logarithmic spiral

mathematics

Also known as: equiangular spiral, logistic spiral

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

June 30, 2026

Quick Summary

The logarithmic spiral, also known as the equiangular spiral, is a curve that winds around a central point while continuously moving farther from it. Discovered by René Descartes in 1638 and later named the spira mirabilis ("miracle spiral") by Jakob Bernoulli, it possesses unique mathematical properties.12

Its general equation is r = ae^{θ cot b}, where r is the radius, a and b are constants, θ is the angle of rotation, and e is the base of the natural logarithm. Unlike the spiral of Archimedes, where turns are equally spaced, the distance between successive turns of a logarithmic spiral increases geometrically. A key characteristic is that every ray from its center intersects every turn at a constant angle, making it equiangular. This spiral is observed in nature, such as in spiderwebs and the shells of the nautilus.1

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

2 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

spiral
The logarithmic, or equiangular, spiral was discovered by the French scientist René Descartes in 1638. In 1692 the Swiss mathematician Jakob Bernoulli named it spira mirabilis (“miracle spiral”) for its mathematical properties; it is carved on his tomb. The general equation of the logarithmic spiral is…
Read More
The logarithmic, or equiangular, spiral was discovered by the French scientist René Descartes in 1638. In 1692 the Swiss mathematician Jakob Bernoulli named it spira mirabilis (“miracle spiral”) for its mathematical properties; it is carved on his tomb. The general equation of the logarithmic spiral is…
Read More

gastropod
Size range and diversity of structure
…shapes, based primarily upon the logarithmic spiral. They can be coiled flatly in one plane, as in Planorbis; become globose with the whorls increasing rapidly in size, as in Pomacea; have the whorls become elongate and rapidly larger, as in Conus and Scaphella; have a few flatly coiled whorls that…
Read More
…shapes, based primarily upon the logarithmic spiral. They can be coiled flatly in one plane, as in Planorbis; become globose with the whorls increasing rapidly in size, as in Pomacea; have the whorls become elongate and rapidly larger, as in Conus and Scaphella; have a few flatly coiled whorls that…
Read More