Mammalian Skull Dimensions and the Golden Ratio (Φ)

Abstract

The Golden Ratio (Phi, or Φ = 1.618…) is a potentially unifying quantity of structure and function in nature, as best observed in phyllotactic patterns in plants. For centuries, Φ has been identified in human anatomy, and in recent decades, Φ has been identified in human physiology as well. The anatomy and evolution of the human skull have been the focus of intense study. Evolving over millenia, the human skull embodies an elegant harmonization of structure and function. The authors explored the dimensions of the neurocranium by focusing on the midline calvarial perimeter between the nasion and inion (nasioiniac arc) and its partition by bregma into 2 sub-arcs. The authors studied 100 human skulls and 70 skulls of 6 other mammalian species and calculated 2 ratios: 1) the nasioiniac arc divided by the parieto-occipital arc (between bregma and inion), and 2) the parieto-occipital arc divided by the frontal arc (between nasion and bregma). The authors report that in humans these 2 ratios coincide (1.64 ± 0.04 and 1.57 ± 0.10) and approximate Φ. In the other 6 mammalian species, these 2 ratios were not only different, but also unique to each species. The difference between the ratios showed a trend toward convergence on Φ correlating with species complexity. The partition of the nasioiniac arc by bregma into 2 unequal arcs is a situation analogous to that of the geometrical division of a line into Φ. The authors hypothesize that the Golden Ratio (Φ) principle, documented in other biological systems, may be present in the architecture and evolution of the human skull.

PhiΦ Breast New Diagnostic Techniques for Breast Cancer Detection

“…The Golden Ratio (Phi, or Φ 1.618...) is a potentially unifying quantity of structure and function in nature, as best observed in phyllotactic patterns in plants. For centuries, Phi (Φ) has been identified in human anatomy, and in recent decades, Φ has been proved in human physiology as well with scientific studies of some authors [9,10]. With the use of Golden Ratio [11] and Fibonacci sequence [12] applied to ultrasonography and mammography, we have experimented and developed a diagnostic map with characteristics of high probability of identifying suspicious lesions at an early stage.…”

A Safe and Effective Lip Augmentation Method: The Step-by-Step Φ (Phi) Technique

“…Over centuries, evidence of its presence in a wide variety of mathematical, biological, and natural systems has been recorded, derived from the idea that it represents the perfect beauty. 17 It has also appeared throughout history in many artistic masterpieces and architectural creations such as Da Vinci’s paintings and the Parthenon, and it is, in brief, an attempt to mathematically and geometrically interpret the captivating mystery of beauty. 18 , 19 Undeterred by its subjective nature, 20 we applied this number in our way: to identify the proper points for injection.…”

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“…A recent discovery is that φ occurs in the dimensions of the human skull. The work of Tamargo and Pindrik [48] took CAT scanned images of human skulls and considered three points; the nasion (where the frontal bone meets the nasal bones), the inion (a protuberance at the back of the skull) and the bregma (a point at the top of the skull where the frontal and parietal lobes meet). Taking the arc joining the nasion and inion, and dividing it into two sections separated by the bregma, Tamargo and Pindrik found that the resulting sections are Golden, i.e., the bregma makes a Golden cut to this arc.…”