logarithmic spiral

In polar coordinates $(r,\; \backslash varphi)$ the logarithmic spiral can be written as{{cite book | title = Divine Proportion: Φ Phi in Art, Nature, and Science | author = Priya Hemenway | isbn = 978-1-4027-3522-6 | publisher = Sterling Publishing Co | year = 2005}}

$$r\; =\; ae^\{k\backslash varphi\},\backslash quad\; \backslash varphi\; \backslash in\; \backslash R,$$

or

$$\backslash varphi\; =\; \backslash frac\{1\}\{k\}\; \backslash ln\; \backslash frac\{r\}\{a\},$$

with $e$ being the base of natural logarithms, and $a\; >\; 0$, $k\backslash ne\; 0$ being real constants.

The logarithmic spiral with the polar equation

$$r\; =\; a\; e^\{k\backslash varphi\}$$

can be represented in Cartesian coordinates $(x=r\backslash cos\backslash varphi,\backslash ,\; y=r\backslash sin\backslash varphi)$ by

$$x\; =\; a\; e^\{k\backslash varphi\}\backslash cos\; \backslash varphi,\; \backslash qquad\; y\; =\; a\; e^\{k\backslash varphi\}\backslash sin\; \backslash varphi.$$

In the complex plane $(z=x+iy,\backslash ,\; e^\{i\backslash varphi\}=\backslash cos\backslash varphi\; +\; i\backslash sin\backslash varphi)$:

$$z=ae^\{(k+i)\backslash varphi\}.$$

Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Jacob Bernoulli wanted such a spiral engraved on his headstone along with the phrase "Eadem mutata resurgo" ("Although changed, I shall arise the same."), but, by error, an Archimedean spiral was placed there instead.{{cite book |last=Livio |first=Mario |year=2002 |title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number |publisher=Broadway Books |location=New York |isbn=978-0-7679-0815-3 |url-access=registration |url=https://archive.org/details/goldenratiostory00livi }}Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes". p. 206.

File:Spiral-log-st-se.svg

File:Logspiral.gif

The logarithmic spiral $r=a\; e^\{k\backslash varphi\}\; \backslash ;,\backslash ;\; k\backslash ne\; 0,$ has the following properties (see Spiral):

• Pitch angle: $\backslash tan\backslash alpha=k\backslash quad\; (\; \{\backslash color\{red\}\{\backslash text\{constant\; !\}\}\}\; )$ {{pb}} with pitch angle $\backslash alpha$ (see diagram and animation).{{pb}}(In case of $k=0$ angle $\backslash alpha$ would be 0 and the curve a circle with radius $a$.)
• Curvature: $\backslash kappa=\backslash frac\{1\}\{r\backslash sqrt\{1+k^2\}\}=\backslash frac\{\backslash cos\; \backslash alpha\}\{r\}$
• Arc length: $L(\backslash varphi\_1,\backslash varphi\_2)=\backslash frac\{\backslash sqrt\{k^2+1\}\}\{k\}\backslash big(r(\backslash varphi\_2)-r(\backslash varphi\_1)\backslash big)=\; \backslash frac\{r(\backslash varphi\_2)-r(\backslash varphi\_1)\}\{\backslash sin\; \backslash alpha\}${{pb}}Especially: $\backslash \; L(-\backslash infty,\backslash varphi\_2)=\backslash frac\{r(\backslash varphi\_2)\}\{\backslash sin\; \backslash alpha\}\backslash quad\; (\{\backslash color\{red\}\{\backslash text\{finite\; !\}\}\})\backslash ;$, if $k\; >\; 0$. {{pb}} This property was first realized by Evangelista Torricelli even before calculus had been invented.{{cite book | title = The history of the calculus and its conceptual development | author = Carl Benjamin Boyer | publisher = Courier Dover Publications | year = 1949 | isbn = 978-0-486-60509-8 | page = 133 | url = https://books.google.com/books?id=KLQSHUW8FnUC&pg=PA133 }}
• Sector area: $A=\backslash frac\{r(\backslash varphi\_2)^2-r(\backslash varphi\_1)^2\}\{4k\}$
• Inversion: Circle inversion ($r\backslash to\; 1/r$) maps the logarithmic spiral $r=a\; e^\{k\backslash varphi\}$ onto the logarithmic spiral $r=\backslash tfrac\{1\}\{a\}\; e^\{-k\backslash varphi\}\; \backslash ,\; .$

File:Spiral-log-a-1-5.svg

• Rotating, scaling: Rotating the spiral by angle $\backslash varphi\_0$ yields the spiral $r=ae^\{-k\backslash varphi\_0\}e^\{k\backslash varphi\}$, which is the original spiral uniformly scaled (at the origin) by $e^\{-k\backslash varphi\_0\}$. {{pb}}Scaling by $\backslash ;e^\{kn2\backslash pi\}\backslash ;\; ,\; n=\backslash pm\; 1,\backslash pm2,...,\backslash ;$ gives the same curve.
• Self-similarity: A result of the previous property: {{pb}}A scaled logarithmic spiral is congruent (by rotation) to the original curve. {{pb}}Example: The diagram shows spirals with slope angle $\backslash alpha=20^\backslash circ$ and $a=1,2,3,4,5$. Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles $-109^\backslash circ,-173^\backslash circ,-218^\backslash circ,-253^\backslash circ$ resp.. All spirals have no points in common (see property on complex exponential function).
• Relation to other curves: Logarithmic spirals are congruent to their own involutes, evolutes, and the pedal curves based on their centers.
• Complex exponential function: The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at $0$: $$z(t)=\backslash underbrace\{(kt+b)\backslash ;\; +it\}\_\{\backslash text\{line\}\}\backslash quad\; \backslash to\backslash quad\; e^\{z(t)\}=e^\{kt+b\}\backslash cdot\; e^\{it\}=\; \backslash underbrace\{e^b\; e^\{kt\}(\backslash cos\; t+i\backslash sin\; t)\}\_\{\backslash text\{log.\; spiral\}\}$$ The pitch angle $\backslash alpha$ of the logarithmic spiral is the angle between the line and the imaginary axis.

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.

{{further|Patterns in nature#Spirals}}

{{Multiple image|total_width=480

|image1=Low pressure system over Iceland.jpg|caption1=An extratropical cyclone over Iceland shows an approximately logarithmic spiral pattern

|image2=Messier51 sRGB.jpg|caption2=The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy}}

File:Nautilus Cutaway with Logarithmic Spiral.png shell showing the chambers arranged in an approximately logarithmic spiral. The plotted spiral (dashed blue curve) is based on growth rate parameter $b\; =\; 0.1759$, resulting in a pitch of $\backslash arctan\; b\; \backslash approx\; 10^\backslash circ$.]]

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:

• The approach of a hawk to its prey in classical pursuit, assuming the prey travels in a straight line. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.{{cite journal |first=Gilbert J. |last=Chin |date=8 December 2000 |title=Organismal Biology: Flying Along a Logarithmic Spiral |journal=Science |volume=290 |issue=5498 |page=1857 |doi=10.1126/science.290.5498.1857c|s2cid=180484583 }}
• The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the Sun (or Moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.

{{cite book

| title = Discovering Moths: Nighttime Jewels in Your Own Backyard

| author = John Himmelman

| publisher = Down East Enterprise Inc

| year = 2002

| isbn = 978-0-89272-528-1

| page = 63

| url = https://books.google.com/books?id=iGn6ohfKhbAC&pg=PA63

}} In the same token, a rhumb line approximates a logarithmic spiral close to a pole.

• The arms of spiral galaxies.

{{cite book

| title = Spiral structure in galaxies: a density wave theory

| author = G. Bertin and C. C. Lin

| publisher = MIT Press

| year = 1996

| isbn = 978-0-262-02396-2

| page = 78

| url = https://books.google.com/books?id=06yfwrdpTk4C&pg=PA78

}} The Milky Way galaxy has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees.

{{cite book

| title = The universal book of mathematics: from Abracadabra to Zeno's paradoxes

| author = David J. Darling

| publisher = John Wiley and Sons

| year = 2004

| isbn = 978-0-471-27047-8

| page = 188

| url = https://books.google.com/books?id=nnpChqstvg0C&pg=PA188

}} However, although spiral galaxies have often been modeled as logarithmic spirals, Archimedean spirals, or hyperbolic spirals, their pitch angles vary with distance from the galactic center, unlike logarithmic spirals (for which this angle does not vary), and also at variance with the other mathematical spirals used to model them.{{cite journal

| last1 = Savchenko | first1 = S. S.

| last2 = Reshetnikov | first2 = V. P.

| date = September 2013

| doi = 10.1093/mnras/stt1627

| issue = 2

| journal = Monthly Notices of the Royal Astronomical Society

| pages = 1074–1083

| title = Pitch angle variations in spiral galaxies

| volume = 436| doi-access = free

| arxiv = 1309.4308

}}

• The nerves of the cornea (this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).C. Q. Yu CQ and M. I. Rosenblatt, "Transgenic corneal neurofluorescence in mice: a new model for in vivo investigation of nerve structure and regeneration,"

Invest Ophthalmol Vis Sci. 2007 Apr;48(4):1535-42.

• The bands of tropical cyclones, such as hurricanes.

{{cite book

| title = Treatise on physics, Volume 1

| author = Andrew Gray

| publisher = Churchill

| year = 1901

| pages = [https://archive.org/details/atreatiseonphys02graygoog/page/n386 356]–357

| url = https://archive.org/details/atreatiseonphys02graygoog

}}

• Many biological structures including the shells of mollusks.

{{cite book

| title = Spiral symmetry

| chapter = The form, function, and synthesis of the molluscan shell

| author = Michael Cortie

| editor = István Hargittai and Clifford A. Pickover

| publisher = World Scientific

| year = 1992

| isbn = 978-981-02-0615-4

| page = 370

| chapter-url = https://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA370

}} In these cases, the reason may be construction from expanding similar shapes, as is the case for polygonal figures.

• Logarithmic spiral beaches can form as the result of wave refraction and diffraction by the coast. Half Moon Bay (California) is an example of such a type of beach.

{{cite book

| title = Beach management: principles and practice

| author = Allan Thomas Williams and Anton Micallef

| publisher = Earthscan

| year = 2009

| isbn = 978-1-84407-435-8

| page = 14

| url = https://books.google.com/books?id=z_vKEMeJXKYC&pg=PA14

}}

{{clear}}

{{multiple image|total_width=400

|image1=A Kerf Canceling Mechanism (bearing).gif|caption1=A kerf-canceling mechanism leverages the self similarity of the logarithmic spiral to lock in place under rotation, independent of the kerf of the cut.{{Cite web|title=kerf-canceling mechanisms |url=https://hpi.de/baudisch/projects/kerf-canceling-mechanisms.html|access-date=2020-12-26 |website=hpi.de |language=en}}

|image2=ILA Berlin 2012 PD 128.JPG|caption2=A logarithmic spiral antenna}}

• Logarithmic spiral antennas are frequency-independent antennas, that is, antennas whose radiation pattern, impedance and polarization remain largely unmodified over a wide bandwidth.{{Cite journal|last=Mayes|first=P.E. |date=1992 |title=Frequency-independent antennas and broad-band derivatives thereof|url=https://ieeexplore.ieee.org/document/119570 |journal=Proceedings of the IEEE|volume=80|issue=1|pages=103–112|doi=10.1109/5.119570|bibcode=1992IEEEP..80..103M }}
• When manufacturing mechanisms by subtractive fabrication machines (such as laser cutters), there can be a loss of precision when the mechanism is fabricated on a different machine due to the difference of material removed (that is, the kerf) by each machine in the cutting process. To adjust for this variation of kerf, the self-similar property of the logarithmic spiral has been used to design a kerf cancelling mechanism for laser cutters.{{Cite book|last1=Roumen|first1=Thijs |last2=Apel|first2=Ingo|last3=Shigeyama|first3=Jotaro|last4=Muhammad|first4=Abdullah|last5=Baudisch|first5=Patrick |title=Proceedings of the 33rd Annual ACM Symposium on User Interface Software and Technology |chapter=Kerf-canceling mechanisms: Making laser-cut mechanisms operate across different laser cutters |date=2020-10-20 |chapter-url=https://dl.acm.org/doi/10.1145/3379337.3415895|language=en |location=Virtual Event USA|publisher=ACM|pages=293–303|doi=10.1145/3379337.3415895 |isbn=978-1-4503-7514-6|s2cid=222805227 }}
• Logarithmic spiral bevel gears are a type of spiral bevel gear whose gear tooth centerline is a logarithmic spiral. A logarithmic spiral has the advantage of providing equal angles between the tooth centerline and the radial lines, which gives the meshing transmission more stability.{{Cite journal|last1=Jiang|first1=Jianfeng |last2=Luo | first2=Qingsheng |last3=Wang|first3=Liting|last4=Qiao|first4=Lijun|last5=Li|first5=Minghao|date=2020|title=Review on logarithmic spiral bevel gear|journal=Journal of the Brazilian Society of Mechanical Sciences and Engineering|language=en|volume=42|issue=8|pages=400|doi=10.1007/s40430-020-02488-y|issn=1678-5878 | doi-access=}}

{{CSS image crop|Image=Camalot number 6.JPG|bSize=400|cWidth=300|oLeft=60|cHeight=200|oTop=40|Description=A spring-loaded camming device, with logarithmic spiral cam surfaces}}

• In rock climbing, spring-loaded camming devices are made from metal cams whose outer gripping surfaces are shaped as arcs of logarithmic spirals. When the device is inserted into a rock crack, the rotation of these cams expands their combined width to match the width of the crack, while maintaining a constant angle against the surface of the rock (relative to the center of the spiral, where force is applied). The pitch angle of the spiral is chosen to optimize the friction of the device against the rock.{{cite book | last = Todesco | first = Gian Marco | editor1-last = Emmer | editor1-first = Michele | editor2-last = Abate | editor2-first = Marco | contribution = Weird gears | doi = 10.1007/978-3-319-93949-0_16 | isbn = 9783319939490 | pages = 179–193 | publisher = Springer International Publishing | title = Imagine Math 6: Between Culture and Mathematics | year = 2018}}
• Soft robots based on the logarithmic spiral were designed for scalable and efficient 3D printing. Using cable-driven actuation, they mimic octopus-like movements for stable and versatile object manipulation.{{Cite journal |last=Wang |first=Zhanchi |last2=Freris |first2=Nikolaos M. |last3=Wei |first3=Xi |date=2024 |title=SpiRobs: Logarithmic spiral-shaped robots for versatile grasping across scales |url=https://linkinghub.elsevier.com/retrieve/pii/S2666998624006033 |journal=Device |language=en |pages=100646 |doi=10.1016/j.device.2024.100646|arxiv=2303.09861 }}

• Archimedean spiral
• Epispiral
• List of spirals
• Mice problem, a geometric problem asking for the path followed by mice chasing one another whose solution is a logarithmic spiral
• Tait–Kneser theorem

{{reflist}}

• {{mathworld|urlname=LogarithmicSpiral|title=Logarithmic Spiral}}
• Jim Wilson, [http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/related%20curves/related%20curves.html Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves], University of Georgia (1999)
• Alexander Bogomolny, [http://www.cut-the-knot.org/Curriculum/Geometry/Mirabilis.shtml Spira Mirabilis - Wonderful Spiral], at cut-the-knot

{{commons category|Logarithmic spirals}}

• [http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html Spira mirabilis] history and math
• {{APOD |date=25 September 2003 |title=Hurricane Isabel vs. the Whirlpool Galaxy}}
• {{APOD |date=17 May 2008 |title=Typhoon Rammasun vs. the Pinwheel Galaxy}}
• [http://SpiralZoom.com/ SpiralZoom.com], an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
• [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Logarithmic_spiral Online exploration using JSXGraph (JavaScript)]
• [https://www.youtube.com/watch?v=NdTVvWrD6r0&ab_channel=Math%2CPhysics%2CEngineering YouTube lecture on Zeno's mice problem and logarithmic spirals]

{{Patterns in nature}}

{{Spirals}}

Category:Spirals

Spiral

Spiral

Category:Plane curves