Overview
This lecture introduces the definition and importance of mathematics, explores patterns in nature, discusses the Fibonacci sequence and the golden ratio, and explains the role of mathematics in organizing patterns and regularities.
Introduction to Mathematics
- Mathematics is the study of relationships among numbers, quantities, and shapes.
- It enhances critical thinking, reasoning, spatial thinking, and creativity.
- Mathematics helps organize patterns and regularities in the world.
Patterns in Nature and the World
- Symmetry: A pattern identical on both halves when folded (e.g., butterfly wings).
- Spiral Pattern: A curved pattern focusing on a central point with circular shapes around it (e.g., some plants).
- Fractal Pattern: Built from simple, repeated shapes that get smaller with each repetition (e.g., romanesco broccoli, spider webs).
- Tessellations: Patterns made of identical shapes that fit together with no gaps (e.g., pineapples, beehives).
Fibonacci Sequence
- Discovered by Leonardo Pisano (Fibonacci); the sequence starts 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
- Each number is the sum of the two preceding numbers.
- November 23 (11/23) is Fibonacci Day because these numbers match the first four non-zero Fibonacci digits.
Golden Ratio
- The golden ratio is denoted by φ (phi) and is approximately 1.618034.
- The ratio of successive Fibonacci numbers approaches the golden ratio but never equals it.
- The larger the Fibonacci numbers used, the closer their ratio is to the golden ratio.
Patterns and Regularities Organized by Mathematics
- The motion of a pendulum shows regularities in timing that can be explained mathematically.
- The law of reflection explains regularities in size and distance in plane mirrors.
Key Terms & Definitions
- Symmetry — A pattern that is identical on both sides when folded.
- Spiral Pattern — Circular pattern focused on a central point.
- Fractal Pattern — Repeating self-similar shapes that decrease in size.
- Tessellation — Identical shapes fitting together without gaps.
- Fibonacci Sequence — A sequence where each number is the sum of the two preceding numbers.
- Golden Ratio (φ) — An irrational number approximately equal to 1.618034, found by the ratio of successive Fibonacci numbers.
Action Items / Next Steps
- Reflect on and answer: What is the application of mathematics in your chosen course?