The modern nautilus's smooth, brown-striped shell houses a small squishy beast with 90 sheathed tentacles. Because the shell's chambers follow the golden ratio of 1:1.6180 in a logarithmic spiral, mathematicians love this creature. Renaissance goldsmiths loved it too, encrusting the shell with jewels and metals to serve as a lavish goblet. Similar shells were often displayed in churches to showcase their beauty as created by God and embellished by Man.
As cephalopods, nautiloids are related on the one hand to octopi and squids, and on the other hand to the extinct ammonites, which have coiled shells serrated like ram's horns.
The nautilus we know and love, N. pompilius, is a survivor of a dazzling prehistoric diversity of shelled cephalopods. Nautiloids have lived in the ocean for over 500 million years. They are older than the dinosaurs, older than the crocodiles, older than the horseshoe crabs.
The fossil record shows a number of fantastic nautilus shells, including straight shells, cones, crescents, and large open coils totally unlike the nautilus we know today. Some shells had spikes, spines, or wings. One creature, Endoceras, was upwards of ten feet long and crawled along the ocean floor. Others, including the very first cephalopod Plectronoceras, moved by squirting water out of its chambers and through an open funnel.
The fleshy parts of the nautiloids (if we may extrapolate from living specimens) also include as many as ninety tentacles, two pairs of gills, simple lensless eyes that work like a pinhole camera, and a leathery hood to close the shell when the animal retreats inside it.
A blood-rich tube of tissue called the siphuncle can secrete gas into the shell's chambers to make the nautilus float, or release gas to allow it to sink. The nautilus adjusts its buoyancy this way, ascending from the depths of the Pacific every night to scavenge near the surface.
As the nautilus grows, it closes off chambers in the shell behind it. Each chamber is slightly larger than the previous one—in N. pompilius they are larger by a factor of (1 + sqrt(5))/2 , which corresponds neatly to the golden ratio. Rene Descartes wrote a treatise on the geometry of N. pompilius, proving the congruence of the segments and finding that the logarithmic growth of the shell represents a fibonacci spiral.
Year after year beheld the silent toil
That spread his lustrous coil;
Still, as the spiral grew,
He left the past year's dwelling for the new,
Stole with soft step its shining archway through,
Built up its idle door,
Stretched in his last-found home, and knew the old no more.
—Oliver Wendell Holmes, "The Chambered Nautilus"