Mathematics in Our World Vocabulary

Objectives and Learning Outcomes

• Identify patterns in nature and regularities in the world.

• Articulate the importance of mathematics in one's life.

• Argue about the nature of mathematics: what it is, how it is expressed, represented, and used.

• Express appreciation for mathematics as a human endeavor.

• Discuss how mathematics serves as a tool for observing patterns and predicting the future behavior of objects in nature.

• Discuss the Fibonacci sequence and related concepts, identifying real-life examples.

• Identify relationships in nature and find their logical connections to form generalizations and make predictions.

• Generate patterns by performing one or several mathematical operations repeatedly.

Philosophical Perspectives on Mathematics: The Blind Men and the Elephant

• The parable "The Blind Men and the Elephant" by John G. Saxe ($1816$–$1887$) illustrates how different individuals perceive aspects of a complex whole based on limited observation:

  • Blind Man $1$ touches the side: "The elephant is like a wall."

  • Blind Man $2$ touches the tusk: "The elephant is like a spear."

  • Blind Man $3$ touches the trunk: "The elephant is like a snake."

  • Blind Man $4$ touches the leg or knee: "The elephant is like a tree."

  • Blind Man $5$ touches the ear: "The elephant is like a fan or mat."

  • Blind Man $6$ touches the tail: "The elephant is like a rope."

• In mathematics, individual perspectives often capture specific facets (such as arithmetic, geometry, or equations) without immediately grasping the complete unified structure.

Fundamentals and Scope of Mathematics

• Conceptual Definitions of Mathematics:

  • A universal language understood across cultures and disciplines.

  • A system encompassing sensory experience: everything eyes see, ears hear, and human perception senses in the physical environment.

  • A multi-faceted communication system expressing ideas through patterns, shapes, music, and symbolic language.

  • The study of hidden, extraordinary numerical patterns created by nature and the universe.

  • Formal definition by Ian Stewart ($1995$): A formal system of thought for recognizing, classifying, and exploiting patterns.

• Role of Natural Patterns:

  • Nature's patterns are not merely aesthetic features to be admired; they provide vital clues to the underlying laws and rules governing physical processes.

  • Cognitive workflow: Recognize clues of patterns in nature $\rightarrow$ Deduce underlying rules and regularities.

• Primary Domains and Notions:

  • Numbers, counting, and fundamental arithmetic operations.

  • Numeric and geometric patterns.

  • Patterns of motion, change, and physical trajectories.

Locations and Applications of Mathematics

• Hints and Clues in Nature:

  • Flower petal counts, leaf arrangements, landscape topographies, coat patterns of giraffes, and water flow dynamics.

• Daily Routine and Time Management:

  • Allocating travel time to avoid tardiness.

  • Tracking time spent showering, eating, dressing, and preparing supplies.

  • Managing personal finances, budgeting for fare, food, school supplies, and leisure.

  • Performing routine activities involves computing numbers knowingly or unknowingly, consciously or unconsciously.

• Professional Work and Human Endeavors:

  • Computer-based analysis, documentation, and office communication.

  • Technical construction, roofing, drilling, and structural engineering.

  • Volunteer work, community organization, and collaborative group labor.

• Educational Events and Social Structures:

  • Interactive math laboratories, geometry construction projects using straws and links, giant dice probability games, and school-wide mathematics weeks.

Primary Purposes of Mathematics

• Unraveling natural puzzles and providing a structured framework for thinking about nature.

• Organizing patterns, regularities, and irregular phenomena.

• Predicting future behaviors, physical events, and trends.

• Assisting in controlling external conditions, such as weather forecasting and epidemic modeling.

• Providing powerful tools for numeric computation and algebraic manipulation.

• Generating new intellectual questions, theories, and avenues of thought.

Elements and Methods of Doing Mathematics

• Core Components:

  • Numbers, symbols, and mathematical notations.

  • Operations, equations, and functional representations ($f(x)$).

  • Abstraction processes and the "thingification" of abstract procedures.

  • Rigorous mathematical proof.

• Mindset and Methodology:

  • Cultivating driven curiosity and an inquisitive search for generalities.

  • Seeking absolute logical truth.

  • Utilizing systematic trial and error.

  • Embracing complex problems and facing new questions without fear.

• User Demographics:

  • Everyday individuals navigating life tasks.

  • Pure and applied mathematicians expanding theoretical or functional frameworks.

  • Natural and social scientists analyzing physical, biological, or human data.

Importance and Historical Foundations

• Foundational Value:

  • Mathematics puts order into disorder.

  • Enhances logical reasoning, helping individuals become better thinkers.

  • Improves societal functioning and makes the world a better place.

  • Reveals natural simplicity and enables general principles to be extracted from simple examples to explain complex real-world dynamics.

• Historical Figures and Breakthroughs:

  • The Pythagoreans: Taught that the universe's nature is fundamentally mathematical, believing whole numbers and their ratios describe and represent all natural occurrences.

  • Johann Carl Friedrich Gauss: Recognized as the "Prince of Mathematicians," making foundational contributions to number theory, statistics, analysis, and differential geometry.

  • Blaise Pascal: Co-developed probability theory and structured Pascal's Triangle for binomial coefficients.

• Isaac Newton: Formulated the universal Laws of Motion and calculus, authoring Philosophiae Naturalis Principia Mathematica.

• Gottfried Wilhelm Leibniz: Independently developed modern calculus and its refined notation system.

• René Descartes: Developed the Cartesian Coordinate System, bridging algebra and geometry to enable mapping and spatial navigation.

• Albert Einstein: Formulated the Theory of Relativity, transforming modern physics and the understanding of spacetime.

• Marie Curie: Conducted pioneering research on radioactivity, discovering the elements Polonium and Radium.

• Johannes Kepler: Discovered the laws of planetary motion, writing: "Those laws [of nature] are within the grasp of the human mind; God wanted us to recognize them by creating us after his own image so that we could share in his own thoughts."

Mathematical Patterns: Symmetry and Rotational Calculations

• Definition of Pattern:

  • Regular, repeated, or recurring forms and designs that allow humans to identify relationships, establish logical connections, form generalizations, and make predictions.

• Bilateral Symmetry:

  • Occurs when an imaginary line (axis of symmetry) drawn across an object divides it into two parts that are exact mirror images.

  • Examples: Butterfly wing structures, the human body as illustrated in Leonardo da Vinci's Vitruvian Man.

• Rotational Symmetry:

  • Occurs when an object can be rotated around a central point by an angle less than $360^\circ$ and still look identical to its original position.

  • Described by the order of rotation ($n$-fold symmetry), meaning $\frac{1}{n}$ of a complete turn leaves the figure unchanged.

• Formula for Angle of Rotation:   $\text{Angle of Rotation} = \frac{360^\circ}{n}$

  • Example (Snowflake): A snowflake exhibits a $6$- fold rotational pattern ($n = 6$).     $\text{Angle of Rotation} = \frac{360^\circ}{6} = 60^\circ$

The Packing Problem: Square vs. Hexagonal Packing

• Concept:

  • The packing problem seeks the most efficient method to fill a given two-dimensional or three-dimensional space with uniform objects (such as circles or spheres).

• Mathematical Comparison:

  • Consider uniform circles of radius $r = 1\,\text{cm}$, each possessing an area of:     $A_{\text{circle}} = \pi r^2 = \pi(1\,\text{cm})^2 = \pi\,\text{cm}^2$

  • Square Packing:

  • Each circle is enclosed within a square cell of side length $s = 2r = 2\,\text{cm}$.

  • Area of square cell:       $A_{\text{square}} = s^2 = (2\,\text{cm})^2 = 4\,\text{cm}^2$

  • Packing Efficiency Density:       $\text{Density}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} \times 100\% = \frac{\pi\,\text{cm}^2}{4\,\text{cm}^2} \times 100\% \approx 78.54\%$

  • Hexagonal Packing:

  • Formed by equilateral triangles of side length $s = 2r = 2\,\text{cm}$.

  • Area of one equilateral triangle:       $A_{\Delta} = s^2 \times \frac{\sqrt{3}}{4} = (2\,\text{cm})^2 \times \frac{\sqrt{3}}{4} = 4\,\text{cm}^2 \times \frac{\sqrt{3}}{4} = \sqrt{3}\,\text{cm}^2$

  • A regular hexagon contains $6$ equilateral triangles:       $A_{\text{hexagon}} = 6 \times A_{\Delta} = 6\sqrt{3}\,\text{cm}^2$

  • Inside this hexagonal unit cell, the combined area of enclosed circle sectors equals $3$ full circles:       $A_{\text{circles enclosed}} = 3\pi\,\text{cm}^2$

  • Packing Efficiency Density:       $\text{Density}_{\text{hexagonal}} = \frac{3\pi\,\text{cm}^2}{6\sqrt{3}\,\text{cm}^2} \times 100\% = \frac{\pi}{2\sqrt{3}} \times 100\% \approx 90.69\%$

• Conclusion: Hexagonal packing fills approximately $90.69\%$ of available space compared to $78.54\%$ for square packing. This maximizes space utilization and minimizes wax usage, explaining why honeybees build hexagonal cells for honeycombs.

The Fibonacci Sequence

• Historical Background:

  • Named after Leonardo Bigollo Pisano (also known as Leonardo of Pisa or Fibonacci, meaning "Leonardo the Traveler from Pisa").

  • In $1202$, after traveling across Arab and Eastern countries, he published Liber Abaci (Book of Calculations).

  • Introduced modus Indorum (the method of the Indians), establishing the Hindu-Arabic numerical system in Europe and showing its efficiency over Roman numerals.

• Fibonacci's Rabbit Problem:

  • Problem Statement: "At the beginning of a month, you are given a pair of newborn rabbits. After a month the rabbits have produced no offspring; however, every month thereafter, the pair of rabbits produces another pair of rabbits. The offspring reproduce in exactly the same manner. If none of the rabbits dies, how many pairs of rabbits will there be at the start of each succeeding month?"

  • Monthly Pair Progression:

  • January (Month $1$): $1$ pair

  • February (Month $2$): $1$ pair

  • March (Month $3$): $2$ pairs

  • April (Month $4$): $3$ pairs

  • May (Month $5$): $5$ pairs

  • June (Month $6$): $8$ pairs

  • July (Month $7$): $13$ pairs

  • August (Month $8$): $21$ pairs

• Formal Recursive Definition:   $F_1 = 1, \quad F_2 = 1$   $F_n = F_{n-1} + F_{n-2} \quad \text{for } n \ge 3$

  • Sequence terms: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots$

• Worked Examples:

  • Example 1: Generate a $7$-term Fibonacci-style sequence where $F_1 = 9$.

  • $F_1 = 9$

  • $F_2 = 9$

  • $F_3 = F_2 + F_1 = 9 + 9 = 18$

  • $F_4 = F_3 + F_2 = 18 + 9 = 27$

  • $F_5 = F_4 + F_3 = 27 + 18 = 45$

  • $F_6 = F_5 + F_4 = 45 + 27 = 72$

  • $F_7 = F_6 + F_5 = 72 + 45 = 117$

  • Resulting Sequence: $9, 9, 18, 27, 45, 72, 117$

  • Example 2: Generate a $5$-term Fibonacci-style sequence where $F_1 = -3$.

  • $F_1 = -3$

  • $F_2 = -3$

  • $F_3 = -3 + (-3) = -6$

  • $F_4 = -6 + (-3) = -9$

  • $F_5 = -9 + (-6) = -15$

  • Resulting Sequence: $-3, -3, -6, -9, -15$

  • Example 3: Given $F_{16} = 987$ and $F_{17} = 1597$, compute $F_{18}$.     $F_{18} = F_{17} + F_{16} = 1597 + 987 = 2584$

  • Example 4: Given $F_{22} = 43$ and $F_{23} = 75$, compute $F_{26}$.

  • $F_{24} = F_{23} + F_{22} = 75 + 43 = 118$

  • $F_{25} = F_{24} + F_{23} = 118 + 75 = 193$

  • $F_{26} = F_{25} + F_{24} = 193 + 118 = 311$

The Golden Ratio (\phi) and Natural Occurrences

• Definition and Value:

  • As $n$ increases, the ratio of consecutive Fibonacci numbers $\frac{F_n}{F_{n-1}}$ approaches the Golden Ratio:     $\phi \approx 1.61803399$

• Consecutive Ratios Progression Summary:

  • $\frac{F_4}{F_3} = \frac{3}{2} = 1.50000000$

  • $\frac{F_5}{F_4} = \frac{5}{3} \approx 1.66666667$

  • $\frac{F_6}{F_5} = \frac{8}{5} = 1.60000000$

  • $\frac{F_7}{F_6} = \frac{13}{8} = 1.62500000$

  • $\frac{F_8}{F_7} = \frac{21}{13} \approx 1.61538462$

  • $\frac{F_9}{F_8} = \frac{34}{21} \approx 1.61904762$

  • $\frac{F_{10}}{F_9} = \frac{55}{34} \approx 1.61818182$

  • $\frac{F_{16}}{F_{15}} = \frac{987}{610} \approx 1.61803445$

  • $\frac{F_{28}}{F_{27}} = \frac{317811}{196418} \approx 1.61803399$

• Natural Manifestations of the Golden Ratio ($\phi$):

  • Seed patterns in sunflowers (counter-rotating spirals).

  • Chamber growth in nautilus shells.

• Spiral leaf growth in succulents and cacti.

• Arm structure in spiral galaxies.

• Human anatomical proportions (such as ear curvature and facial proportions).

• DNA double helix structural dimensions (width-to-turn ratio yielding approximately $1.00$ to $0.618$ and $0.618$ to $0.382$ proportions).

Explicit Calculations: Binet's Formula

• Historical Background:

  • Named after French mathematician, physicist, and astronomer Jacques Philippe Marie Binet ($1786$–$1856$).

• Formula Statement:

  • For $n \ge 1$, the $n^{\text{th}}$ Fibonacci number $f_n$ is calculated directly without recursion using:     $f_n = \frac{1}{\sqrt{5}} \left[ \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right]$

• Worked Example ($n = 40$):

  • Find $f_{40}$ using Binet's Formula:     $f_{40} = \frac{1}{\sqrt{5}} \left[ \left( \frac{1 + \sqrt{5}}{2} \right)^{40} - \left( \frac{1 - \sqrt{5}}{2} \right)^{40} \right]$

  • Computing exponential terms:     $\left( \frac{1 + \sqrt{5}}{2} \right)^{40} \approx 228,826,127$     $\left( \frac{1 - \sqrt{5}}{2} \right)^{40} \approx 0$

  • Division by $\sqrt{5}$:     $f_{40} \approx \frac{228,826,127}{\sqrt{5}} = 102,334,155$

  • Recursive equivalent:     $f_{40} = f_{39} + f_{38} = 102,334,155$

Mathematical Modeling: Population Growth

• Global Population Context:

  • Global human population exceeded $8,045,311,447$ as of mid-year $2023$.

• Exponential Growth Model Formula:   $A = P e^{r t}$

  • Where:

  • $A$ = final population size after growth.

  • $P$ = initial population size.

  • $r$ = annual rate of growth.

  • $t$ = elapsed time in years.

  • $e$ = Euler's constant ($\approx 2.718$).

• Worked Example (City Population Model):

  • Let the population of a city in the Philippines (in thousands) be modeled by $A = 30 e^{0.02 t}$, where $t$ is the number of years after $1995$.

  • Problem (A): Find the city population in $1995$ ($t = 0$).     $A = 30 e^{0.02(0)} = 30 e^0 = 30(1) = 30$

  • Conclusion: The city population in $1995$ was $30\,\text{thousand}$ ($30,000$ people).

  • Problem (B): Find the city population in $2023$ ($t = 2023 - 1995 = 28$).     $A = 30 e^{0.02(28)} = 30 e^{0.56}$     $e^{0.56} \approx 1.7506725$     $A = 30 \times 1.7506725 = 52.520175$

  • Conclusion: The city population in $2023$ ($28$ years after $1995$) is approximately $52.52\,\text{thousand}$ ($52,520$ people).