Overview
This lecture explains how the Golden Ratio and Fibonacci Numbers appear in natural patterns, especially in plants, and explores their mathematical properties and significance.
Spiral Patterns in Plants
- Plants often grow new cells, seeds, or leaves in spiral patterns to maximize space and minimize gaps.
- Each new cell forms after turning a specific fraction of a rotation.
The Optimal Turn: Golden Ratio
- If the turn is a simple fraction of a circle, patterns develop gaps or alignments.
- The ideal turn is approximately 0.618 of a rotation (or 222.5°), related to the Golden Ratio.
- The Golden Ratio is denoted by the Greek letter Phi (Φ), about 1.618.
Properties of the Golden Ratio
- The Golden Ratio is an irrational number, meaning it cannot be written as a simple fraction.
- Its continued fraction form is: Φ = 1 + 1/Φ.
- It is uniquely effective at avoiding repeated alignments that create gaps.
Irrational Numbers and Patterns
- Other irrational numbers like π (pi) and e do not create efficient plant patterns because their decimals are close to simple fractions.
Fibonacci Numbers and the Golden Ratio
- Fibonacci Numbers are a sequence where each number is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
- Ratios of successive Fibonacci Numbers approach the Golden Ratio as the numbers increase.
- In many plants, the number of spiral arms or petals matches Fibonacci Numbers.
Applications in Nature
- Leaves, petals, and branches often grow at successive Fibonacci fraction rotations (e.g., 1/2, 3/5, 5/8).
- This arrangement ensures minimum shading and maximum exposure to sunlight and rain.
The Golden Angle
- The Golden Angle is approximately 137.5°, derived from dividing a full circle by the Golden Ratio.
- Many plants exhibit this angle between successive leaves or seeds.
Not Universal
- Not all plants follow these patterns; nature uses various methods for growth and survival.
Key Terms & Definitions
- Golden Ratio (Φ) — An irrational number (~1.618) that describes a unique proportion often found in nature.
- Fibonacci Numbers — A sequence where each number is the sum of the previous two.
- Irrational Number — A number that cannot be written as a simple fraction.
- Golden Angle — About 137.5°, the angle related to the Golden Ratio, frequently observed in plant growth.
Action Items / Next Steps
- Observe plants and measure the angle between leaves, petals, or seeds to find examples of the Golden Ratio and Fibonacci numbers.
- Record your findings, including plant name, spiral pattern, rotations, angles, and petal counts.