List of spirals information

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

Name	First described	Equation	Comment
circle		$r=k$	The trivial spiral
Archimedean spiral (also arithmetic spiral)	c. 320 BC	$r=a+b\cdot \theta$
Fermat's spiral (also parabolic spiral)	1636^[1]	$r^{2}=a^{2}\cdot \theta$
Euler spiral (also Cornu spiral or polynomial spiral)	1696^[2]	$x(t)=\operatorname {C} (t),\,$ $y(t)=\operatorname {S} (t)$	using Fresnel integrals^[3]
hyperbolic spiral (also reciprocal spiral)	1704	$r={\frac {a}{\theta }}$
lituus	1722	$r^{2}\cdot \theta =k$
logarithmic spiral (also known as equiangular spiral)	1638^[4]	$r=a\cdot e^{b\cdot \theta }$	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		$r=\varphi ^{\frac {2\cdot \theta }{\pi }}\,$	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	$x(t)=r(\cos(t+a)+t\sin(t+a)),$ $y(t)=r(\sin(t+a)-t\cos(t+a))$	involutes of a circle appear like Archimedean spirals
helix		$r(t)=1,\,$ $\theta (t)=t,\,$ $z(t)=t$	a 3-dimensional spiral
Rhumb line (also loxodrome)			type of spiral drawn on a sphere
Cotes's spiral	1722	${\frac {1}{r}}={\begin{cases}A\cosh(k\theta +\varepsilon )\\A\exp(k\theta +\varepsilon )\\A\sinh(k\theta +\varepsilon )\\A(k\theta +\varepsilon )\\A\cos(k\theta +\varepsilon )\\\end{cases}}$	Solution to the two-body problem for an inverse-cube central force
Poinsot's spirals		$r=a\cdot \operatorname {csch} (n\cdot \theta ),\,$ $r=a\cdot \operatorname {sech} (n\cdot \theta )$
Nielsen's spiral	1993^[5]	$x(t)=\operatorname {ci} (t),\,$ $y(t)=\operatorname {si} (t)$	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		$r=\mu ^{t}\cdot a,\,$ $\theta =t,\,$ $z=\mu ^{t}\cdot c$	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]	$r=\operatorname {sn} (s,k),\,$ $\theta =k\cdot s$ $z=\operatorname {cn} (s,k)$	spiral curve on the surface of a sphere using the Jacobi elliptic functions^[7]
Tractrix spiral	1704^[8]	${\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}$
Pappus spiral	1779	${\begin{cases}r=a\theta \\\psi =k\end{cases}}$	3D conical spiral studied by Pappus and Pascal^[9]
doppler spiral		$x=a\cdot (t\cdot \cos(t)+k\cdot t),\,$ $y=a\cdot t\cdot \sin(t)$	2D projection of Pappus spiral^[10]
Atzema spiral		$x={\frac {\sin(t)}{t}}-2\cdot \cos(t)-t\cdot \sin(t),\,$ $y=-{\frac {\cos(t)}{t}}-2\cdot \sin(t)+t\cdot \cos(t)$	The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.^[11]
Atomic spiral	2002	$r={\frac {\theta }{\theta -a}}$	This spiral has two asymptotes; one is the circle of radius 1 and the other is the line $\theta =a$ ^[12]
Galactic spiral	2019	${\begin{cases}dx=R\cdot {\frac {y}{\sqrt {x^{2}+y^{2}}}}d\theta \\dy=R\cdot \left[\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2}}}}\right]d\theta \end{cases}}{\begin{cases}x=\sum dx\\\\\\y=\sum dy+R\end{cases}}$	The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases: $\rho <1,\rho =1,\rho >1$ , the spiral patterns are decided by the behavior of the parameter $\rho$ . For $\rho <1$ , spiral-ring pattern; $\rho =1,$ regular spiral; $\rho >1,$ 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 ( $-\theta$ ) for plotting.^[13]

^ Cite error: The named reference Fermat was invoked but never defined (see the help page).
^ Weisstein, Eric W. "Cornu Spiral". mathworld.wolfram.com. Retrieved 2023-11-22.
^ Weisstein, Eric W. "Fresnel Integrals". mathworld.wolfram.com. Retrieved 2023-01-31.
^ Cite error: The named reference Descartes was invoked but never defined (see the help page).
^ Cite error: The named reference Nelsen was invoked but never defined (see the help page).
^ Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com. Retrieved 2023-01-31.
^ Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com. Retrieved 2023-01-31.
^ Cite error: The named reference tractrix was invoked but never defined (see the help page).
^ Cite error: The named reference Pappus was invoked but never defined (see the help page).
^ Cite error: The named reference doppler was invoked but never defined (see the help page).
^ Cite error: The named reference Atzema was invoked but never defined (see the help page).
^ Cite error: The named reference atomic was invoked but never defined (see the help page).
^ Cite error: The named reference galactic-1 was invoked but never defined (see the help page).

[Fermat-1] Cite error: The named reference Fermat was invoked but never defined (see the help page).

[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.

[Descartes-4] Cite error: The named reference Descartes was invoked but never defined (see the help page).

[Nelsen-5] Cite error: The named reference Nelsen was invoked but never defined (see the help page).

[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.

[tractrix-8] Cite error: The named reference tractrix was invoked but never defined (see the help page).

[Pappus-9] Cite error: The named reference Pappus was invoked but never defined (see the help page).

[doppler-10] Cite error: The named reference doppler was invoked but never defined (see the help page).

[Atzema-11] Cite error: The named reference Atzema was invoked but never defined (see the help page).

[atomic-12] Cite error: The named reference atomic was invoked but never defined (see the help page).

[galactic-1-13] Cite error: The named reference galactic-1 was invoked but never defined (see the help page).

List of spirals information

and 25 Related for: List of spirals information

List of spirals

List of Spiral episodes

Golden spiral

Logarithmic spiral

Magellanic spiral

Archimedean spiral

List of spiral galaxies

List of Spiral characters

Spiral of Theodorus

List of curves

Euler spiral

Bethungra Spiral

Spiral galaxy

Pitch angle of a spiral

Spiral bridge

Lists of astronomical objects

List of spiral DRAGNs

Spirals of Silence

The Downward Spiral

Flocculent spiral galaxy

List of galaxies

List of ancient spiral stairs

Spiral of silence

Lists of galaxies

Spiral Dynamics