The **equiangular spiral** (also called the* Bernoulli spiral*, *logarithmic spiral*, *logistique*, or *Spira Mirabilis*) is a family of spirals defined as a monotonic curve that cuts all radii vectors at a constant angle [1]. In other words, the spiral forms a constant angle between a line drawn from the origin to any point on the curve and the tangent line at that point [2]. It is this fact — equal angles — that gives the curve its name.

The curve is also sometimes called the

*geometrical spiral*, because a radius’s angle increases in geometrical progression as its polar angle increases in arithmetical progression [3].

## Equations for the Equiangular Spiral

The equiangular spiral has polar equation r = r * k^{θ}, where:

- r = initial radius,
- k = a constant > 1 or < 1,
- θ = the angle.

The parametric equation is [1]:

- x = e
^{(t * cot(α))}* cos(t) - y = e
^{(t * cot(α))}* sin(t)

The Cartesian equation is:

x^{2} + y^{2} = e^{(θ*cot(α))}.

Various natural phenomena have the shape of an equiangular spiral, including chambered nautilus shells, the Milky Way galaxy, and arrangements of sunflower seeds on the sunflower.

## History of the Equiangular Spiral

The first known construction of the equiangular spiral was in Durer’s 1525 book Udterweysung [4]. The formula was discovered by Descartes [1]. It was later studied by Jacques Bernoulli, who dubbed the spiral *spira mirabilis*, “the wonderful spiral.” Bernoulli was so enamored with the spiral that he has it engraved on his tomb with the phrase “*Eadem mutata resurgo*” (Though changed, I rise again the same.).

## References

[1] Tully, D. Equiangular Spiral, Logarithmic Spiral, Bernoulli Spiral. Retrieved February 23, 2022 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/Fall98/DarrenT/EquiangularSpiral.html

[2] Spiral. Retrieved February 23, 2022 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/Sp98/GabeP/Spiral.htm

[3] Erbas, A. MATH 7200-Foundations of Geometry. Retrieved February 23, 2022 from: http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html

[4] Albrecht Dürer (1525). Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen