GEC002 – Mathematics in the Modern World: Fibonacci, Golden Ratio, and Mathematics in Nature

Course Framework and Administrative Details

• Course Code & Name: GEC002 – Mathematics in the Modern World
• Catalogue Description
  • Investigates mathematics as a language, an art, and a set of tools for daily‐life decision-making.
  • Emphasises pattern recognition, inductive & deductive reasoning, aesthetic appreciation, and applications to finance, voting, coding, resource division, etc.
  • Based on CHED CMO No. 20 s. 2013.
• Intended Learning Outcomes
  • Demonstrate mathematical ideas in analysing real-life problems.
  • Construct persuasive, ethically guided arguments via mathematical & statistical reasoning.
• Macro Schedule
  • Preliminary: Mathematics in our World; Mathematical Language & Symbols; Problem Solving & Reasoning.
  • Midterm: Geometric Designs; Mathematics of Graphs; Mathematical Systems; Linear Programming.
  • Final: Data Management; Voting & Apportionment; Mathematics of Finance.
• Assessment Schema
  • 70 % – Examinations
  • 30 % – Other Teaching/Learning Activities & Assessment Tasks.
• Instructor: Angel Constantine “AC” Bajana, LPT, MAEd, Asst. Prof III, College of Arts & Humanities
  • BSED Physics (MSU–IIT); MAEd Teaching Mathematics (Univ. of Mindanao); MS Teaching Math (CAR) – MSU.
  • CAH Coordinator for Community Development Services.

Mathematics: Essence, Purpose, and Reach

• What is Mathematics?
  • Abstract study of numbers, quantities, shapes, patterns.
  • Rooted in logical reasoning & systematic inquiry.
  • Main branches cited: algebra, geometry, calculus, etc.
  • Major historical figures: Euclid, Pythagoras, Newton.
• What is Mathematics For?
  • Universal language for expressing/analysing patterns & structures.
  • Enables modelling & solution of complex problems across physics, engineering, economics, and virtually all sciences.
• Where is Mathematics?
  • Omnipresent in nature: snowflake symmetry, seashell spirals, planetary orbits.
  • Transcends cultural/geographical boundaries.
• What is Mathematics About?
  • Discovering patterns → forming conjectures → proving theorems.
  • Involves abstraction, reasoning, and problem-solving.
  • Reflects humanity’s quest to uncover governing principles of the universe.
• Who Uses Mathematics?
  • Scientists, engineers, economists, architects, artists, etc.
  • Illustrative icons: Carl Friedrich Gauss (multi-field contributions), Ada Lovelace (computing).
• Why Learn Mathematics?
  • Develops critical thinking, logical reasoning, and problem-solving mindset.
  • Essential for thriving in a tech-driven, data-rich world.
  • Historical milestones—from ancient Greek formalisation to Renaissance revival—show its enduring necessity.
• How is Mathematics Done?
  • Blend of creativity (intuition) and rigour (proof).
  • Process: explore → conjecture → verify → refine.
  • Collaboration and cumulative evolution, e.g., rise of calculus.

Mathematics Embedded in Nature

• General Principle: Natural forms often optimise growth, packing, or aesthetics, leading to recurring mathematical patterns.
• Representative Patterns
  • Spiral galaxies → logarithmic spirals.
  • Tree branching → fractal‐like recursion.
  • Phyllotaxis (leaf arrangement) → Fibonacci angles.

The Fibonacci Sequence

• Definition & Recurrence
  • Starts: $0,\ 1,$ then $F<em>{n+2}=F</em>{n+1}+F_n$.
  • First few: $0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ 89,\ 144,\ 233,\ 377,\ 610,\ 987,\ \dots$
• Origin Story
  • Modelled rabbit population in Fibonacci’s 1202 text Liber Abbaci.
• Biographical Snapshot: Leonardo “Fibonacci” Pisano
  • Born ≈ $1175\,\text{AD}$ in Pisa; educated under North‐African Moors.
  • Travelled Mediterranean, studied merchant arithmetic.
  • Recognised superiority of Hindu–Arabic numerals over Roman numerals.
  • Authored five surviving works: Liber Abbaci, Practica Geometriae, Flos, Liber Quadratorum, and a letter to Master Theodorus.
• Key Contribution: Introduced Hindu–Arabic number system (digits $1!\text{–}!9,0$, place value, decimal point) to Europe; persuaded contemporaries to abandon cumbersome Roman numerals.
• Appearances in Nature
  • Flower petal counts:
  – Lilies/Irises = $3$ petals
  – Buttercups/Wild roses = $5$ petals
  – Corn marigolds = $13$ petals
  – Black‐eyed Susans = $21$ petals
  • Fruit & cones:
  – Bananas show $3$ or $5$ ridges.
  – Pineapple scales spiral in sets of $8,\ 13,\ 21$.
  • Shells (snail, nautilus) exhibit Fibonacci spiral growth.

The Golden Ratio (Golden Section)

• Numeric Value & Formula
  • Denoted $\varphi$ (“phi”): the positive root of $x^2-x-1=0$
  $\varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803398875$.
  • Relation to Fibonacci: $\displaystyle \lim<em>{n\to\infty} \frac{F</em>{n+1}}{F_n}=\varphi$.
• Segment Definition
  • For line $AB$ divided at $G$: $\frac{AG}{GB}=\frac{AB}{AG}=\varphi$.
  • Diagram referenced: whole = $1$, longer part = $g$, shorter part = $1-g$.
• Golden Rectangle
  • Sides in ratio $\varphi:1$ (≈ $1.618:1$).
  • Empirically judged as “most pleasing” in quick‐choice rectangle tests.
  • Common in playing cards, windows, book covers, file cards, and skyscraper façades.

Aesthetic & Structural Applications

• Design & Fine Arts
  • Artists deliberately embed golden rectangles for visual harmony.
  • Classical Greek statuary often proportioned total height : navel height ≈ $\varphi$.
• Architecture
  • Parthenon façade: width ≈ $0.618 \times$ height (golden section).
  • Great Pyramid of Giza: $\frac{\text{perimeter}}{2\times\text{height}}\approx\varphi$.
  • Notre Dame (Paris) rose window & elevation grids.
  • Modern examples: United Nations Secretariat Building.
• Music
  • Instrument-making:
  – Stradivari placed violin f-holes using golden ratio.
  – Baginsky designed contour & arch via $\varphi$.
  • Composition structure:
  – Mozart split sonata form at $\varphi$: Exposition $=38$ measures, Development + Recapitulation $=62$ measures, $\frac{38}{38+62}\approx0.38$ and $\frac{62}{100}=0.62$ (complements of $\varphi^{-1}$).
  – Beethoven’s 5th Symphony: major thematic returns occur at $0.618$ and $\varphi$ fractions through movements.

Ethical, Philosophical, and Practical Reflections

• Mathematics teaches disciplined reasoning that underpins ethical argumentation.
• The universality of patterns (e.g., Fibonacci, $\varphi$) fosters philosophical awe at the cosmos’ intelligibility.
• Practical mastery—whether budgeting, voting‐systems design, or cyber-security—relies on the same core mathematical literacy nurtured in this course.

Key Formulae, Constants, and Notation Reference

• Fibonacci recursion: $F<em>{n+2}=F</em>{n+1}+F_n$.
• Golden ratio: $\varphi = \frac{1+\sqrt{5}}{2}\approx1.61803$.
• Convergence property: $\displaystyle \lim<em>{n\to\infty}\frac{F</em>{n+1}}{F_n}=\varphi$.
• Quadratic generating $\varphi$: $x^2 - x - 1 = 0$.

Study Tips Based on Lecture Emphasis

• Memorise early Fibonacci terms and practise locating them in natural objects.
• Re-derive $\varphi$ from the quadratic to solidify algebraic manipulation skills.
• Sketch and label a golden rectangle; practise dividing arbitrary segments into the golden section.
• When analysing design or architecture, compute proportion ratios to check for golden sections.
• Listen to classical works with score in hand to perceive golden-ratio timing—strengthening both mathematical and musical pattern recognition.