A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral". 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.
Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Although this curve had already been named by other mathematicians, the specific name was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Jacob Bernoulli wanted such a spiral engraved on his headstone along with the phrase "Eadem mutata resurgo", but, by error, an Archimedean spiral was placed there instead.
Properties
The logarithmic spiral has the following properties :
Polar slope:
Curvature:
Arc length:
Sector area:
Inversion:Circle inversion maps the logarithmic spiral onto the logarithmic spiral
Rotating, scaling: Rotating the spiral by angle yields the spiral, which is the original spiral uniformly scaled by.
Self-similarity: A result of the previous property:
Relation to other curves: Logarithmic spirals are congruent to their own involutes, evolutes, and the pedal curves based on their centers.
The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation. It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.
In nature
In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:
The approach of a hawk to its prey in classical pursuit, assuming the prey travels in a straight line. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.
The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun is the only light source and flying that way will result in a practically straight line.
The arms of spiral galaxies. Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees.
Many biological structures including the shells of mollusks. In these cases, the reason may be construction from expanding similar shapes, as is the case for polygonal figures.
Logarithmic spiral beaches can form as the result of wave refraction and diffraction by the coast. Half Moon Bay is an example of such a type of beach.
