Overview
This lecture explores the golden ratio (phi), its mathematical properties, historical origins, links to nature, and addresses common myths about its ubiquity in art, architecture, and biology.
The Golden Ratio: Definition and Properties
- The golden ratio (phi, φ) is approximately 1.618 and is an irrational number, meaning its decimals never repeat or terminate.
- Defined as the ratio where a line is divided so that the whole to the longer part is the same as the longer to the shorter part.
- Phi has unique algebraic properties: φ² = φ + 1 and 1/φ = φ - 1.
- Golden rectangles and golden triangles use phi in their side length ratios.
Historical Background
- Euclid first described phi as the "extreme and mean ratio" around 300 BC.
- The term "golden ratio" and the symbol phi were adopted much later.
- Fibonacci introduced the sequence that leads to the golden ratio in his book, Liber Abaci (1202), through a rabbit population problem.
The Fibonacci Sequence and its Connection to Phi
- The Fibonacci sequence: each number is the sum of the two preceding numbers (1, 1, 2, 3, 5, 8, 13, ...).
- As the sequence progresses, the ratio of consecutive terms approaches phi.
- Johannes Kepler later established the connection between the Fibonacci sequence and phi.
Myths vs. Reality: Phi in Nature and Art
- Claims that phi appears in human bodies or ancient structures (Pyramids, Parthenon, DNA) are often inaccurate or coincidental.
- Many supposed examples of phi are simply close approximations, not exact matches.
- Our pattern-seeking brains can lead to seeing phi's influence where it doesn't actually exist.
Phi’s Genuine Role in Nature: Phyllotaxis
- Some plants arrange leaves, seeds, or petals according to the golden angle (≈137.5°), derived from phi, to maximize sun exposure.
- Fibonacci numbers often describe the number of spirals in pineapples, sunflowers, pinecones, and artichokes.
- The configuration is driven by evolutionary efficiency, not mathematical perfection.
Beauty, Art, and the Golden Ratio
- The golden ratio has been used intentionally by some artists and architects, but many beautiful works do not involve phi.
- Beauty is subjective and not governed by mathematical rules alone.
Key Terms & Definitions
- Golden Ratio (phi, φ) — An irrational number ≈ 1.618 with unique proportional properties.
- Fibonacci Sequence — A sequence where each term is the sum of the two preceding terms.
- Golden Rectangle — A rectangle whose side lengths are in the ratio of phi:1.
- Golden Angle — Approximately 137.5°, derived from phi, important in plant leaf arrangement (phyllotaxis).
- Irrational Number — A number that cannot be expressed as a ratio of two integers.
Action Items / Next Steps
- Review examples of the golden ratio and Fibonacci numbers in plants.
- Reflect on how mathematical patterns can appear in nature due to evolutionary advantages.
- Optional: Practice identifying exact vs. approximate uses of the golden ratio in art and nature.