Logarithmic spiral (pitch 10°)
A section of the Mandelbrot set following a logarithmic spiral

A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie").[1][2] More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.


In polar coordinates the logarithmic spiral can be written as[3]

with being the base of natural logarithms, and , being real constants.

In Cartesian coordinates

The logarithmic spiral with the polar equation

can be represented in Cartesian coordinates by
In the complex plane :

Spira mirabilis and Jacob Bernoulli

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.[4][5]


Definition of slope angle and sector
Animation showing the constant angle between an intersecting circle centred at the origin and a logarithmic spiral.

The logarithmic spiral has the following properties (see Spiral):

Examples for

Special cases and approximations

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (polar slope 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.

In nature

Further information: Patterns in nature § Spirals

An extratropical cyclone over Iceland shows an approximately logarithmic spiral pattern
The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy
Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral. The plotted spiral (dashed blue curve) is based on growth rate parameter , resulting in a pitch of .

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

In engineering applications

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.[16]
A logarithmic spiral antenna

See also


  1. ^ Albrecht Dürer (1525). Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen.
  2. ^ Hammer, Øyvind (2016). "Dürer's dirty secret". The Perfect Shape: Spiral Stories. Springer International Publishing. pp. 173–175. doi:10.1007/978-3-319-47373-4_41.
  3. ^ Priya Hemenway (2005). Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co. ISBN 978-1-4027-3522-6.
  4. ^ Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books. ISBN 978-0-7679-0815-3.
  5. ^ Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes". p. 206.
  6. ^ Carl Benjamin Boyer (1949). The history of the calculus and its conceptual development. Courier Dover Publications. p. 133. ISBN 978-0-486-60509-8.
  7. ^ Chin, Gilbert J. (8 December 2000), "Organismal Biology: Flying Along a Logarithmic Spiral", Science, 290 (5498): 1857, doi:10.1126/science.290.5498.1857c, S2CID 180484583
  8. ^ John Himmelman (2002). Discovering Moths: Nighttime Jewels in Your Own Backyard. Down East Enterprise Inc. p. 63. ISBN 978-0-89272-528-1.
  9. ^ G. Bertin and C. C. Lin (1996). Spiral structure in galaxies: a density wave theory. MIT Press. p. 78. ISBN 978-0-262-02396-2.
  10. ^ David J. Darling (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley and Sons. p. 188. ISBN 978-0-471-27047-8.
  11. ^ Savchenko, S. S.; Reshetnikov, V. P. (September 2013). "Pitch angle variations in spiral galaxies". Monthly Notices of the Royal Astronomical Society. 436 (2): 1074–1083. doi:10.1093/mnras/stt1627.
  12. ^ 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.
  13. ^ Andrew Gray (1901). Treatise on physics, Volume 1. Churchill. pp. 356–357.
  14. ^ Michael Cortie (1992). "The form, function, and synthesis of the molluscan shell". In István Hargittai and Clifford A. Pickover (ed.). Spiral symmetry. World Scientific. p. 370. ISBN 978-981-02-0615-4.
  15. ^ Allan Thomas Williams and Anton Micallef (2009). Beach management: principles and practice. Earthscan. p. 14. ISBN 978-1-84407-435-8.
  16. ^ "kerf-canceling mechanisms". hpi.de. Retrieved 2020-12-26.
  17. ^ Mayes, P.E. (1992). "Frequency-independent antennas and broad-band derivatives thereof". Proceedings of the IEEE. 80 (1): 103–112. Bibcode:1992IEEEP..80..103M. doi:10.1109/5.119570.
  18. ^ Roumen, Thijs; Apel, Ingo; Shigeyama, Jotaro; Muhammad, Abdullah; Baudisch, Patrick (2020-10-20). "Kerf-Canceling Mechanisms". Proceedings of the 33rd Annual ACM Symposium on User Interface Software and Technology. Virtual Event USA: ACM. pp. 293–303. doi:10.1145/3379337.3415895. ISBN 978-1-4503-7514-6. S2CID 222805227.
  19. ^ Jiang, Jianfeng; Luo, Qingsheng; Wang, Liting; Qiao, Lijun; Li, Minghao (2020). "Review on logarithmic spiral bevel gear". Journal of the Brazilian Society of Mechanical Sciences and Engineering. 42 (8): 400. doi:10.1007/s40430-020-02488-y. ISSN 1678-5878.