Overview
This lecture introduces the Golden Ratio and Fibonacci sequence, highlighting their appearances in mathematics, art, nature, architecture, music, and human design.
Introduction to the Golden Ratio and Fibonacci Sequence
- The Golden Ratio (approximately 1.618), denoted as "phi," is a proportion found in mathematics, art, and nature.
- The Fibonacci sequence is a series where each number is the sum of the two preceding numbers (e.g., 1, 1, 2, 3, 5, 8).
- Fibonacci numbers are commonly found in natural and man-made structures.
Architectural Examples
- The Great Pyramids of Giza incorporate the Golden Ratio in their triangular geometry and grid-based placement, forming golden spirals from an aerial view.
- The Taj Mahal's architecture uses the Golden Ratio (phi matrix) in its base and other buildings within its premises.
Art and Design
- Leonardo da Vinci’s Mona Lisa applies golden rectangles to facial and body proportions, fitting the golden spiral.
- Da Vinci's Vitruvian Man and The Last Supper also feature the Golden Ratio.
Nature and the Cosmos
- Spiral galaxies, including the Milky Way, form golden spirals; even hurricanes show this pattern.
- The radius of the Moon plus Earth equals the square root of five, relating to phi in their spatial relationship.
- Flower petals often occur in Fibonacci numbers; rose petal and pine cone arrangements show golden spirals.
- DNA dimensions (34 by 21 angstroms) are both Fibonacci numbers; DNA cross-sections show phi relationships.
Music and Human Perception
- Music compositions (e.g., Beethoven’s Fifth Symphony, Mozart’s sonatas) utilize the Golden Ratio for timing and structure.
- Human facial symmetry is perceived as attractive due to alignment with the Golden Ratio; used in plastic surgery and facial recognition software.
Advertising, Branding, and Web Design
- Advertisers use faces and designs following the Golden Ratio to attract customers.
- Logos (e.g., Toyota, Pepsi) and web templates often use golden rectangles or grids for visual appeal.
Anatomy of Living Things
- The human body shows the Golden Ratio in proportions (head, shoulders, fingertips, forearm).
- The number five recurs in human anatomy (five senses, fingers, etc.) and in the segmented bodies of insects.
Key Terms & Definitions
- Golden Ratio (phi, ≈ 1.618) — a mathematical proportion often found in nature, art, and architecture.
- Fibonacci Sequence — a series where each number equals the sum of the two preceding numbers.
- Golden Spiral — a spiral that increases in size by the factor of the Golden Ratio for each quarter turn.
Action Items / Next Steps
- Explore your surroundings and identify examples of the Golden Ratio or Fibonacci sequence in objects or designs.
- Review the definitions and examples of the Golden Ratio for further study.