Nautilus pompilius
spiraalvormige schelp
As an illustration of the golden section in nature one often points to the Nautilus pompilius, an octopus that occurs in the Indian and the Pacific Ocean. It can grow 20 cm long.
Obviously the shell of the animal is a spiral.
If you want to examine which spiral (approximately) matches the shell, you have to find the exact center, because just a small difference will lead to clear differences of the spiral.
This being said the numbers in the applet below are approximate.
shape of the spiral
A rectangle tangent to the shell defines the factor of the increasing radius every 90°.
Since we do not know exactly the pole of the shell, we can only define approximately the tangent points that define the rectangle. .
However this approximation is more than clear to illustrate the difference with the golden rectangle.
No, the spiral shell of the Nautilus pompilius is not a golden spiral.
The radius of the spiral does not increase with factor each 90°
'approximate' alternative
As Chris Impens writes in Golden Maths and Myths, since at least 1838 it's established that the shell of the octopus has the shape of a logaritmic spiral in which every 360° the radius approximately tripples.
The factor 1.301 each 90° that we found in the tangent rectangle matches a factor 2.86 each 360°
Now there's a 'golden' interpretatie of the shell using .
The website goldennumber.net writes:
"There is, however, more than one way to create spirals with golden ratio proportions of 1.618 in their dimensions. The traditional golden spiral (aka Fibonacci spiral) expands the width of each section by the golden ratio with every quarter (90 degree) turn. However, there's another golden spiral that expands with golden ratio proportions with every full 180 degree rotation."
How 'approximate' can a difference be? 2.62 still differs 7.6% from the measured 2.86 and much from a trippling that's measured in most studied. Sure you can make calculations with until you find anything that appromates the measured value.
![spiral with factor [math]\Phi^2=2.62[/math] each 360°](https://www.geogebra.org/resource/hzwxvqqx/U9BCleiqLLCJqqJv/material-hzwxvqqx.png)