This list is incomplete; you can help by adding missing items. (February 2019)

This list of spirals includes named spirals that have been described mathematically.

Image Name First described Equation Comment circle $r=k$ The trivial spiral Archimedean spiral c. 320 BC $r=a+b\cdot \theta$ Also known as the arithmetic spiral Euler spiral (also Cornu spiral or polynomial spiral) $x(t)=\operatorname {C} (t),\,$ $y(t)=\operatorname {S} (t)$ using Fresnel integrals Fermat's spiral (also parabolic spiral) 1636 $r^{2}=a^{2}\cdot \theta$  hyperbolic spiral 1704 $r={\frac {a}{\theta ))$ also reciprocal spiral lituus 1722 $r^{2}\cdot \theta =k$  logarithmic spiral 1638 $r=a\cdot e^{b\cdot \theta )$ Approximations of this are found in nature; also known as the equiangular spiral. Fibonacci spiral circular arcs connecting the opposite corners of squares in the Fibonacci tiling approximation of the golden spiral golden spiral $r=\varphi ^{\frac {2\cdot \theta }{\pi ))\,$ special case of the logarithmic spiral Spiral of Theodorus Also known as the Pythagorean spiral; an polygonal spiral composed of contiguous right triangles that approximates the Archimedean spiral helix $r(t)=1,\,$ $\theta (t)=t,\,$ $z(t)=t$ a 3-dimensional spiral Cotes's spiral 1722 ${\frac {1}{r))={\begin{cases}A\cosh(k\theta +\varepsilon )\\A\exp(k\theta +\varepsilon )\\A\sinh(k\theta +\varepsilon )\\A(k\theta +\varepsilon )\\A\cos(k\theta +\varepsilon )\\\end{cases))$ Solution to the two-body problem for an inverse-cube central force Poinsot's spirals $r=a\cdot \operatorname {csch} (n\cdot \theta ),\,$ $r=a\cdot \operatorname {sech} (n\cdot \theta )$  Nielsen's spiral 1993 $x(t)=\operatorname {ci} (t),\,$ $y(t)=\operatorname {si} (t)$ A variation of Euler spiral, using sine integral and cosine integrals Conchospiral $r=\mu ^{t}\cdot a,\,$ $\theta =t,\,$ $z=\mu ^{t}\cdot c$ three-dimensional spiral on the surface of a cone.
Seiffert's spiral 2000 $r=\operatorname {sn} (s,k),\,$ $\theta =k\cdot s$ $z=\operatorname {cn} (s,k)$ spiral curve on the surface of a sphere

using the Jacobi elliptic functions Tractrix spiral 1704 ${\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases))$ Pappus spiral 1779 ${\begin{cases}r=a\theta \\\psi =k\end{cases))$ 3D conical spiral studied by Pappus and Pascal doppler spiral $x=a\cdot (t\cdot \cos(t)+k\cdot t),\,$ $y=a\cdot t\cdot \sin(t)$ 2D projection of Pappus spiral Atzema spiral $x={\frac {\sin(t)}{t))-2\cdot \cos(t)-t\cdot \sin(t),\,$ $y=-{\frac {\cos(t)}{t))-2\cdot \sin(t)+t\cdot \cos(t)$ The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral. Atomic spiral 2002 $r={\frac {\theta }{\theta -a))$ This spiral has two asymptotes; one is the circle of radius 1 and the other is the line $\theta =a$  Galactic spiral 2019 ${\begin{cases}dx=R*{\frac {y}{\sqrt {x^{2}+y^{2))))d\theta \\dy=R*{\Bigl [}\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2)))){\Bigr ]}d\theta \end{cases)){\begin{cases}x=\sum dx\\\\\\y=\sum dy+R\end{cases))$ The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases:$\rho <1,\rho =1,\rho >1$ , the spiral patterns are decided by the behavior of the parameter $\rho$ . For $\rho <1$ , spiral-ring pattern; $\rho =1,$ regular spiral; $\rho >1,$ loose spiral. R is the distance of spiral starting point (0, R) to the center. The calculated x and y have to be rotated backward by ($-\theta$ ) for plotting.