Overview
This lecture examines the Golden Ratio (phi), its mathematical properties, historical background, appearances in nature, its role in aesthetics, common misconceptions, and its real biological significance.
Introduction to the Golden Ratio
- The Golden Ratio (phi, ≈1.618) is considered a "perfect" number in mathematics and nature.
- Phi is an irrational number, meaning it cannot be written as a ratio of two integers.
- The Golden Ratio is visually embodied in the golden rectangle, considered by some as the most aesthetically pleasing rectangle.
Historical Context
- Euclid, around 300 BC, described phi as the "extreme and mean ratio."
- The term "Golden Ratio" was introduced much later, near the 20th century.
- Leonardo of Pisa (Fibonacci) popularized Hindu-Arabic numerals and introduced the Fibonacci sequence.
Mathematical Properties of Phi
- In the golden rectangle, the ratio of the long side to the short side is phi.
- Phi has unique properties: phi × phi = 1 + phi, and 1/phi = phi - 1.
- Other shapes like the golden triangle and golden spiral are constructed using phi.
Fibonacci Sequence & Its Link to Phi
- The Fibonacci sequence progresses by summing the two previous terms (e.g., 1, 1, 2, 3, 5, 8, 13...).
- Ratios of consecutive Fibonacci numbers approach phi as the sequence continues.
- Johannes Kepler later established the connection between Fibonacci numbers and the Golden Ratio.
Phi in Nature and Art: Myths vs. Facts
- Claims that phi governs proportions in the human body, architecture, and certain shells are often exaggerated or incorrect.
- Many measurements cited as examples of phi are not mathematically exact matches.
- Despite common beliefs, the nautilus shell and DNA helix do not perfectly fit the Golden Ratio.
Real Occurrences of Phi and Fibonacci in Nature
- Phi and Fibonacci numbers genuinely appear in the arrangement of plant spirals, such as those on pinecones, pineapples, and sunflowers.
- The number of spirals in opposing directions on these plants often correspond to Fibonacci numbers.
- The "golden angle" (≈137.5°) helps plants maximize sunlight and minimize leaf overlap, resulting in Fibonacci spiral patterns.
Biological Explanation
- Plant spiral patterns are influenced by growth hormones and physical repulsion between new growths, not by mathematical calculation.
- Evolution favors arrangements that maximize light capture or space efficiency, sometimes leading to Fibonacci patterns.
Beauty, Art, and Perception
- While some artists and architects have intentionally used the Golden Ratio, many beautiful works do not use phi.
- Human brains are wired to recognize patterns, which can lead to overestimating the prevalence of phi in nature and art.
Key Terms & Definitions
- Golden Ratio (phi) — An irrational number approximately equal to 1.618, with unique mathematical properties.
- Golden Rectangle — A rectangle whose side lengths are in the golden ratio.
- Fibonacci Sequence — A sequence where each number is the sum of the two preceding ones.
- Golden Spiral — A spiral that increases by a factor of phi for each quarter turn.
- Golden Angle — About 137.5°, derived from dividing a circle by phi squared, important in plant growth patterns.
- Irrational Number — A number that cannot be written as a simple fraction.
Action Items / Next Steps
- Review class notes on irrational numbers, ratios, and geometric constructions.
- Optional: Count spirals on a plant (e.g., sunflower, pinecone) and compare to Fibonacci numbers.
- Read about phyllotaxis and the role of hormones in plant growth patterns.