Mathematics in Our World: Comprehensive Study Notes (Transcript)
Golden Ratio in Architecture
- 1. Great Pyramid of Giza (Egypt, 4700\,\text{BC}).
- 2. Notre Dame (Paris, 1163-1250).
- 3. Taj Mahal (India, 1648).
- 4. Chartres Cathedral (France).
- 5. UN Building (New York).
- 6. Eiffel Tower (Paris, 1889).
- 7. CN Tower (Toronto, Canada).
Note: These buildings are cited as examples where Golden Ratio proportions are observed or implied in their design aesthetics and spatial layout.
Behavior of Nature (Natural Regularities)
- Patterns in nature include symmetry, fractals, spirals, waves, trees, foams, cracks, meanders, stripes, and spots.
- Examples of natural regularities:
- Honeycombs: hexagonal structure as an efficient storage/packing shape.
- Zebra coat pigment rule: specific stripe patterns correlate with species and function.
- Spider webs: geometric patterns with reinforcing symmetry.
- Tree rings: age dating via dendrochronology.
- Turtles’ scutes: hexagonal growth ring patterns.
- Lightning: fractal branching patterns.
- Foam: bubbles and trapped gas forming structured patterns.
- River meanders: winding curves shaped by flow and sediment.
Applications of Mathematics
- 1. Forensic Science: calculus for image clarification, optimization.
- 2. Medicine: protein modeling, drug development, disease understanding.
- 3. Engineering/Physics: fluid dynamics, quantum mechanics.
- 4. Information Technology: logic, sets, graph theory, recursion, algorithms.
- 5. Cryptography: ATM security, passwords, e-commerce protections.
- 6. Archaeology: carbon dating, statistical analysis.
- 7. Social Sciences: economics, sociology, psychology, linguistics (game theory, probability).
- 8. Economics: matrices, probability, statistics, modeling.
- 9. Political Science: voting analysis, conflict resolution (game theory).
- Music & Arts: rhythm, permutations, set theory, number theory.
Quick Review
Leonardo of Pisa (Fibonacci) introduced the Fibonacci sequence in Liber Abaci (1202).
Luca Pacioli wrote De Divina Proportione (1509) about the Golden Ratio.
Leonardo da Vinci illustrated the Golden Ratio in the Vitruvian Man and in the Mona Lisa.
Michelangelo used the Golden Ratio in The Creation of Adam.
Fibonacci Sequence: 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\dots where each number is the sum of the two previous numbers. Equivalently, F1=1,\ F2=1,\ Fn = F{n-1}+F_{n-2}\quad (n\ge3).
Golden Ratio: \phi = \frac{1+\sqrt{5}}{2} \approx 1.618…
Patterns in Nature:
- Spirals: shells (nautilus), galaxies, hurricanes, sunflower seeds.
- Fractals (self-similar patterns): lightning, tree branches, broccoli, coastlines.
- Symmetry:
- Radial: starfish, jellyfish, snowflakes.
- Bilateral: humans, butterflies, orchids.
- Stripes/Spots: zebra, tiger, giraffe, leopard.
- Honeycomb hexagons (efficient storage).
- Pattern rules in coats and patterns (e.g., zebra coat pigment rule).
- Spider webs: geometric patterns.
- Tree rings: dendrochronology dating.
- Turtles’ scutes: hexagonal growth rings.
- Lightning fractals.
- Foam bubbles: trapped gas patterns.
- River meanders: winding curves.
Applications of Mathematics (expanded):
- Forensic science: calculus for image clarity and optimization.
- Medicine: protein modeling, drug development.
- Engineering/Physics: fluid dynamics, quantum mechanics.
- IT: logic, sets, graph theory, recursion, algorithms.
- Cryptography: ATM security, passwords, e-commerce.
- Archaeology: carbon dating, statistical analysis.
- Social sciences: economics, sociology, psychology, linguistics (game theory, probability).
- Economics: matrices, probability, statistics, modeling.
- Political science: voting analysis, conflict resolution (game theory).
- Music & Arts: rhythm, permutations, set theory, number theory.
Patterns in Nature & the World
- Patterns are visible regularities of form in the natural world.
- Found in stars, weather, animals, plants, seas, skies, galaxies.
- Serve as clues to natural processes.
Examples
1. Stars move in circles across the sky.
2. Weather seasons cycle each year; snowflakes have sixfold symmetry.
3. Fish patterns: trunkfish, puffer, stingray, eel, grouper, lionfish, boxfish, angelfish.
4. Animal skins:
- Stripes: zebras, tigers, cats, snakes.
- Spots: leopards, hyenas.
- Blotches: giraffes.
5. Natural waves: ocean ripples, sand dunes, typhoons, water drops.
6. Animal formations: mackerel ball, V-formation of geese, tornado formation of starlings.
Bilateral Symmetry: left & right are mirror images (insects, animals, plants, orchids).
Mathematics in Our World (Chapter 1 Review)
What is Mathematics?
- Study of patterns and structures.
- Fundamental in sciences, engineering, IT, economics, and social sciences.
- Helps us quantify, organize, control, and predict phenomena.
Where is Mathematics?
- Found in nature, our world, and daily life.
- Helps explain patterns and occurrences.
Role of Mathematics in the World
- Organizes patterns & regularities.
- Predicts behavior of nature & phenomena.
- Controls nature & occurrences.
- Has countless applications indispensable.
Patterns in Nature & the World (detailed recap)
- Patterns are visible regularities of form in the natural world.
- Found in stars, weather, animals, plants, seas, skies, galaxies.
- Serve as clues to natural processes.
Examples
1. Stars move in circles across the sky.
2. Weather seasons cycle each year; snowflakes have sixfold symmetry.
3. Fish patterns: trunkfish, puffer, stingray, eel, grouper, lionfish, boxfish, angelfish.
4. Animal skins:
- Stripes: zebras, tigers, cats, snakes.
- Spots: leopards, hyenas.
- Blotches: giraffes.
5. Natural waves: ocean ripples, sand dunes, typhoons, water drops.
6. Animal formations: mackerel ball, V-formation of geese, tornado formation of starlings.
Bilateral Symmetry: left & right are mirror images (insects, animals, plants, orchids).