List of spirals

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

Image  | Name  | First described  | Equation  | Comment

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| circle |  | [math]\displaystyle{ r= k }[/math] | The trivial spiral

| Archimedean spiral (also arithmetic spiral) | -300 c. 320 BC | [math]\displaystyle{ r=a+b \cdot \theta }[/math] |

| Fermat's spiral (also parabolic spiral) | 1636[1] | [math]\displaystyle{ r^{2} = a^{2} \cdot \theta }[/math] |  |

| Euler spiral (also Cornu spiral or polynomial spiral) | 1696[2] | [math]\displaystyle{ x(t) = \operatorname{C}(t),\, }[/math][math]\displaystyle{ y(t) = \operatorname{S}(t) }[/math] | using Fresnel integrals[3] |

| hyperbolic spiral (also reciprocal spiral) | 1704 | [math]\displaystyle{ r = \frac{a}{\theta} }[/math] |

| lituus | 1722 | [math]\displaystyle{ r^{2} \cdot \theta = k }[/math] |

| logarithmic spiral (also known as equiangular spiral) | 1638[4] | [math]\displaystyle{ r=a\cdot e^{b \cdot \theta} }[/math] | 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 |  | [math]\displaystyle{ r = \varphi^{\frac{2 \cdot \theta}{\pi}}\, }[/math] | 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 | [math]\displaystyle{ x(t)=r(\cos(t+a)+t\sin(t+a)), }[/math]

[math]\displaystyle{ y(t)=r(\sin(t+a)-t \cos(t+a)) }[/math]

| involutes of a circle appear like Archimedean spirals

| helix |  | [math]\displaystyle{ r(t) = 1,\, }[/math] [math]\displaystyle{ \theta(t) = t,\, }[/math] [math]\displaystyle{ z(t) = t }[/math] | a 3-dimensional spiral

| Rhumb line (also loxodrome) |  |  | type of spiral drawn on a sphere

| Cotes's spiral | 1722 | [math]\displaystyle{ \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} }[/math] | Solution to the two-body problem for an inverse-cube central force

| Poinsot's spirals |  | [math]\displaystyle{ r = a \cdot \operatorname{csch}(n \cdot \theta),\, }[/math]
[math]\displaystyle{ r = a \cdot \operatorname{sech}(n \cdot \theta) }[/math] |

| Nielsen's spiral | 1993[5] | [math]\displaystyle{ x(t) = \operatorname{ci}(t),\, }[/math]
[math]\displaystyle{ y(t) = \operatorname{si}(t) }[/math] | A variation of Euler spiral, using sine integral and cosine integrals

| Polygonal spiral |  |  | special case approximation of logarithmic spiral

| Fraser's Spiral | 1908 |  | Optical illusion based on spirals

| Conchospiral |  | [math]\displaystyle{ r = \mu^{t} \cdot a,\, }[/math][math]\displaystyle{ \theta = t,\, }[/math][math]\displaystyle{ z = \mu^{t} \cdot c }[/math] | three-dimensional spiral on the surface of a cone.

| Calkin–Wilf spiral |  |  |

| Ulam spiral (also prime spiral) | 1963 |  |

| Sack's spiral | 1994 |  | variant of Ulam spiral and Archimedean spiral.
| Seiffert's spiral | 2000[6] | [math]\displaystyle{ r = \operatorname{sn}(s, k),\, }[/math][math]\displaystyle{ \theta = k \cdot s }[/math][math]\displaystyle{ z = \operatorname{cn}(s, k) }[/math] | spiral curve on the surface of a sphere

using the Jacobi elliptic functions[7]

| Tractrix spiral | 1704[8] | [math]\displaystyle{ \begin{cases} r = A \cos(t) \\\ \theta = \tan(t ) -t\end{cases} }[/math] |  |
| Pappus spiral | 1779 | [math]\displaystyle{ \begin{cases} r=a \theta \\\ \psi = k \end{cases} }[/math] | 3D conical spiral studied by Pappus and Pascal[9]

| doppler spiral |  | [math]\displaystyle{ x = a \cdot ( t \cdot \cos(t) + k \cdot t),\, }[/math][math]\displaystyle{ y = a \cdot t \cdot \sin(t) }[/math] | 2D projection of Pappus spiral[10]

| Atzema spiral |  | [math]\displaystyle{ x = \frac{\sin(t)}{t} - 2 \cdot \cos(t) - t \cdot \sin(t),\, }[/math][math]\displaystyle{ y = -\frac{\cos(t)}{t} - 2 \cdot \sin(t) + t \cdot \cos(t) }[/math] | The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.[11]

| Atomic spiral | 2002 | [math]\displaystyle{ r = \frac{\theta}{\theta - a} }[/math] | This spiral has two asymptotes; one is the circle of radius 1 and the other is the line [math]\displaystyle{ \theta=a }[/math][12]

| Galactic spiral | 2019 | [math]\displaystyle{ \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} }[/math] | The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases:[math]\displaystyle{ \rho \lt 1, \rho = 1, \rho \gt 1 }[/math], the spiral patterns are decided by the behavior of the parameter [math]\displaystyle{ \rho }[/math]. For [math]\displaystyle{ \rho \lt 1 }[/math], spiral-ring pattern; [math]\displaystyle{ \rho = 1, }[/math] regular spiral; [math]\displaystyle{ \rho \gt 1, }[/math] 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 ([math]\displaystyle{ -\theta }[/math]) for plotting.[13] |

## See also

* Catherine wheel (firework)
* List of spiral galaxies
* Parker spiral
* Spirangle
* Spirograph

## References

1. ↑ "Fermat spiral - Encyclopedia of Mathematics". https://www.encyclopediaofmath.org/index.php/Fermat_spiral. Retrieved 18 February 2019.
2. ↑ Weisstein, Eric W.. "Cornu Spiral" (in en). https://mathworld.wolfram.com/.
3. ↑ Weisstein, Eric W.. "Fresnel Integrals" (in en). https://mathworld.wolfram.com/.
4. ↑ Weisstein, Eric W.. "Logarithmic Spiral" (in en). Wolfram Research, Inc.. http://mathworld.wolfram.com/LogarithmicSpiral.html. Retrieved 18 February 2019.
5. ↑ Weisstein, Eric W.. "Nielsen's Spiral" (in en). Wolfram Research, Inc.. http://mathworld.wolfram.com/NielsensSpiral.html. Retrieved 18 February 2019.
6. ↑ Weisstein, Eric W.. "Seiffert's Spherical Spiral" (in en). https://mathworld.wolfram.com/.
7. ↑ Weisstein, Eric W.. "Seiffert's Spherical Spiral" (in en). https://mathworld.wolfram.com/.
8. ↑ "Tractrix spiral". http://www.mathcurve.com/courbes2d.gb/spiraletractrice/spiraletractrice.shtml.
9. ↑ "Conical spiral of Pappus". https://www.mathcurve.com/courbes3d.gb/spiraleconic/pappus.shtml. Retrieved 28 February 2019.
10. ↑ "Doppler spiral". https://www.mathcurve.com/courbes2d.gb/doppler/doppler.htm. Retrieved 28 February 2019.
11. ↑ "Atzema spiral". http://www.2dcurves.com/spiral/spiralat.html. Retrieved 11 March 2019.
12. ↑ "atom-spiral". http://www.2dcurves.com/spiral/spiralas.html. Retrieved 11 March 2019.
13. ↑ Pan, Hongjun. "New spiral". https://arpgweb.com/pdf-files/ajams7(2)66-76.pdf. Retrieved 5 March 2021.

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Spirals|

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* On Spirals
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