Logarithmic Spiral - Properties

Properties

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling them gives the same result as rotating them). Scaling by a factor gives the same as the original, without rotation. They are also congruent to their own involutes, evolutes, and the pedal curves based on their centers.

Starting at a point and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as goes toward is finite. This property was first realized by Evangelista Torricelli even before calculus had been invented. The total distance covered is, where is the straight-line distance from to the origin.

The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0. (Up to adding integer multiples of to the lines, the mapping of all lines to all logarithmic spirals is onto.) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.

The function, where the constant is a complex number with non-zero imaginary part, maps the real line to a logarithmic spiral in the complex plane.

One can construct a golden spiral, a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch about 17.03239 degrees), or approximate it using Fibonacci numbers.