Skip to main content

Nautilus Spirals and the Meta-Golden Ratio Chi

Abstract

The Nautilus shell is the popular iconic image for a logarithmic spiral. It is also frequently cited as an example of a golden ratio logarithmic spiral in nature. Evidently, this not the case. Contrarian studies have proposed that the Nautilus spiral is actually in the 4:3 ratio. Yet, these recommendations are based on one, or just a few shells. In this research, to compare the mean aspect ratio of Nautilus shells to the 4:3 ratio and the meta-golden ratio Chi, eighty Nautilus shells were measured in the Smithsonian collection. The results show that the Nautilus genus is clearly not the widely quoted 4:3 (1.333), but averaged 1.310. However, there was one species that was remarkably different, the Crusty Nautilus averaging 1.356 which is an excellent match for the Meta-golden ratio Chi.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Notes

  1. 1.

    Markowsky (2005: 347) also made this point in his review of Livio (2003).

  2. 2.

    Cf. the Nautilus mean aspect ratio of 1.31 in Table 1. (The meta-golden ratio χ4 is equal to 3.38).

  3. 3.

    √φ is a significant ratio in geometric pictorial composition where the four ‘eyes’ are in golden ratio proportions, vertically, horizontally and diagonally to all sides of the canvas. (Bartlett 2007: 103).

  4. 4.

    Starting with a Phi ratio rectangle, using what Le Corbusier called l’angle droit; the partition created divides the rectangle into a similar Phi proportioned rectangle and a square. This procedure can be used with any rectangle. If the rectangle, for example, were in a 1:1.310 ratio then the intersecting diagonals would subdivide it into a similar smaller 1.310 rectangle and its derived rectangle. The meta-golden ratio Chi rectangle (1:1.356) is unique in that it divides into another Chi rectangle and a Phi rectangle.

  5. 5.

    E.g. aspect ratios over the Nautilus and Allonautilus species ranged from 1.261 to 1.348.

  6. 6.

    Nautilidae populations are endangered because of commercial over-fishing for its shell. “Save the Nautilus Foundation” leads conservation efforts.

  7. 7.

    With mean aspect ratios of 1.310 and 1.356 respectively (Table 2), the Nautilus and Allonautilus genera show a morphological variance in spiral expansion rate of about 0.045 or 3.46%. Some corroboration of this was actually recorded by paleobiologists, Shapiro and Saunders (1987: 533 Table II). Respective measures of the whorl expansion rates between these two genera on a small sample were 2.975 and 3.016, a variance of 0.041 and similar to 0.045 observed here.

  8. 8.

    It also shows an accurate septal neck bisection, critical for accuracy, which is missing in some studies.

  9. 9.

    Unfortunately, photographs of the exhibition were prohibited. Besides a graphic sketch (used as a book cover for Who was Le Corbusier), he also painted the shell in an illustration of his poem on proportion in Le Poeme de l'Angle Droit.

References

  1. Bartlett, C. and D. Huylebrouck. 2013. Art and Math of the 1.35 Ratio Rectangle. Symmetry: Culture and Science 24: 451-462.

  2. Bartlett, C. 2007. Decoding Fairfield Porter’s July Interior. American Art Journal 21(1): 98-100.

  3. Bartlett, C. 2014. The Design of the Great Pyramid of Khufu. Nexus Network Journal 16(2): 299-311.

  4. Brown, D. 2003. The Da Vinci Code. USA. New York: Doubleday.

  5. Du Sautoy, M. 2011. The Secrets of the Nautilus Shell - The Code - Episode 1 - BBC Two. https://www.youtube.com/watch?v=Ysw3iM3ENQA.

  6. Devlin, K. 2007. June. The Myth That Will Not Go Away. https://www.maa.org/external_archive/devlin/devangle.html. accessed 6.10.18.

  7. Falbo, C. 2005. The golden ratio: A contrary viewpoint. College Mathematics Journal 36(2): 123-134.

  8. Fathauer, R and N. Selikoff, eds. 2018. Exhibition of Mathematical Art. JMM: Tessellations Publishing.

  9. Fletcher, R. 1988. Proportion and the Living World. Parabola 13(1): 36-51.

  10. Fonseca, R. 1993. Shape and Order in Organic Nature: The Nautilus Pompilius. Leonardo. MIT Press. 26 (3): 201-204. accessed: 03-02-2016 20:53 UTC.

  11. Grulke, W. 2016. Nautilus: Beautiful Survivor. UK: At One Communications.

  12. Huylebrouck, D. 2013. Some Mathematical Properties of the Bartlett Chi Ratio. arXiv:1310.5611v1 [math.HO]. accessed 02.05.2016.

  13. Huylebrouck, D. 2014. The Meta-golden Ratio Chi. Proceedings Bridges Seoul: 151-158.

  14. Impens, C. 2016. Debunking golden ratio shells, 1& 2. Chris Impens @ Valvas http://ci47.blogspot.be/2016/08/debunking-golden-ratio-shells-1.html. accessed 10.26.17.

  15. Le Corbusier. 1961. The Modulor. A Harmonious Measure to the Human Scale. Universally Applicable to Architecture and Mechanics. London: Faber and Faber Limited.

  16. Livio, M. 2003. The Golden Ratio. New York: Broadway.

  17. Markowsky, G. 2005. The golden ratio, a book review. Notices of the AMS 52(3).

  18. McMahon, T. A. and J. T. Bonner. 1983. On Size and Life. New York: Scientific American.

  19. Meisner, G. 2014. Feb. 8. Is the Nautilus shell spiral a golden spiral? https://www.goldennumber.net. accessed 11.19.17.

  20. Padovan, R. 1999. Proportion. London: Routledge.

  21. Redondo Buitrago, Antonia. 2013. On the Ratio 1.3 and Related Numbers. Proceedings of 9th ISIS Congress Festival Symmetry: Art and Science. Panormo, Crete, Greece: 76-81.

  22. Redondo Buitrago, Antonia. 2014. Approximations and Applications of the 1.3 Ratio and Related Numbers. Symmetry: Art and Science. (submitted 22-10-2013, admitted 16-07-2014).

  23. Ryan, M. 2016. Geometry for Dummies. Hoboken: Wiley.

  24. Sharp, J. 2002. Spirals and the Golden Section. Nexus Network Journal 4(1): 59-82.

  25. Saunders W. B. and N. H. Landman. eds. 1987. Nautilus, The Biology and Paleobiology of a Living Fossil. New York and London: Plenum Press.

  26. Shapiro, E.A. and W.B Saunders. 1987. Nautilus Shell Hydrostatics. In: Saunders W. B. and N. H. Landman. eds. 1987. Nautilus, The Biology and Paleobiology of a Living Fossil, Saunders, W. B, New York. Plenum Press.

  27. Smithsonian Ocean Portal website. https://ocean.si.edu/ocean-life/invertebrates/couple-nautiluses. accessed 12.11.2017.

  28. Strogatz, S. 2012. Sep. 24. Me, Myself and Math, Proportion Control. https://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/ accessed 9.5.17.

  29. Tajika A, Morimoto N, Wani R, Naglik C, Klug C. (2015) Intraspecific variation of phragmocone chamber volumes throughout ontogeny in the modern nautilid Nautilus and the Jurassic ammonite Normannites. PeerJ 3:e1306. https://doi.org/10.7717/peerj.1306. accessed 8.22.18.

  30. Thompson, D’Arcy. 1992. On Growth and Form. New York: Dover.

  31. Trott, M. 2015. 11/18. From Aspect Ratios in Art: What Is Better Than Being Golden? Being Plastic, Rooted, or Just Rational? Investigating Aspect Ratios of Old vs. Modern Paintings. Blog.wolfram.com. accessed 8.12.17.

  32. Vandepas, L. E. et al. 2016. A revisited phylogeography of Nautilus pompilius. Ecology and Evolution. John Wiley & Sons. 6(14): 4924–4935. https://doi.org/10.1002/ece3.2248. accessed 3.04.18.

  33. Ward, P. D. & W. B Saunders, 1997. Allonautilus: A New Genus of Living Nautiloid Cephalopod and Its Bearing on Phylogeny of the Nautilida. Journal of Paleontology, 71(6): 1054-1064. http://www.jstor.org.proxy-tu.researchport.umd.edu/stable/1306604 accessed 4.5.18.

Download references

Acknowledgements

I wish to thank the following: Dr. Mike Veccione and staff at NMNH; Dr. Neil Landman and staff at AMNH; Dr. René Hoffmann, Ruhr-University Bochum; Dr. Amane Tajika; Dr. Greg Barord; Christina Piotrowski, CAS; Prof. Yunwei Cui for statistical analyses; and Ms. Robin Williams for her unwavering support.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Christopher Bartlett.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bartlett, C. Nautilus Spirals and the Meta-Golden Ratio Chi. Nexus Netw J 21, 641–656 (2019). https://doi.org/10.1007/s00004-018-0419-3

Download citation

Keywords

  • Meta-golden Ratio Chi
  • Nautilus spiral
  • Logarithmic spirals
  • Golden ratio spiral
  • Myth
  • Proportion
  • Diagonals
  • Geometry