The Golden Ratio and Phi Number: 1.618

The golden ratio, also known as the Phi (φ) number, is a mathematical concept with a value of 1.618. It is one of the most fascinating topics in mathematics, alongside Fibonacci numbers. Leonardo Fibonacci, a 13th-century Italian mathematician, introduced this concept.

Fibonacci Sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765…

This sequence follows a simple rule: Each number (except the first two) is the sum of the two preceding numbers.
For example, dividing the number of female bees by the number of male bees in a beehive consistently gives the golden ratio of 1.618.
Similarly, Leonardo da Vinci’s famous “Vitruvian Man” illustration also demonstrates the golden ratio.

Applications and Occurrences of the Golden Ratio

  1. Sunflower: The number of spirals in the seeds of a sunflower, counted from the center outward in both clockwise and counterclockwise directions, aligns with the golden ratio.
  2. Daisy: Like the sunflower, the arrangement of petals in a daisy follows the golden ratio.
  3. Human Head: The hair growth from the crown of the head forms a spiral pattern whose curvature corresponds to the golden ratio.
  4. Human Body:
    • Arms: The ratio of the upper arm to the forearm and the entire arm to the upper arm gives the golden ratio.
    • Fingers: The ratio of the top segment to the middle segment, and the entire finger to the top segment, equals the golden ratio.
  5. Rabbits: Similar to the human head, the spiral patterns in rabbits align with the golden ratio.
  6. Pyramids of Egypt: The ratio of the base to the height of each pyramid equals the golden ratio.
  7. Leonardo da Vinci:
    • Mona Lisa: The ratio of the painting’s height to its width equals the golden ratio.
    • Saint Jerome: Similarly, the ratio of the height to the width in this painting reflects the golden ratio.
  8. Picasso: Like Leonardo, Picasso also utilized the golden ratio in his works.
  9. Pinecones: The spirals in a pinecone adhere to the golden ratio.
  10. Seashells: The spiral in a seashell, when measured, gives the golden ratio.
  11. Tobacco Plant: The arrangement of leaves on a tobacco plant follows a spiral pattern aligned with the golden ratio.
  12. Fern: Similar to the tobacco plant, ferns display the same golden spiral pattern.
  13. Electrical Circuits: In physics, the golden ratio is used to determine the optimal resistance for maximum efficiency in circuits.
  14. Snails: When the shell of a snail is flattened, the resulting rectangle forms the “golden rectangle,” where the ratio of length to width is the golden ratio.
  15. Architect Sinan: In his masterpieces such as the Süleymaniye and Selimiye Mosques, the minarets demonstrate the golden ratio.

The Golden Ratio in the Human Body

The human body exhibits countless examples of the golden ratio. This innate symmetry may explain why humans instinctively find golden ratio-based designs aesthetically pleasing. Here are a few examples:

  • Height / Leg length
  • Body length / Length from shoulder to waist
  • Full arm length / Upper arm length
  • Arm span / Distance from fingertip to elbow
  • Belly button to shoulder / Belly button to waist

The Golden Ratio in the Human Face

An ideal human face features numerous examples of the golden ratio:

  • Face height / Face width
  • Hairline to eyes / Eyes to chin
  • Eyes to chin / Nose to chin
  • Forehead width / Nose width
  • Eyes to lips / Nose length
  • Nose base to chin / Lips to chin
  • Face width / Distance between pupils
  • Distance between pupils / Mouth width
  • Mouth width / Nose width

References

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