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List of spirals

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

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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 {\displaystyle r=k} The trivial spiral
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 Archimedean spiral (also arithmetic spiral) c. 320 BC r = a + b ⋅ θ {\displaystyle r=a+b\cdot \theta }
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 Fermat's spiral (also parabolic spiral) 16361 r 2 = a 2 ⋅ θ {\displaystyle r^{2}=a^{2}\cdot \theta } Encloses equal area per turn
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 Doyle spiral 1980—19902 circle packing, using circles of structured radii
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 Euler spiral (also Cornu spiral or polynomial spiral) 16963 x ( t ) = C ⁡ ( t ) , {\displaystyle x(t)=\operatorname {C} (t),\,} y ( t ) = S ⁡ ( t ) {\displaystyle y(t)=\operatorname {S} (t)} Using Fresnel integrals4
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 Hyperbolic spiral (also reciprocal spiral) 1704 r = a θ {\displaystyle r={\frac {a}{\theta }}}
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 Lituus 1722 r 2 ⋅ θ = k {\displaystyle r^{2}\cdot \theta =k}
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 Logarithmic spiral (also known as equiangular spiral) 16385 r = a ⋅ e b ⋅ θ {\displaystyle r=a\cdot e^{b\cdot \theta }} Constant pitch angle. Approximations of this are found in nature
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 Fibonacci spiral Circular arcs connecting the opposite corners of squares in the Fibonacci tiling Approximation of the golden spiral
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 Golden spiral r = φ 2 ⋅ θ π {\displaystyle r=\varphi ^{\frac {2\cdot \theta }{\pi }}\,} Special case of the logarithmic spiral
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 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
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 Involute 1673 x ( t ) = r ( cos ⁡ ( t + a ) + t sin ⁡ ( t + a ) ) , {\displaystyle x(t)=r(\cos(t+a)+t\sin(t+a)),}

y ( t ) = r ( sin ⁡ ( t + a ) − t cos ⁡ ( t + a ) ) {\displaystyle y(t)=r(\sin(t+a)-t\cos(t+a))}

Involutes of a circle appear like Archimedean spirals
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 Helix r ( t ) = 1 , {\displaystyle r(t)=1,\,} θ ( t ) = t , {\displaystyle \theta (t)=t,\,} z ( t ) = t {\displaystyle z(t)=t} A three-dimensional spiral
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 Rhumb line (also loxodrome) Type of spiral drawn on a sphere
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 Cotes's spiral 1722 1 r = { A cosh ⁡ ( k θ + ε ) A exp ⁡ ( k θ + ε ) A sinh ⁡ ( k θ + ε ) A ( k θ + ε ) A cos ⁡ ( k θ + ε ) {\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}}} Solution to the two-body problem for an inverse-cube central force
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 Poinsot's spirals r = a ⋅ csch ⁡ ( n ⋅ θ ) , {\displaystyle r=a\cdot \operatorname {csch} (n\cdot \theta ),\,}
r = a ⋅ sech ⁡ ( n ⋅ θ ) {\displaystyle r=a\cdot \operatorname {sech} (n\cdot \theta )}
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 Nielsen's spiral 19936 x ( t ) = ci ⁡ ( t ) , {\displaystyle x(t)=\operatorname {ci} (t),\,}
y ( t ) = si ⁡ ( t ) {\displaystyle y(t)=\operatorname {si} (t)} 		A variation of Euler spiral, using sine integral and cosine integrals
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 Polygonal spiral Special case approximation of arithmetic or logarithmic spiral
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 Fraser's Spiral 1908 Optical illusion based on spirals
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 Conchospiral r = μ t ⋅ a , {\displaystyle r=\mu ^{t}\cdot a,\,} θ = t , {\displaystyle \theta =t,\,} z = μ t ⋅ c {\displaystyle z=\mu ^{t}\cdot c} A three-dimensional spiral on the surface of a cone.
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 Calkin–Wilf spiral
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 Ulam spiral (also prime spiral) 1963
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 Sacks spiral 1994 Variant of Ulam spiral and Archimedean spiral.
Seiffert's spiral 20007 r = sn ⁡ ( s , k ) , {\displaystyle r=\operatorname {sn} (s,k),\,} θ = k ⋅ s {\displaystyle \theta =k\cdot s} z = cn ⁡ ( s , k ) {\displaystyle z=\operatorname {cn} (s,k)} Spiral curve on the surface of a sphere using the Jacobi elliptic functions8
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 Tractrix spiral 17049 { r = A cos ⁡ ( t ) θ = tan ⁡ ( t ) − t {\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}}
Pappus spiral 1779 { r = a θ ψ = k {\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}} 3D conical spiral studied by Pappus and Pascal10
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 Doppler spiral x = a ⋅ ( t ⋅ cos ⁡ ( t ) + k ⋅ t ) , {\displaystyle x=a\cdot (t\cdot \cos(t)+k\cdot t),\,} y = a ⋅ t ⋅ sin ⁡ ( t ) {\displaystyle y=a\cdot t\cdot \sin(t)} 2D projection of Pappus spiral11
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 Atzema spiral x = sin ⁡ ( t ) t − 2 ⋅ cos ⁡ ( t ) − t ⋅ sin ⁡ ( t ) , {\displaystyle x={\frac {\sin(t)}{t}}-2\cdot \cos(t)-t\cdot \sin(t),\,} y = − cos ⁡ ( t ) t − 2 ⋅ sin ⁡ ( t ) + t ⋅ cos ⁡ ( t ) {\displaystyle 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.12
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 Atomic spiral 2002 r = θ θ − a {\displaystyle r={\frac {\theta }{\theta -a}}} This spiral has two asymptotes; one is the circle of radius 1 and the other is the line θ = a {\displaystyle \theta =a} 13
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 Galactic spiral 2019 { d x = R ⋅ y x 2 + y 2 d θ d y = R ⋅ [ ρ ( θ ) − x x 2 + y 2 ] d θ { x = ∑ d x y = ∑ d y + R {\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}}} 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 {\displaystyle \rho <1,\rho =1,\rho >1} , the spiral patterns are decided by the behavior of the parameter ρ {\displaystyle \rho } . For ρ < 1 {\displaystyle \rho <1} , spiral-ring pattern; ρ = 1 , {\displaystyle \rho =1,} regular spiral; ρ > 1 , {\displaystyle \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 ( − θ {\displaystyle -\theta } ) for plotting.14
See also

See also

References

References

  1. "Fermat spiral - Encyclopedia of Mathematics". www.encyclopediaofmath.org. 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". mathworld.wolfram.com. Retrieved 2023-11-22.
  4. Weisstein, Eric W. "Fresnel Integrals". mathworld.wolfram.com. Retrieved 2023-01-31.
  5. Weisstein, Eric W. "Logarithmic Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
  6. Weisstein, Eric W. "Nielsen's Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
  7. Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com. Retrieved 2023-01-31.
  8. Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com. Retrieved 2023-01-31.
  9. "Tractrix spiral". www.mathcurve.com. Retrieved 2019-02-23.
  10. "Conical spiral of Pappus". www.mathcurve.com. Retrieved 28 February 2019.
  11. "Doppler spiral". www.mathcurve.com. Retrieved 28 February 2019.
  12. "Atzema spiral". www.2dcurves.com. Retrieved 11 March 2019.
  13. "atom-spiral". www.2dcurves.com. Retrieved 11 March 2019.
  14. Pan, Hongjun. "New spiral" (PDF). www.arpgweb.com. Retrieved 5 March 2021.