List of spirals

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

Image  | Name  | First described  | Equation  | Comment

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| Circle |  | r=k | The trivial spiral
| Archimedean spiral (also arithmetic spiral)  | -300 c. 320 BC

| r=a+b⋅θ |
| Fermat's spiral (also parabolic spiral)  | 1636[1] | r2=a2⋅θ | Encloses equal area per turn
| Doyle spiral | 1980—1990[2] |  | circle packing, using circles of structured radii
| Euler spiral (also Cornu spiral or polynomial spiral)  | 1696[3] | x(t)=C⁡(t),y(t)=S⁡(t) | Using Fresnel integrals[4]
| Hyperbolic spiral (also reciprocal spiral)  | 1704  | r=aθ |
| Lituus | 1722  | r2⋅θ=k |
| Logarithmic spiral (also known as equiangular spiral)  | 1638[5] | r=a⋅eb⋅θ | Constant pitch angle. 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=φ2⋅θπ | Special case of the logarithmic spiral
| Spiral of Theodorus (also known as Pythagorean spiral)  | -500 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)+tsin⁡(t+a)),

y(t)=r(sin⁡(t+a)−tcos⁡(t+a))

| Involutes of a circle appear like Archimedean spirals
| Helix |  | r(t)=1, θ(t)=t, z(t)=t | A three-dimensional spiral
| Rhumb line (also loxodrome)  |  |  | Type of spiral drawn on a sphere
| Cotes's spiral | 1722  | 1r={Acosh⁡(kθ+ε)Aexp⁡(kθ+ε)Asinh⁡(kθ+ε)A(kθ+ε)Acos⁡(kθ+ε) | Solution to the two-body problem for an inverse-cube central force
| Poinsot's spirals |  | r=a⋅csch⁡(n⋅θ),
r=a⋅sech⁡(n⋅θ) |
| Nielsen's spiral  | 1993[6] | x(t)=ci⁡(t),
y(t)=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=μt⋅a,θ=t,z=μt⋅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[7] | r=sn⁡(s,k),θ=k⋅sz=cn⁡(s,k) | Spiral curve on the surface of a sphere using the Jacobi elliptic functions[8]
| Tractrix spiral  | 1704[9] | {r=Acos⁡(t)θ=tan⁡(t)−t |
| Pappus spiral  | 1779  | {r=aθψ=k | 3D conical spiral studied by Pappus and Pascal[10]
| Doppler spiral  |  | x=a⋅(t⋅cos⁡(t)+k⋅t),y=a⋅t⋅sin⁡(t) | 2D projection of Pappus spiral[11]
| Atzema spiral  |  | x=sin⁡(t)t−2⋅cos⁡(t)−t⋅sin⁡(t),y=−cos⁡(t)t−2⋅sin⁡(t)+t⋅cos⁡(t) | The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.[12]
| Atomic spiral  | 2002  | r=θθ−a | This spiral has two asymptotes; one is the circle of radius 1 and the other is the line θ=a[13]
| Galactic spiral  | 2019  | {dx=R⋅yx2+y2dθdy=R⋅[ρ(θ)−xx2+y2]dθ{x=∑dxy=∑dy+R | The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases:ρ<1,ρ=1,ρ>1, the spiral patterns are decided by the behavior of the parameter ρ. For ρ<1, spiral-ring pattern; ρ=1, regular spiral; ρ>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 (−θ) for plotting.[14]Template:Pred

## 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. ↑ Carter, Ithiel; Rodin, Burt (December 1992). "An Inverse Problem for Circle Packing and Conformal Mapping". Transactions of the American Mathematical Society 334 (2): 861. doi:10.2307/2154486.
3. ↑ Weisstein, Eric W.. "Cornu Spiral" (in en). https://mathworld.wolfram.com/.
4. ↑ Weisstein, Eric W.. "Fresnel Integrals" (in en). https://mathworld.wolfram.com/.
5. ↑ Weisstein, Eric W.. "Logarithmic Spiral" (in en). Wolfram Research, Inc.. http://mathworld.wolfram.com/LogarithmicSpiral.html. Retrieved 18 February 2019.
6. ↑ Weisstein, Eric W.. "Nielsen's Spiral" (in en). Wolfram Research, Inc.. http://mathworld.wolfram.com/NielsensSpiral.html. Retrieved 18 February 2019.
7. ↑ Weisstein, Eric W.. "Seiffert's Spherical Spiral" (in en). https://mathworld.wolfram.com/.
8. ↑ Weisstein, Eric W.. "Seiffert's Spherical Spiral" (in en). https://mathworld.wolfram.com/.
9. ↑ "Tractrix spiral". http://www.mathcurve.com/courbes2d.gb/spiraletractrice/spiraletractrice.shtml.
10. ↑ "Conical spiral of Pappus". https://www.mathcurve.com/courbes3d.gb/spiraleconic/pappus.shtml. Retrieved 28 February 2019.
11. ↑ "Doppler spiral". https://www.mathcurve.com/courbes2d.gb/doppler/doppler.htm. Retrieved 28 February 2019.
12. ↑ "Atzema spiral". http://www.2dcurves.com/spiral/spiralat.html. Retrieved 11 March 2019.
13. ↑ "atom-spiral". http://www.2dcurves.com/spiral/spiralas.html. Retrieved 11 March 2019.
14. ↑ Pan, Hongjun. "New spiral". https://arpgweb.com/pdf-files/ajams7(2)66-76.pdf. Retrieved 5 March 2021.

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Spirals|

* Archimedean
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* Epispiral
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* Poinsot's
* Euler
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* Golden
* Involute
* List
* Logarithmic
* On Spirals
* Theodorus

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