#Fibonacci sequence in seashells full#

Fonseca calculated that her specimen actually averaged a growth ratio of radius vectors for θ = 90° in the order of 1.316, not 1.272, thus yielding a constant angle β of about 80°. Four tangents are necessary to establish the correct aspect ratio. Footnote 3 In discussing her analysis, Fonseca ( 1993) points out that her shell’s spiral is inscribed in the √φ rectangle with only three sides at a tangent. Meisner’s specimen falls just within that range and so his hypothesis based on his single shell may have some validity.Īnother interpretation employing the √φ (1.272) ratio was advocated by Rachel Fletcher ( 1988: 36–51: Fig. Nonetheless, the range of ratios noted in Table 1, for all sampled Nautilus species, is between 1.261 and 1.348.

#Fibonacci sequence in seashells full#

For a full rotation, it would be 2.94 as indicated above, not 2.618. If the mean value for the aspect ratio of Nautilus shells is actually 1.310, the growth ratio for a half rotation would be 1.716. Over a complete revolution of 360°, the growth factor would be 1.272 4 or 2.618. Therefore, the expansion over 180° would be √φ 2 or 1.618 as he asserts. For every 90°, the shell spiral would have grown logarithmically by 1.272. This implies that his specific Nautilus specimen has an aspect ratio of 1.272 (the square root of the golden ratio). Meisner ( 2014) alternatively suggests that the Nautilus shell spiral expands by a golden ratio proportion, not by a full revolution as the common myth claims but by 1.618 every 180°. Footnote 2 Sharp gave the tangent angle as about 80.08°, agreeing with 80★′ set forth by Thompson ( 1992: 791), a far cry from a φ ratio spiral with a radius vector ratio of about 7, and a tangent angle of 72.97°. His average of 2.94 would indicate the aspect ratio of his individual shell was 1.31 (1.31 4 ≈ 2.94). British mathematician, John Sharp ( 2002: 79) also found that the ratio of the distance between two consecutive curves (whorls) of the spiral on radius vectors for four different 360° rotations of his Nautilus specimen spiral ranged from 2.83 to 3.02 and averaged 2.94 or about 3. Moseley had observed that the whorls of the Nautilus spiral grow by about a factor of three (Thompson 1992: 770). One strong bulwark to the still prevalent golden ratio myth is Falbo’s ( 2005: 127) paper that concludes “Anyone with access to such a shell can see immediately that the ratio is somewhere around 4 to 3… not phi”. Although graphic designers might claim some innocence for their role in using the visual appeal of the Nautilus shell to grace the covers and pages of publications, their authors may not be able to avoid responsibility so easily. Even Smithsonian’s own science Facebook page (Ocean Portal) describes the Nautilus as “a natural example of Fibonacci’s sequence…”. Ironically, ‘Geometry for Dummies’ (Ryan 2016: 371) instructs us that “the golden rectangle spiral…happens to be the same shape as the spiraling shell of the Nautilus…”. Astrophysicist Livio ( 2003: 9) in ‘The Golden Ratio’ would also lead us to believe this false notion. Novelist Dan Brown ( 2003: 94) famously popularized this myth in ‘The Da Vinci Code’. Writing in ‘The Myth That Will Not Go Away’ (Devlin 2007), he candidly acknowledged that while debunking various other applications of the golden ratio, he inadvertently claimed that the Nautilus shell growth was governed by the golden ratio. Respected mathematician, Keith Devlin sets the stage.

Recent examples demonstrate how deep this fallacy penetrates. Footnote 1 Even at a cursory glance, any one of the myriad images of the shell compared or superimposed over a golden ratio spiral demonstrates that it simply does not match. Certainly, a logarithmic spiral, as a special case, can be drawn in golden ratio proportions, but it also can be drawn in various other proportions. These sources have stated or inferred that the Nautilus spiral is an excellent natural example of a logarithmic spiral and as such must also be a notable example of the golden ratio spiral.

The facts are also misrepresented in blogs, by museums, and by influential authors, even including mathematicians and scientists.

Examples on the Internet and in literature blindly perpetuate the common fallacy that the Nautilus logarithmic spiral shape is also a golden ratio or Fibonacci logarithmic spiral. According to Richard Padovan ( 1999: 314), “by the early years of this century, the notions of ‘beauty of proportion’ and ‘the golden section’ seemed to have become virtually synonymous.” So, it’s not surprising that the striking natural beauty of the Nautilus shell has been easily linked to the golden ratio.

The exquisite chambered Nautilus shell has long been a source of inspiration for artists and architects.