Golden spiral in the context of Nautilus shell

In the geometry of spirals, the pitch angle or pitch of a spiral is the angle made by the spiral with a circle through one of its points, centered at the center of the spiral. Equivalently, it is the complementary angle to the angle made by the vector from the origin to a point on the spiral, with the tangent vector of the spiral at the same point. Pitch angles are used to characterize the steepness of spirals, such as in astronomy to describe spiral galaxies.

Logarithmic spirals, for example, are characterized by the property that the pitch angle remains invariant for all points of the spiral. Two logarithmic spirals are congruent when they have the same pitch angle, but otherwise are not congruent. For instance, only the golden spiral has pitch angle$${\displaystyle \arctan \left({\frac {\ln \varphi }{\pi /2}}\right)\approx 17^{\circ },}$$where ${\displaystyle \varphi }$ denotes the golden ratio; logarithmic spirals with other angles are not golden spirals.

The chambered nautilus (Nautilus pompilius), also called the pearly nautilus, is the best-known species of nautilus. The shell, when cut away, reveals a lining of lustrous nacre and displays a nearly perfect equiangular spiral, although it is not a golden spiral. The shell exhibits countershading, being light on the bottom and dark on top. This is to help avoid predators, because when seen from above, it blends in with the darkness of the sea, and when seen from below, it blends in with the light coming from above.

The range of the chambered nautilus encompasses much of the south Pacific; It has been found near reefs and on the seafloor off the coasts of Australia, Japan, and Micronesia.