# List of spirals

This list of spirals includes named spirals that have been described mathematically.
| Name | First described | Equation | Comment |
| Circle | | $r=k$ | The trivial spiral |
| Archimedean spiral (also arithmetic spiral) | c. 320 BC | $r=a+b\cdot \theta$ | |
| Fermat's spiral (also parabolic spiral) | 1636 | $r^{2}=a^{2}\cdot \theta$ | |
| Euler spiral (also Cornu spiral or polynomial spiral) | 1696 | $x(t)=\operatorname {C} (t),\,$$y(t)=\operatorname {S} (t)$ | Using Fresnel integrals |
| Hyperbolic spiral (also reciprocal spiral) | 1704 | $r={\frac {a}{\theta }}$ | |
| Lituus | 1722 | $r^{2}\cdot \theta =k$ | |
| Logarithmic spiral (also known as equiangular spiral) | 1638 | $r=a\cdot e^{b\cdot \theta }$ | Approximations of this are found in nature |
| Fibonacci spiral | | Circular arcs connecting the opposite corners of squares in the Fibonacci tiling | Approximation of the golden spiral |
| Golden spiral | | $r=\varphi ^{\frac {2\cdot \theta }{\pi }}\,$ | Special case of the logarithmic spiral |
| Spiral of Theodorus (also known as Pythagorean spiral) | c. 500 BC | Contiguous right triangles composed of one leg with unit length and the other leg being the hypotenuse of the prior triangle | Approximates the Archimedean spiral |
| Involute | 1673 | $x(t)=r(\cos(t+a)+t\sin(t+a)),$ $y(t)=r(\sin(t+a)-t\cos(t+a))$ | Involutes of a circle appear like Archimedean spirals |
| Helix | | $r(t)=1,\,$ $\theta (t)=t,\,$ $z(t)=t$ | A three-dimensional spiral |
| Rhumb line (also loxodrome) | | | Type of spiral drawn on a sphere |
| Cotes's spiral | 1722 | ${\frac {1}{r}}={\begin{cases}A\cosh(k\theta +\varepsilon )\\A\exp(k\theta +\varepsilon )\\A\sinh(k\theta +\varepsilon )\\A(k\theta +\varepsilon )\\A\cos(k\theta +\varepsilon )\\\end{cases}}$ | Solution to the two-body problem for an inverse-cube central force |
| Poinsot's spirals | | $r=a\cdot \operatorname {csch} (n\cdot \theta ),\,$ $r=a\cdot \operatorname {sech} (n\cdot \theta )$ | |
| Nielsen's spiral | 1993 | $x(t)=\operatorname {ci} (t),\,$ $y(t)=\operatorname {si} (t)$ | A variation of Euler spiral, using sine integral and cosine integrals |
| Polygonal spiral | | | Special case approximation of arithmetic or logarithmic spiral |
| Fraser's Spiral | 1908 | | Optical illusion based on spirals |
| Conchospiral | | $r=\mu ^{t}\cdot a,\,$$\theta =t,\,$$z=\mu ^{t}\cdot c$ | A three-dimensional spiral on the surface of a cone. |
| Calkin–Wilf spiral | | | |
| Ulam spiral (also prime spiral) | 1963 | | |
| Sacks spiral | 1994 | | Variant of Ulam spiral and Archimedean spiral. |
| Seiffert's spiral | 2000 | $r=\operatorname {sn} (s,k),\,$$\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 | ${\begin{cases}dx=R\cdot {\frac {y}{\sqrt {x^{2}+y^{2}}}}d\theta \\dy=R\cdot \left[\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2}}}}\right]d\theta \end{cases}}{\begin{cases}x=\sum dx\\\\\\y=\sum dy+R\end{cases}}$ | 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. |