The Golden ratio principle is present in the architecture and evolution of the human skull, suggests a new study by researchers from the Johns Hopkins University School of Medicine and Nationwide Children’s Hospital.
In human skulls, division of the nasioiniac arc (from nasion to inion, NI) by bregma into a shorter frontal arc (from nasion to bregma, NB) and longer parietooccipital arc (from bregma to inion BI), creates a geometrical relationship in which the ratio of the nasioiniac arc over the bregma-inion arc (NI/BI) coincides with the ratio of the bregma-inion arc over the nasion-bregma arc (BI/NB), both 1.6. The subdivision of the nasioiniac arc by bregma into two unequal arcs emulates the geometrical division of a line into the Golden ratio. Image credit: Tamargo & Pindrik, doi: 10.1097/SCS.0000000000005610.
Represented by the Greek upper-case letter phi (Φ), the Golden ratio belongs to the set of irrational numbers and expands in a decimal form as 1.618033…
The first written description and illustration of how to obtain the Golden ratio geometrically came from Euclid of Alexandria around 300 BCE.
Eighteen centuries after Euclid, in 1509, Franciscan friar and mathematician Fra Luca Bartolomeo de Pacioli dedicated an entire book to the Golden ratio and titled it De Divina Proportione (The Divine Proportion).
Leonardo da Vinci, a friend and mathematics pupil of Pacioli, illustrated De Divina Proportione. He used the Golden ratio in paintings during his career, starting as early as 1472 to 1475 when he painted Annunciation.
Over centuries, scientists have demonstrated the presence of the Golden ratio in a wide variety of mathematical, biological, and natural systems.
In a new study investigating whether mammalian skull shape follows the Golden ratio, Professor Rafael Tamargo and Dr. Jonathan Pindrik compared 100 human skulls to 70 skulls from six other mammals (lion, tiger, rhesus monkey, domestic dog, blue monkey, and Eastern cottontail rabbit).
“We 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 two sub-arcs,” they explained.
“We studied 100 human skulls and 70 skulls of 6 other mammals and calculated two ratios:
(i) the nasioiniac arc divided by the parietooccipital arc (between bregma and inion),
(ii) and the parietooccipital arc divided by the frontal arc (between nasion and bregma).”
Among the 7 species surveyed, only in humans these two ratios were essentially identical (1.64 and 1.57, respectively) and closely approximated Φ (1.618) within 1 standard deviation.
The skulls of dogs, two kinds of monkeys, rabbits, lions and tigers, however, diverged from Φ.
“The other mammals we surveyed actually have unique ratios that approach the Golden ratio with increased species sophistication,” Professor Tamargo said.
“We believe that this finding may have important anthropological and evolutionary implications.”
The findings were published in the September 2019 issue of the Journal of Craniofacial Surgery.
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Rafael Tamargo & Jonathan Pindrik. 2019. Mammalian Skull Dimensions and the Golden Ratio (Φ). Journal of Craniofacial Surgery 30 (6): 1750-1755; doi: 10.1097/SCS.0000000000005610
