Logarithmic Spiral

An interactive logarithmic spiral — the self-similar spiral seen in nautilus shells and spiral galaxies, whose distance from the center grows exponentially with the angle. Looking for a different curve? Browse the full list of interactive curves.

Logarithmic Spiral

Drag the sliders to change how tightly the spiral winds ($b$) and its overall scale ($a$).

Logarithmic Spiral — The Details

A logarithmic spiral (also called an equiangular spiral, or spira mirabilis — "miraculous spiral", a name coined by Jacob Bernoulli) is a self-similar curve: rotate it by any fixed angle and rescale it by the matching constant factor, and it lands exactly back on itself. Unlike an Archimedean spiral, whose turns are always the same fixed distance apart, a logarithmic spiral's turns spread out exponentially, and the curve winds inward toward its center infinitely often without ever quite reaching it. This growth pattern shows up throughout nature — most famously in the chambered nautilus shell, but also in spiral galaxies and the path a hawk takes homing in on prey while keeping its head turned at a fixed angle. In polar coordinates it follows the equation:

$r = ae^{b\theta}$