - it uses only high-school mathematics.

References #

Next: A seashell spiral