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	c. 320 BC	${\displaystyle r=a+b\cdot \theta }$
	Euler spiral			also called Cornu spiral or polynomial spiral
	Fermat's spiral (also parabolic spiral)	1636[1]	${\displaystyle r^{2}=a^{2}\theta }$
	hyperbolic spiral	1704	${\displaystyle r=a/\theta }$	also reciprocal spiral
	lituus	1722	${\displaystyle r^{2}\theta =k}$
	logarithmic spiral	1638[2]	${\displaystyle r=a\cdot e^{b\theta }}$	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		${\displaystyle r=\varphi ^{\theta {\frac {2}{\pi }}}\,}$	special case of the logarithmic spiral
	Spiral of Theodorus (also Pythagorean spiral)			an polygonal spiral composed of contiguous right triangles, that approximates the Archimedean spiral
	involute	1673
	helix		${\displaystyle r(t)=1,\,}$ ${\displaystyle \theta (t)=t,\,}$ ${\displaystyle h(t)=t.\,}$	a 3-dimensional spiral
	Rhumb line (also loxodrome)			type of spiral drawn on a sphere
	Cotes's spiral	1722
	Poinsot's spirals		${\displaystyle r=a\operatorname {csch} (n\theta ),\,}$ ${\displaystyle r=a\operatorname {sech} (n\theta )}$
	Nielsen's spiral	1993[3]	${\displaystyle x(t)=\operatorname {ci} (t),\,}$ ${\displaystyle y(t)=\operatorname {si} (t)}$	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		${\displaystyle {\begin{cases}r=\mu ^{t}a\\\theta =t\\z=\mu ^{t}c\end{cases}}}$	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			spiral curve on the surface of a sphere
	Tractrix spiral	1704[4]	${\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}}$
	Pappus spiral	1779	${\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}}$	3D conical spiral studied by Pappus and Pascal[5]
	doppler spiral		${\displaystyle {\begin{cases}x=a(t\cos(t)+kt)\\y=at\sin(t)\end{cases}}}$	2D projection of Pappus spiral[6]
	Atzema spiral		${\displaystyle {\begin{cases}x=\sin(t)/t-2\cos(t)-t\sin(t)\\y=-\cos(t)/t-2\sin(t)+t\cos(t)\end{cases}}}$	The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.[7]
	Atomic spiral	2002	${\displaystyle r=\theta /(\theta -a)}$	This spiral has two asymptotes; one is the circle of radius 1 and the other is the line ${\displaystyle \theta =a}$[8]

See also

• 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. "Logarithmic Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
3. Weisstein, Eric W. "Nielsen's Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
4. "Tractrix spiral". www.mathcurve.com. Retrieved 2019-02-23.
5. "Conical spiral of Pappus". www.mathcurve.com. Retrieved 28 February 2019.
6. "Doppler spiral". www.mathcurve.com. Retrieved 28 February 2019.
7. "Atzema spiral". www.2dcurves.com. Retrieved 11 March 2019.
8. "atom-spiral". www.2dcurves.com. Retrieved 11 March 2019.