Image Name First described Equation Comment Circle r = k {\displaystyle r=k} The trivial spiral Archimedean spiral (also arithmetic spiral ) c. 320 BCr = a + b ⋅ θ {\displaystyle r=a+b\cdot \theta } Fermat's spiral (also parabolic spiral) 1636 [ 1] r 2 = a 2 ⋅ θ {\displaystyle r^{2}=a^{2}\cdot \theta } Euler spiral (also Cornu spiral or polynomial spiral) 1696 [ 2] x ( t ) = C ( t ) , {\displaystyle x(t)=\operatorname {C} (t),\,} y ( t ) = S ( t ) {\displaystyle y(t)=\operatorname {S} (t)} Using Fresnel integrals [ 3] Hyperbolic spiral (also reciprocal spiral ) 1704 r = a θ {\displaystyle r={\frac {a}{\theta }}} Lituus 1722 r 2 ⋅ θ = k {\displaystyle r^{2}\cdot \theta =k} Logarithmic spiral (also known as equiangular spiral ) 1638 [ 4] r = a ⋅ e b ⋅ θ {\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 r = φ 2 ⋅ θ π {\displaystyle r=\varphi ^{\frac {2\cdot \theta }{\pi }}\,} Special case of the logarithmic spiral Spiral of Theodorus (also known as Pythagorean spiral ) c. 500 BCContiguous 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 ) ) , {\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 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 Rhumb line (also loxodrome) Type of spiral drawn on a sphere 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 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 )} Nielsen's spiral 1993 [ 5] 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 Polygonal spiral Special case approximation of arithmetic or logarithmic spiral Fraser's Spiral 1908 Optical illusion based on spirals 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. Calkin–Wilf spiral Ulam spiral (also prime spiral) 1963 Sacks spiral 1994 Variant of Ulam spiral and Archimedean spiral. Seiffert's spiral 2000 [ 6] 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 functions [ 7] Tractrix spiral 1704 [ 8] { 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 Pascal [ 9] 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 spiral [ 10] 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. [ 11] 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} [ 12] 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} ρ {\displaystyle \rho } ρ < 1 {\displaystyle \rho <1} ρ = 1 , {\displaystyle \rho =1,} ρ > 1 , {\displaystyle \rho >1,} − θ {\displaystyle -\theta } [ 13] [ predatory publisher ] 
