This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.
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The same phenomena of Phi that is found in nature’s objects from snail shells to the spirals of galaxies is found also in the design and structure of the human body. For example, the cochlea of the ear is a Fibonacci spiral as is the spiral of the umbilical cord. The progression of the Fibonacci numbers and ratio are well suited to describing organic growth in the human body because they have the properties of self-similarity and of “gnomonic growth;” that is, only the size changes while the shape remains constant. The majority of organs in the human body maintain their overall shape and proportions as they grow (Sacco).
Simple observation confirms that Fibonacci numbers are represented by many human parts: one trunk, one head, one heart, etc. Then there are pairs: arms, legs, eyes, ears. Three is represented by the number of bones in each leg and arm and the three main parts of the hand: wrist, metacarpus and set of fingers consisting of three phalanxes, main, mean and nail. Considering each finger individually, the lengths the phalangeal bones relate to each other according to the rule of golden proportion (Akhtaruzzaman and Shafie). These dimensions allow for the flawless execution and adaptability of “transitory movements of the digits” when grasping. Specifically, radiological and anatomical studies show that “the lengths for the index, long, and ring fingers follow a newly identified, specific mathematical pattern, the Littler series, which is closely related to the Fibonacci series” (Yetkin, et al.; Hutchison and Hutchison).
Five appendages adjoin the torso: the arms, legs and a head; five appendages are on each of these: five fingers on hands and foot; and there are five openings on the face. Through these, five senses equip the body to interact with the world around it: sight, sound, touch, taste and smell. Back to the hand, five fingers are connected to five metacarpal bones forming the basis of the palm, which is connected to the wrist structure. Continuing the count, the human arm together with fingers consists of eight parts. There are 12 pairs of ribs but some claim (without scientific evidence) that man in the past had 13 pairs of ribs. Fourteen facial bones, six middle ear bones and the throat total 21 bones. Human backbone with the skull consists of 34 bones: Eight skull bones (Crania), 24 Vertebrae, one Sacrum and one Coccyx. The base column of human body structure therefore totals 55 (21 + 34 = 55) bones (Akhtaruzzaman and Shafie). All of these numbers – 1, 2, 3, 5, 8, 13, 21, 34 and 55 – are numbers in the Fibonacci series.
Many features of the “ideal” human face are said to have ratios equal to φ; the dimension relationships between the eyes, ears, mouth and nose, for instance. The ratio of the height of the whole head to that of the head above the nose is also said to be Phi (Akhtaruzzaman and Shafie). Other examples supposedly include the ratio between the total height of the body and the distance from head to the finger tips, and “the distances from head to naval and naval to hill.” Then there is the proportion between the forearm and upper arm and the one between the hand and forearm; all of these are said to follow the rule of Golden Ratio (Akhtaruzzaman and Shafie).
Dentists interested in the health of their patients study the relationships between dental aesthetics and the golden proportion. According to Dr. Stephen Marquardt, an eminent oral surgeon in California, “the height of the central incisor is in Golden Proportion to the width of the two central incisors.” Dentists have used this information when addressing “a host of dental aesthetic problems.” Golden Proportion grids have been developed which show Golden Rectangle relationships between the widths and heights of eight teeth of the anterior aesthetic segment, the incisors. In addition, the four front teeth, from central incisor to premolar, are in Golden Proportion to each other. In “Maxillary and Mandibular Teeth Widths” (1985), Dr. McArthur explained that “the average ratio of upper central incisor to lower central incisor is 1.62.” Soon after, in 1987, Shoemaker “wrote a series of articles promoting the use of the Golden Proportion as an adjunct to cosmetic Dentistry.” The article, “Le nombre d’or” (1989) by Amoric showed “many Golden Proportions in cephalometric tracings at various stages of facial growth and also included geometrical propositions.”
In that same year, The Annals of Plastic Surgery (1989) featured the investigations of Kawakami et al. who measured for Golden Proportion balance between the eyes, nose and mouth in the facial appearance of typical Japanese individuals and compared the ratios to measurements in Caucasians. “Each ratio was then used for pre and post-operative aesthetic analysis” (Kawakami).
Similar purpose motivated dentist Yosh Jefferson in 1996 to provide Golden Proportion diagrams, cephalometric tracings, and computer-generated photographs in “Facial Beauty – Establishing a Universal Standard” to depict an ideal structure for the human head. He believed “there are possibly billions of examples of divine proportion within the human body” and “all living organisms, including humans, are genetically encoded to develop and conform to the divine proportion.” The purpose of Dr. Jefferson’s study was to establish a universal standard for facial beauty regardless of race, age, or sex; but his stated intent was not to empower or support discrimination. His hope was that his work would simplify the diagnoses and treatments of facial-skeletal structure misalignments in patients, thereby improving not only their dental and physical health but (as consequence) also their emotional and psychological health (Jefferson).
Elsewhere in the human body, internal organs also exhibit Golden Ratio relationships. Belgian gynecologist Jasper Verguts, at the University Hospital Leuven, says “gynecologists can instantly tell whether a uterus looks normal or not based on its relative dimensions” (Bellos). Uterine size varies in relation to age and gravidity (the number of times a woman has been pregnant). Dr. Verguts conducted ultrasound measurements of uteruses in 5,466 non-pregnant women and created a table showing the average ratio of a uterus’s length to its width for different age bands (Verguts).
The data shows that the mean length/width ratio averages 1.857 at birth, decreasing to 1.452 at the age of 91 years. At the age when women are at their most fertile, between the ages of 16 and 20, the ratio of length to width is 1.6, “a very good approximation to the Golden Ratio” (Verguts, et al.; Bellos).
According to some, even the human heart beats in Golden Proportion rhythm. Doctors Gulay Yetkin, Nasir Sivri, Kenan Yalta, and Ertan Yetkin assessed the ratio of cardiac phases (diastole and systole) in 162 healthy subjects aged 20 years to 40 years after they had rested in the supine position for fifteen minutes and found that the diastolic time interval to systolic time interval ratio was 1.611 and the R– R/diastole ratio was 1.618 (Yetkin, et al.).
In addition to its activity, the human cardiovascular system is structured according to Golden Ratio design. Ashrafian and Athanasiou found that coronary arteries are distributed sequentially in a pattern that follows the Fibonacci series, resembling phyllotaxis seen in other branches in nature. Moreover, “data from 36 species has shown that the association of cardiac diameters by the sum of the diameters of all 13 branches across these species is in the order of the Golden Ratio, 1.618” (Yetkin, et al.; Ashrafian and Athanasiou). Even “diseased atherosclerotic lesions in coronary arteries follow a Fibonacci distribution” (Yetkin, et al; Gibson, et al).
On the molecular level, the nucleotide spirals of human DNA have Fibonacci proportions. Further research is needed to discover the way in which “the crystallographic structure of DNA, stress patterns in nanomaterials, the stability of atomic nuclides and the periodicity of atomic matter depend on the Golden Ratio” (Boeyens and Thackeray). Meanwhile, recent genetic research has determined that the cross-section of microscopic double helix of DNA illustrates the Phi ratio. Each spiral of the double helix traces the shape of a pentagon. The DNA molecule “measures 34 angstroms long and 21 angstroms wide for each full cycle of its double helix spiral model” so its ratio is 1.6190476, close to the ratio of Phi, 1.61803. Since the primary DNA structure molecule is formed according to Fibonacci sequence, it is assumed that linker segments between molecules are also formed according to this mathematical regularity (Shabalkin, et al.).
Just as beautiful art and music reflect harmony in nature, so, too, does the most efficient human walking pattern (gait). Roman professor and neurophysiologist Marco Iosa used a stereophotogrammetric system with 25 retroreflective markers located on the skin to analyze the spatiotemporal gait parameters of 25 healthy human subjects. Repetitive gait phases following the Golden Ratio during physiological walking were found to be most energy efficient; the ones which were “in repetitive proportions with each other,” revealed “an intrinsic harmonic structure.” The conclusion was that this Golden Ratio “harmony could be the key for facilitating the control of repetitive walking.” Thus, harmony is not only an important component for establishing balance in art and music, it also plays a part in facilitating the maximum effective, harmonic rhythm of walking for humans (Iosa, Fusco, et al.; Iosa, Morone et al.).
The use of Fibonacci numbers to inform computer science engineering has the power to improve the lives of countless humans. Computer science and automation engineer from University of Florence (Italy) Claudio Fantacci conducted a case study involving the testing of a model of malware propagation in a computer network. He used a random generalized Fibonacci sequence to test random rates of computer infection within finite time frames because “several systems both in biology and economy are well represented by Fibonacci binary random trees” (Farina, Fantacci, and Frasca). This research is expected to help robotics engineers better anticipate and prevent disruptions in humanoid robot kinematic platforms, or robot-assisted human applications (such as the development of prostheses for loss of limb patients). Physicist Zexian Cao and colleagues from the Chinese Academy of Sciences in China have performed stress engineering to create Fibonacci-sequence spirals on microstructures grown in the lab, and they think they have discovered the reason why the Fibonacci sequence is so ubiquitous in nature – it is a natural consequence of stress minimization (Cartwright).
They coated a curved “core” material of silver with a SiO2 shell material at a high temperature. Because material thermal expansion differs, when the composite is then cooled in a restricted geometric shape, “selective parts of the shell buckle under stress, causing patterns to form.” They created microstructures just 12 μm across and discovered that shells directed into spherical shapes during cooling developed triangular stress patterns. Forced conical shapes, however, caused spiral stress patterns to be formed. These spiral patterns had dimensions governed by the Fibonacci series, “Fibonacci spirals” (Cartwright). This tendency may be related to something the physicist J. J. Thomson researched in 1904 when he sought “how a collection of like-charges would arrange themselves on a conducting sphere so as to minimize energy. Physicists have since calculated that the charges would take on triangular patterns – similar to Cao’s spherical microstructures.” Therefore, Cao’s team conclude that the Fibonacci spirals on the conical microstructures must be the equivalent minimum-energy (and hence minimum-stress) configuration for a cone. Further research and calculations need to be conducted to prove their theory (Cartwright).
Cao’s experiment using pure inorganic materials may serve as the first concrete proof of the theory long held by biologists that “the branching of trees and other occurrences of the Fibonacci sequence in nature is simply a reaction to minimize stress.” Creating Fibonacci patterns from stress engineering invites possible applications in photonics; Cao says, “Fibonacci spirals are a special lattice; I would say they are both ordered and disordered. If the lattice points were some materials of a proper ‘dielectric,’ it may provide a new photonic crystal that displays some interesting properties” (Cartwright). Photonic crystals can be used to develop biosensor technologies and materials capable of artificial touch in relation to humanoid robotics (Android) structural engineering.
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Paperback: 128 pages
Author: Shelley Allen, M.A.Ed.
Publisher: Fibonacci Inc.; 1st edition (2019)