This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.
(Previous Section: Fibonacci and Insects)
The ubiquity of logarithmic spirals in the animal, bird, and plant kingdoms presents a convincing case for a cosmic character of the Golden Ratio (Boeyens and Thackeray). Livio says Fibonacci numbers are “a kind of Golden Ratio in disguise,” as they are found in even microscopic places, such as in the microtubules of an animal cell. These structures are “hollow cylindrical tubes of a protein polymer” which make up the cytoskeleton. This is the structure that “gives shape to the cell and also appears to act as a kind of cell ‘nervous system.’ Each mammalian microtubule is typically made up of thirteen columns, arranged in five right-handed and eight left-handed structures (5, 8, and 13 are all Fibonacci numbers).” Occasionally there are double microtubules with an outer envelope consisting of 21 columns, the next Fibonacci number. Some investigators argue that microtubules are more efficient “information processors” because they are structured this way rather than with other possible structures; however, Livio admits that “the apparent connection with the Fibonacci series may be coincidental” (Livio). German psychologist Adolf Zeising (1810–1876) studied the skeletons of animals and the branching of their veins and nerves. He observed that there are a lot of examples of the Golden Section or Divine Proportion found in animals, fishes, and birds, in addition to insects and snails. For example, the natural design of a Peacock’s feather hints at the Golden Ratio, the eye, fins and tail of a dolphin appear to fall at Golden Sections of the length of its body, and a penguin body exhibits Golden Ratio properties (Akhtaruzzaman and Shafie).
Some see the Golden Spiral in the shape of the horns of both the ram and the kudu and in the curvature of elephant tusks (Boeyens and Thackeray; Masran; D’Agnese). Animal biology would seem to follow the same spontaneous growth patterns exhibited by plants. It must be noted, however, that many contemporary scholars dismiss such (and similar) claims by Golden Ratio adherents like Ghyka, the Romanian novelist, mathematician, historian, and philosopher who said, “Diagrams of proportions, however diversely arranged, can be deciphered by the same [Golden Ratio] key.” Ghyka, for example, offered a “harmonic analysis” of a thoroughbred horse which showed that ratios between the length of the leg to the vertical thickness of the body are φ. These claims should be viewed with a “healthy degree of skepticism in the absence of full and replicated scientific reports,” however (Green).
Md. Akhtaruzzaman and Shafie mention many sea creatures which exhibit the Golden Proportion in one form or another. For example, they describe the body of the Rainbow Trout fish as having a shape on which “three Golden Rectangles together can be fitted” and “the tail fin falls in the reciprocal Golden Rectangles and square.” Additionally, the individual fins also have the Golden Section properties” (Akhtaruzzaman and Shafie). A wide variety of sea creatures also exhibit pentagonal symmetry. For example, the sea star (Astropecten Aurantiacus), the star fish (Ophiotrix capillaris) and the sand dollar (Echinarachnius parma) exhibit five-fold symmetry (Trinajstic) which Md. Akhtaruzzaman says “has a close intimacy with Golden Ratio.”
In addition, the growth patterns of natural shells like Conch Shell, Moon Snail Shell, and Atlantic Sundial Shell show logarithmic spiral growth patterns of Golden Section properties or golden spiral form (Akhtaruzzaman and Shafie). The chambered nautilus (Nautilus pompilius) is a specific example of one of the marine creatures whose structure represents a spontaneous logarithmic spiral growth pattern. This pelagic marine mollusk of the cephalopod family displays the self-similarity characteristic of an intrinsic response to environmental constraint, growing larger on each spiral by phi, according to Wright (Boeyens and Thackeray; Wright).
The logarithmic spiral is equiangular: if you draw a straight line from the center of the spiral (pole) to any point on the spiral curve, the line always cuts the curve at precisely the same angle. Vance A. Tucker, a biologist at Duke University in Durham, North Carolina, found that falcons appear to capitalize upon this fact by keeping a slightly curved diving trajectory while hunting prey. Rather than plummeting in a straight line, the predatory birds keep “their heads straight while keeping their target in view from the most advantageous angle,” naturally following “the curve of a (highly drawn-out) logarithmic spiral” in their angle of descent (Livio 120). Hawks appear to take the same (or a similar) approach when hunting their prey (Chin).
Did you know you can download a FREE copy of Master Fibonacci with a free membership on Fibonacci.com?
Paperback: 128 pages
Author: Shelley Allen, M.A.Ed.
Publisher: Fibonacci Inc.; 1st edition (2019)