The Nautilus shell

The chambered pearly shell of Nautilus pompilius, named by C. Linnaeus in 1758, is not only pretty but its spiral growth is an excellent example of logarithmic spiral growth, similar to the spiral bands of clouds in a hurricane or the arms of a spiral galaxy (i.e., Google "logarithmic spiral" for more information on this subject).

The Nautilus is a cephalopod, and this group of animals also includes the squid, cuttlefish, and octopus. Living specimens of N. pompilius can only be viewed in their natural state at a few locales in tropical waters in the southwest Pacific Ocean, or in controlled environments in public or private aquariums. Nautilus shells can be found as beach drift on some beaches. 

Adult shell of Nautilus pompilius (swimming mode orientation);
 maximum diameter 5.5 inches (14 cm).
 

Juvenile shell of Nautilus pompilius shell, 
maximum diameter 2.9 inches (7.3 cm) 

Notice that the juvenile Nautilus is fully covered with stripes, whereas the adult shell only has the stripes on its early part. The stripes provide camouflage for the juvenile because it spends its time on or near the ocean floor. The stripes allow it to blend in. The adult spends most of its time swimming or floating in the water column, and stripes are not needed, at least, on the ventral part of its shell. If a predator looks at the adult Nautilus shell from below, the shell looks like the sun-lit waters near the surface of the ocean.

Cut-away (median-longitudinal) section of adult Nautilus pompilius shell
 showing interior structures; diameter 6.3 inches (16 cm).

As shown in the above picture, the early part of the shell has numerous, closely spaced chambers called camera (single chamber = camerum), which provide great strength to the shell when the animal sinks into the depths (several hundred feet deep) of the ocean during the day. If the shell did not have this added strength, it would implode.

The camera are filled with nitrogen gas, which gives buoyancy to the shell. The siphuncle is a fleshy tube that connects all the camera and serves as a conduit for the transfer of the gaseous contents. The buoyancy also affects the shells after death of the animal. The empty shells can drift long distances. If you submerge an empty Nautilus pompilius shell in a bucket of water, the shell will bop up, rather than sink.

The interior of the N. pompilius shell consists of "mother-of-pearl" shelly material, which is the biomineral aragonite. This mineral  was secreted by the animal as the shell grew, and that is why the term "biomineral" is used here. 

Nautilus is one of only two genera of extant (living) cephalopods known as nautiloids. Fossil nautiloids have a geologic record that goes back to the Cambrian Period, 550 million years ago, although shells did not become common until the subsequent Ordovician Period. These early nautiloids had a straight shell and are called orthocone nautilioids, as opposed to the more modern, coiled nautiloids, like N. pompilius.

Example of Eutrephoceras, shell incomplete (dorsal margin partially missing).
 Maximum diameter 1 inch (2.54 cm).

Eutrephoceras, an extinct coiled nautiloid whose geologic range is Late Jurassic to Miocene, differs in its morphology from Nautilus by having straighter septa (also called sutures).