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Nautilus Spirals and the Meta-Golden Ratio Chi


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.

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  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.


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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.

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Bartlett, C. Nautilus Spirals and the Meta-Golden Ratio Chi. Nexus Netw J 21, 641–656 (2019).

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