Overview
This lecture introduces mathematics as the study of relationships among numbers, quantities, and shapes, discussing patterns found in nature, the Fibonacci sequence, the golden ratio, and how math organizes real-world regularities.
What is Mathematics?
- Mathematics is the study of relationships among numbers, quantities, and shapes.
- It enhances critical thinking, reasoning, spatial thinking, and creativity.
- Mathematics helps us organize patterns and regularities in the world.
Patterns and Numbers in Nature
- Patterns found in nature include symmetry, spiral, fractal, and tessellation.
- Symmetry refers to designs identical on both halves when folded (e.g., butterfly wings).
- Spiral patterns consist of curves revolving around a point (e.g., plant arrangements).
- Fractal patterns use simple shapes repeated in smaller sizes (e.g., romanesco broccoli, spider web).
- Tessellations are patterns of identical shapes fitting together with no gaps (e.g., pineapple, beehive).
Fibonacci Sequence
- The Fibonacci sequence was discovered by Leonardo Pisano ("Fibonacci").
- The sequence starts as 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
- Each number is the sum of the two preceding numbers.
- November 23 is called Fibonacci Day, representing the digits 1, 1, 2, 3.
The Golden Ratio
- The golden ratio (denoted by phi, φ) is approximately 1.618034.
- The ratio of two successive Fibonacci numbers approaches the golden ratio as the numbers increase.
- The ratio never exactly equals the golden ratio, only comes closer with larger numbers.
Mathematical Patterns and Regularities in the World
- The motion of a pendulum shows mathematical regularity in its swing time.
- The reflection in a mirror follows mathematical laws, preserving size and distance.
Key Terms & Definitions
- Symmetry — Pattern with two identical halves when folded.
- Spiral Pattern — Curved pattern revolving around a central point.
- Fractal Pattern — Repeated simple shapes, each smaller than the previous.
- Tessellation — Identical shapes fitting together without gaps.
- Fibonacci Sequence — Sequence where each number is the sum of the two preceding ones.
- Golden Ratio (φ) — An irrational number, approximately 1.618034, prominent in nature and art.
Action Items / Next Steps
- Reflect on the application of mathematics in your chosen course.