Patterns and Numbers in Nature - Study Notes

Objectives

• Identify patterns in nature and regularities in the world

• Articulate the importance of Mathematics in one’s life

• Argue about Mathematics: what it is, how it is expressed, represented, and used

• Express appreciation for mathematics as a human endeavor

Focus of the Topic

• Learn new things about mathematics

• Visual cues on Page 3 show numbers: 60, 98, 20, 20 (illustrative figures in a focus section)

Focus of the Topic: Ask New Questions about Mathematics

• Thought-provoking question: How can abstract mathematics, which often develops without any immediate practical application, later become essential in solving real-world problems?

• Example: number theory, once considered purely theoretical, is now the backbone of modern cryptography and internet security

What Mathematicians Do

• Page shows some mathematical expressions; content is partially garbled in the transcript (e.g., fragments like S'rs St, 2. x210, 10, (x)=x, EX, +(2-3, =+=08, 33,x+, 240)-(00)).

• Overall theme: mathematicians explore functions, equations, and patterns; engage in problem solving and abstraction. Specific details in this slide are not fully legible.

What is Math? Is it Just About Arithmetic? Is it Just About Numbers? Is it Just a Body of Formulas and Rules?

• Population growth is exponential over time

• Growth rates by period:

  • 1960s: about 2% per year

  • 1990s: about 1.5% per yearMathematics is more than arithmetic or numbers alone; it is a living discipline of patterns, structures, and ideas

  • It is not merely a useless obstacle course in school; it has beauty, creativity, and power

  • It expresses deep connections beyond rote rules

  Is Mathematics a Formal and Boring Science? Its Beauty, Simplicity, and Creativity

  • Mathematics has its own beauty, creativity, and power

  • It reveals patterns, structures, and connections that shape our understanding of the world

  • Beyond formulas and rigid rules, there is imagination and insight

  Patterns and Numbers in Nature and the World

  • Snowflakes exhibit six-fold symmetry and form a wide variety of intricate shapes; no two snowflakes are exactly alike

  • The concept of six-fold symmetry is a key pattern in natural forms

  Kepler and Descartes on Snow Crystals

  • Johannes Kepler studied snowflakes and concluded they can be seen as 6-corner patterns

  • René Descartes described snow crystals using graphs to describe structure

  Koch Snowflakes (Fractal Concept)

  • Koch snowflake demonstrates infinity via self-similarity (fractal property)

  • Shows how simple rules at small scales produce complex, repeating patterns

  Honeycombs

  • Honeycombs illustrate space-filling/close packing of cells with no gaps

  • Efficient tiling by hexagons in a plane

  Pappus of Alexandria

  • Greek mathematician who highlighted that triangles, squares, and hexagons tile the plane without gaps

  • These shapes were used to maximize honey storage/material use in building

  Tiger Stripes: Natural Pattern Modeling

  • Tiger stripes are parallel and evenly spaced; orientation can be modeled by a master equation

  • Harvard researchers developed a master model with three key conditions controlling stripe orientation:

    • Production Gradient: substance amplifies stripe pattern density

    • Parameter Gradient: substance changes one of the parameters involved in forming stripes

    • Physical Change in the direction of the cellular origin of the stripes

  World’s Population (Patterns in Nature and Data Rates)

  • 2015: about 1% per year

• Projections: by 2050, as many as 10 billion people on Earth

Weather Prediction (Computational Power and Data)

• Weather prediction involves 2.8 quadrillion mathematical calculations per second by computers the size of a school bus

• Data collected by these computers include:

  • Temperature, Air pressure, Moisture, Wind speed, Water levels

Fibonacci Numbers and Their Natural Occurrence

• Definition: every number after the first two is the sum of the two preceding

• Sequence example: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$

• General recurrence: $F<em>n = F</em>{n-1} + F<em>{n-2},\quad F</em>1 = 1,\ F_2 = 1$

Applications of Fibonacci Numbers

• Sunflower: the number of clockwise and counterclockwise spirals are consecutive Fibonacci numbers

• Snail shells and the golden spiral: logarithmic spirals appearances tied to the golden mean

• Flower petals: many petals counts follow Fibonacci numbers:

  • Lilies and Irises — 3 petals

  • Buttercups — 5 petals

  • Delphiniums — 8 petals

  • Corn marigolds — 13 petals

  • Aster — 21 petals

  • Daisies — 34, 55, or 89 petals

Patterns in Nature: Universal Language of Mathematics

• Mathematics helps explain patterns in nature, science, and everyday life

• Mathematics is described as a universal language that reveals order beneath apparent chaos

• Examples across domains:

  • Nature: Fibonacci spirals in shells, branching in trees, honeycombs in bees

  • Science: Planetary orbits, sound waves, atomic structures

  • Everyday life: Time, finance, architecture, art, technology

Musical and Mathematical Symbols

• Musical symbols represent musical ideas; mathematical symbols represent mathematical ideas

• Common symbols shown include:

  • = (equal), ≠ (not equal), < (less than), > (greater than)

  • / (division), π (pi), ∑, ∞ (infinity), √ (square root), ∛, ∜ (roots), parentheses (), brackets [], braces {}

  • plus +, minus -, multiplication (× or implicit), and related operators

  • superscripts and subscripts used for exponents and indexing

The Relationship: Physical World vs Abstract Mathematics

• We live in a physical world full of mathematical patterns, yet mathematics resides in the abstract realm of ideas

• The physical world displays patterns described by mathematics (nature, physics, technology), while mathematics itself is an abstract construct

• Examples of the physical-to-abstract relationship:

  • Nature: Spirals, branching, symmetry, Fibonacci sequences

  • Physics: Gravity, waves, motion follow precise mathematical laws

  • Technology: Computers, AI, and engineering rely on mathematical algorithms

Which One is Different? (Practice Exercise)

• Visual exercise: identify the different item among A, B, C (three slides labeled A, B, C)

• Repeats on Pages 27–29 with slight variations; intended as a quick pattern-recognition check

Hallmarks of Mathematics

• The defining features that make mathematics unique:
  1) Logical Reasoning
  2) Abstraction
  3) Precision and Rigor
  4) Pattern Recognition
  5) Universality
  6) Creativity and Beauty
  7) Applicability

Examples of Hallmarks

• Logical Reasoning: e.g., if a = b and b = c, then a = c (deductive reasoning)

• Precision and Rigor: e.g., the Pythagorean theorem is always true for right-angled triangles; truth by proof, not opinion

Mathematics as a Lens on the World

• Mathematics provides tools to make sense of the physical/perceptual world

• Applications by domain:

  • Physics: gravity, motion, energy

  • Biology: population models, genetics, patterns in living things

  • Psychology: statistics to analyze behavior and decision-making

  • Daily life: budgeting, architecture, music, weather prediction

What Is Reality? How Do We Know What Is Real?

• The classic philosophical problem: the chair’s chair-ness and the problem of universals and essence

• The deeper question: what makes a chair a chair—the shape, the function, the material, or an abstract essence?

Senses and Perception: Can They Deceive?

• Senses can deceive; perception is brain-constructed interpretation, not a perfect mirror of reality

• Examples of limits:

  • Dreams feel real until waking

  • Optical illusions (e.g., a stick appearing bent in water)

  • Taste and smell alter when sick

The Role of the Mind: Reason, Intuition, and Mathematics

• To uncover the ideal world, we rely on intellect, reason, intuition, and mathematics

• Plato’s theory of Forms: physical objects are imperfect copies of ideal forms

• For mathematics, the ideal world contains eternal, unchanging truths

Plato’s Two Levels of Reality

• Physical World: Imperfect, temporary, changing

• Ideal World (World of Forms): Perfect, eternal, unchanging truths

Circle: Physical Drawing vs Ideal Circle

• Physical circle: can be drawn but may be imperfect (rough edges, measurement error)

• Ideal circle: set of points at a fixed distance from the center: ${ x \in \mathbb{R}^2 : |x - O| = r }$

Exercise: What Is a 4-Dimensional Object? Can We Create It in Our Minds?

• Reflection prompt to explore higher dimensions and mental visualization

Two Views on Mathematics

• Platonism: Mathematical objects are real; mathematicians discover mathematics

• Formalism: Mathematics is created by humans from axioms and symbolic rules; mathematics is a formal game of deduction

Platonism (Mathematical Objects Are Real)

• Math exists independently of humans; we discover mathematical truths

• Example: The value of $\pi$ existed before humans; its properties are objective

• Supporting quote: “If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.” – Richard Feynman

Formalism (Mathematics Is Created)

• Idea: Mathematics builds from axioms and logic; mathematics as a constructed system

• Historical roots: Bertrand Russell and Alfred North Whitehead (Principia Mathematica) and formal logical foundations

• Gödel’s incompleteness: 1931 result showing that any sufficiently powerful formal system cannot prove all truths and is incomplete; holes in the system

• Famous quotes: mathematical beauty and deductive rigour are central to formalism

Mathematics in the Real World: Selected Examples

• Shannon Sampling Theorem: signals can be reconstructed from discrete samples if sampling rate exceeds twice the maximum frequency (Nyquist rate)

  • Formula (conceptual): $x(t) = \sum_{n=-\infty}^{\infty} x(nT) \cdot \mathrm{sinc}\left(\frac{t - nT}{T}\right), \quad T \le \frac{1}{2B}$ where the signal is bandlimited to $B$ Hz

• Applications: communication systems (cell phones), data transmission, digital audio/video

• Heart models: mathematical models of the heart to design and implement artificial valves

• Flocking and collective motion: mathematics explains how bird flocks fly cohesively without collisions via equations governing alignment and spacing

• Networks and graphs: the brain as a neural network; the Tokyo subway system uses graph theory, networks, and queuing theory for safe, efficient scheduling

• Round manholes: safety reason – round covers cannot fall through their openings; squares/rectangles could in diagonal orientations

• Wheels: efficiency of rolling motion reduces friction and energy use

• Typhoons: pattern analysis and prediction (listed as a topic, specifics not elaborated in the transcript)

• Mona Lisa: mathematical curiosity; geometry and perception themes (mentioned but not elaborated in detail)

• Three-legged chair stability: center of gravity analysis; a three-legged chair is statically stable on uneven ground, whereas four-legged chairs can wobble if not perfectly level

• Map coloring: color maps (e.g., Japan) require color assignments with constraints; example given of coloring the map of Japan

Colorful Symbols and Notation in Mathematics

• Understanding how symbols encode ideas is central to mathematical thinking

• Examples of symbols and their purposes include equality, inequalities, arithmetic operations, roots, and various brackets and parentheses

Final Takeaway: Mathematics as Pattern, Structure, and Tool

• Mathematics helps us understand patterns in nature, science, and society

• It serves as a universal language to describe order hidden beneath apparent chaos

• The discipline blends beauty, rigor, creativity, and practical applicability across domains