Logarithmic spiral

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A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which 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".



In polar coordinates (r,θ) the curve can be written as[1]


with e being the base of natural logarithms, and a and b being arbitrary positive real constants.

In parametric form, the curve is

with real numbers a and b.

The spiral has the property that the angle φ between the tangent and radial line at the point (r,θ) is constant. This property can be expressed in differential geometric terms as

The derivative of \mathbf{r}(\theta) is proportional to the parameter b. In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that b = 0 (\textstyle\phi = \frac{\pi}{2}) the spiral becomes a circle of radius a. Conversely, in the limit that b approaches infinity (φ → 0) the spiral tends toward a straight half-line. The complement of φ is called the pitch.

Spira mirabilis and Jacob Bernoulli

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 ("miraculous" or "marvelous" spiral) 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. Jakob Bernoulli wanted such a spiral engraved on his headstone along with the phrase "Eadem mutata resurgo" ("Although changed, I shall arise the same."), but, by error, an Archimedean spiral was placed there instead.[2][3]

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