List of spirals

This list is incomplete; you can help by . (February 2019)

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		${\displaystyle r=\varphi ^{\frac {2\theta }{\pi }}\,}$	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
				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]	${\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}}$
		1779	${\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}}$	3D conical spiral studied by Pappus and Pascal[5]
			${\displaystyle {\begin{cases}x=a(t\cos(t)+kt)\\y=at\sin(t)\end{cases}}}$	2D projection of Pappus spiral[6]
			${\displaystyle {\begin{cases}x=\sin(t)/t-2\cos(t)-t\sin(t)\\y=-\cos(t)/t-2\sin(t)+t\cos(t)\end{cases}}}$	The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.[7]
		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]
		2019	${\displaystyle {\begin{cases}dx=R*{\frac {y}{\sqrt {x^{2}+y^{2}}}}d\theta \\dy=R*{\Bigl [}\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2}}}}{\Bigr ]}d\theta \end{cases}}{\begin{cases}x=\sum dx\\\\\\y=\sum dy+R\end{cases}}}$	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. Please check the references for the detail[9]

See also[]

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

References[]