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Archimedean, Logarithmic and Euler spirals − intriguing and ubiquitous patterns in nature

  • Danilo R. Diedrichs (a1)


Spirals are among the most intriguing of geometrical patterns, frequently encountered in the world around us at all scales, from the cosmic spiral arms of galaxies to the microscopic structure of the DNA molecule. For centuries, humans have studied these patterns, classifying them, giving them different names, and describing them mathematically. The most common definition of a two-dimensional spiral is a curve on the plane traced by a point that winds around a certain fixed point (called the spiral's pole), while monotonically approaching or receding from it, depending on the direction of motion.



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1. Mitchison, G. J., Phyllotaxis and the Fibonacci Series: an explanation is offered for the characteristic spiral leaf arrangement found in many plants, Science, New Series, 196 (4287) (1977) pp. 270275.
2. Michael Naylor, Golden, and π flowers: a spiral story, Mathematics Magazine, 75 (3) (2002) pp.163172.
3. Ivars Peterson, Golden blossoms, Pi flowers, Science News (2002)
4. Peterson, Ivars, Fermat's natural spirals, Science News (2005)
5. Vogel, Helmut, A better way to construct the sunflower head, Mathematical Biosciences, 44 (1979) pp. 179189.10.1016/0025-5564(79)90080-4
6. Tung, K. K., Topics in mathematical modeling, Princeton University Press (2007).10.1515/9781400884056
7. Okabe, Takuya., Biophysical optimality of the golden angle in phyllotaxis, Scientific Reports, 5:15358 (2015).10.1038/srep15358
8. Davis, Benjamin L. et al, Measurement of galactic logarithmic spiral arm pitch angle using two-dimensional fast Fourier transform decomposition. The Astrophysical Journal Supplement Series, 199(33) (2012) pp. 120.10.1088/0067-0049/199/2/33
9. Bremner, J. M., Properties of logarithmic spiral beaches with particular reference to Algoa Bay, in Sandy beaches as ecosystems: based on the Proceedings of the First International Symposium on Sandy Beaches, held in Port Elizabeth, South Africa, Springer Netherlands (1983) pp. 97113.10.1007/978-94-017-2938-3_6
10. Chapman, D. M., Zetaform or logarithmic spiral beach, Australian Geographer, 14(1) (1978) pp. 4445.10.1080/00049187808702733
11. LeBlond, Paul H., An explanation of the logarithmic spiral plan shape of headland-bay beaches. Journal of Sedimentary Petrology, 49(4) (1979) pp. 10931100.
12. Mohammad, T. Nejad, S. Iannaccone, W. Rutherford, P. M. Iannaccone, and C. D. Foster, Mechanics and spiral formation in the rat cornea, Biomechanics and Modeling in Mechanobiology, 14(1) (January 2015) pp. 2238.
13. Rhee, Jerry, Talisa Mohammad Nejad, Olivier Comets, Seam Flannery, Eine Begum Gulsoy, Philip Iannaccone and Craig Foster, Promoting convergence: The Phi spiral in abduction of mouse corneal behaviours, Complexity, 20(3) (January 2015) pp. 2238.10.1002/cplx.21562
14. Tucker, Vance., Gliding flight: drag and torque of a hawk and a falcon with straight and turned heads, and a lower value for the parasite drag coefficient, The Journal of Experimental Biology, 203(Pt 24):3733-44 (December 2000).
15. Boyarzhiev, Khristo N., Spirals and conchospirals in the flight of insects. The College Mathematics Journal, 30(1) (1999) pp. 2331.10.1080/07468342.1999.11974025
16. Meisner, Gary, Is the Nautilus shell spiral as a golden spiral? (2014)
17. Meisner, Gary, Spirals and the Golden Ratio (2012)
18. Livio, Mario, The golden ratio: the story of Phi, the world's most astonishing number, Broadway Books (2002).
19. Lord, Nick, Euler, the clothoid and , Math. Gaz., 100 (July 2016) pp. 266273.10.1017/mag.2016.63
20. Jameson, G. J. O., Evaluating Fresnel-type integrals, Math. Gaz., 99 (November 2015) pp. 491498.10.1017/mag.2015.86
21. Brouillard, Adam, Perfect corner: a driver's step-by-step guide to finding their own optimal line through the physics of racing, Paradigm Shift Motorsports Books (2016).

Archimedean, Logarithmic and Euler spirals − intriguing and ubiquitous patterns in nature

  • Danilo R. Diedrichs (a1)


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