List of spirals

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

Image	Name	First described	Equation	Comment
	circle		${\displaystyle r=k}$	The trivial spiral
	Archimedean spiral (also arithmetic spiral)	c. 320 BC	${\displaystyle r=a+b\cdot \theta }$
	Fermat's spiral (also parabolic spiral)	1636 [1]	${\displaystyle r^{2}=a^{2}\cdot \theta }$
	Euler spiral (also Cornu spiral or polynomial spiral)	1696 [2]	${\displaystyle x(t)=\operatorname {C} (t),\,}$${\displaystyle y(t)=\operatorname {S} (t)}$	using Fresnel integrals [3]
	hyperbolic spiral (also reciprocal spiral)	1704	${\displaystyle r={\frac {a}{\theta }}}$
	lituus	1722	${\displaystyle r^{2}\cdot \theta =k}$
	logarithmic spiral (also known as equiangular spiral)	1638 [4]	${\displaystyle r=a\cdot e^{b\cdot \theta }}$	Approximations of this are found in nature
	Fibonacci spiral		circular arcs connecting the opposite corners of squares in the Fibonacci tiling	approximation of the golden spiral
	golden spiral		${\displaystyle r=\varphi ^{\frac {2\cdot \theta }{\pi }}\,}$	special case of the logarithmic spiral
	Spiral of Theodorus (also known as Pythagorean spiral)	c. 500 BC	contiguous right triangles composed of one leg with unit length and the other leg being the hypotenuse of the prior triangle	approximates the Archimedean spiral
	involute	1673	${\displaystyle x(t)=r(\cos(t+a)+t\sin(t+a)),}$ ${\displaystyle y(t)=r(\sin(t+a)-t\cos(t+a))}$	involutes of a circle appear like Archimedean spirals
	helix		${\displaystyle r(t)=1,\,}$${\displaystyle \theta (t)=t,\,}$${\displaystyle z(t)=t}$	a 3-dimensional spiral
	Rhumb line (also loxodrome)			type of spiral drawn on a sphere
	Cotes's spiral	1722	${\displaystyle {\frac {1}{r}}={\begin{cases}A\cosh(k\theta +\varepsilon )\\A\exp(k\theta +\varepsilon )\\A\sinh(k\theta +\varepsilon )\\A(k\theta +\varepsilon )\\A\cos(k\theta +\varepsilon )\\\end{cases}}}$	Solution to the two-body problem for an inverse-cube central force
	Poinsot's spirals		${\displaystyle r=a\cdot \operatorname {csch} (n\cdot \theta ),\,}$ ${\displaystyle r=a\cdot \operatorname {sech} (n\cdot \theta )}$
	Nielsen's spiral	1993 [5]	${\displaystyle x(t)=\operatorname {ci} (t),\,}$ ${\displaystyle y(t)=\operatorname {si} (t)}$	A variation of Euler spiral, using sine integral and cosine integrals
	Polygonal spiral			special case approximation of logarithmic spiral
	Fraser's Spiral	1908		Optical illusion based on spirals
	Conchospiral		${\displaystyle r=\mu ^{t}\cdot a,\,}$${\displaystyle \theta =t,\,}$${\displaystyle z=\mu ^{t}\cdot c}$	three-dimensional spiral on the surface of a cone.
	Calkin–Wilf spiral
	Ulam spiral (also prime spiral)	1963
	Sack's spiral	1994		variant of Ulam spiral and Archimedean spiral.
	Seiffert's spiral	2000 [6]	${\displaystyle r=\operatorname {sn} (s,k),\,}$${\displaystyle \theta =k\cdot s}$${\displaystyle z=\operatorname {cn} (s,k)}$	spiral curve on the surface of a sphere using the Jacobi elliptic functions [7]
	Tractrix spiral	1704 [8]	${\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}}$
	Pappus spiral	1779	${\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}}$	3D conical spiral studied by Pappus and Pascal [9]
	doppler spiral		${\displaystyle x=a\cdot (t\cdot \cos(t)+k\cdot t),\,}$${\displaystyle y=a\cdot t\cdot \sin(t)}$	2D projection of Pappus spiral [10]
	Atzema spiral		${\displaystyle x={\frac {\sin(t)}{t}}-2\cdot \cos(t)-t\cdot \sin(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. [11]
	Atomic spiral	2002	${\displaystyle r={\frac {\theta }{\theta -a}}}$	This spiral has two asymptotes; one is the circle of radius 1 and the other is the line ${\displaystyle \theta =a}$ [12]
	Galactic spiral	2019	${\displaystyle {\begin{cases}dx=R\cdot {\frac {y}{\sqrt {x^{2}+y^{2}}}}d\theta \\dy=R\cdot \left[\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2}}}}\right]d\theta \end{cases}}{\begin{cases}x=\sum dx\\\\\\y=\sum dy+R\end{cases}}}$	The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases:${\displaystyle \rho <1,\rho =1,\rho >1}$, the spiral patterns are decided by the behavior of the parameter ${\displaystyle \rho }$. For ${\displaystyle \rho <1}$, spiral-ring pattern; ${\displaystyle \rho =1,}$ regular spiral; ${\displaystyle \rho >1,}$ 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 (${\displaystyle -\theta }$) for plotting. [13]

See also

• Mathematics portal

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

References

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