Mathematics in the Modern World: Comprehensive Study Guide

Institution: Norzagaray College (established $2007$ ), Norzagaray, Bulacan.
Course Title: Mathematics in the Modern World.
Subject Instructor: Ms. Ma. Ricarhaya C. Mangahas, LPT.
Introductory Reflection: Students are prompted to reflect on how they feel about math in one word and identify their personal relationships with the subject.
Personal Reflection Prompts:
- "What's one positive experience you've had with math?"
- "What's one challenging or confusing experience?"

Ancient Mathematics Around the World:
- Egyptians: Credited with the early development of Geometry.
- Babylonian Mathematics: Developed a number system using base $60$ . The transcript includes a visual table of Babylonian cuneiform numerals from $1$ to $59$ .
- Chinese Mathematics: Associated with the use of the Chinese Abacus for calculation.
Greek Mathematical Contributions:
- The Greeks moved beyond simple calculation to study shapes and logic, leading to the development of formal proofs.
- Euclid of Alexandria: Known as the "Father of Geometry."
- Aristotle: Known as the "Father of Logic."
- Archimedes: Known as the "Father of Mathematics."

Dictionary Definition (Oxford Languages):
- Mathematics is the abstract science of number, quantity, and space.
- It can be categorized into two main branches:
  - Pure Mathematics: Studied in its own right.
  - Applied Mathematics: Applied to other disciplines such as physics and engineering.
- The term can refer to the mathematical aspects of specific concepts (e.g., "the mathematics of general relativity").
Etymological Roots:
- The term "Mathematics" is derived from two Greek words:
  - manthanein: Meaning "learning."
  - techne: An art of technique.
- Collectively, mathematics is defined as the "art of learning related to disciplines."

Definition of Patterns: Patterns in nature are visible, regular forms found in the natural world and the universe.
Significance: Nature patterns are not merely for aesthetic admiration; they serve as vital clues to the rules governing natural processes.
Snowflakes: Presented as a primary example of complex geometric patterns in nature.
Types of Patterns:
- Symmetry: A sense of harmonious and beautiful proportion of balance, or a state where an object is invariant to transformations such as reflection, rotation, or scaling.
  - Bilateral Symmetry: A symmetry in which the left and right sides of an organism can be divided into approximately mirror images of each other along a midline.
  - Radial Symmetry (Rotational Symmetry): Symmetry around a fixed point known as the center. It can be classified as either cyclic or dihedral.
- Fractals: A never-ending pattern found in nature. This involves a process called "self-similarity," where the exact same shape is replicated and repeats itself over and over at different scales.
- Spirals: A curved pattern focusing on a center point with a series of circular shapes revolving around it.

Definition: The Fibonacci sequence is a series of numbers where each number is found by adding up the two numbers preceding it ( $F_n = F_{n-1} + F_{n-2}$ ).
Historical Background:
- Named after Fibonacci, also known as Leonardo of Pisa or Leonardo Pisano.
- These numbers were first introduced in his work Liber Abbaci (Book of Calculation) in $1202$ .
Fibonacci in Nature (Flower Petals):
- $1$ Petal: White Calla Lily.
- $2$ Petals: Euphorbia.
- $3$ Petals: Trillium.
- $5$ Petals: Columbine.
- $8$ Petals: Bloodroot.
- $13$ Petals: Black-eyed Susan.
- $21$ Petals: Shasta Daisy.
- $34$ Petals: Field Daisies.
Sunflowers: Sunflower seeds convey the Fibonacci sequence through two sets of spirals moving in opposing directions (clockwise and counterclockwise). The number of these spirals are consecutive Fibonacci numbers, typically containing $34$ and $55$ seeds.

Golden Rectangle: A "perfect rectangle" that can be broken down into squares the size of the next Fibonacci number. Dividing each square with an arc creates a Fibonacci spiral.
The Golden Ratio (Phi):
- First called the "Divine Proportion" in the early $1500s$ .
- Explored by Luca Pacioli in the work De Divina Proportione ( $1509$ ), which featured illustrations by Leonardo da Vinci.
- Mathematical Expression:
  - $\frac{a+b}{a} = \frac{a}{b} = \phi$
  - The numerical value is approximately: $x = \frac{1+\sqrt{5}}{2} = 1.618034...$
Aesthetics and the Golden Ratio (Human Face):
- The Golden Ratio is used to rate facial symmetry and proportions.
- Example: Bella Hadid's facial features and their proximity to the Golden Ratio:
  - Chin: $99.7\%$
  - Eye Position: $97.65\%$
  - Forehead: $97\%$
  - Nose Base/Lip Width: $95.8\%$
  - Face Shape: $94.5\%$
  - Lips: $94.1\%$
  - Nose Width/Length: $93.4\%$
  - Eyebrow: $89\%$
  - Brow Area: $88\%$
  - Overall Golden Ratio Rating: $94.35\%$

Binet's Formula: Used to find the $n^{th}$ Fibonacci number without calculating all the preceding numbers:
- $F(n) = \frac{(\frac{1+\sqrt{5}}{2})^n - (\frac{1-\sqrt{5}}{2})^n}{\sqrt{5}}$
Sequence Exercises:
- Finding the $8^{th}$ Fibonacci number: $Fib(8)$ .
- Finding the $10^{th}$ Fibonacci number: $Fib(10)$ .
- Pattern Recognition Examples:
  - Alphanumeric: $A1, B2, C3, D4, ...$ (Next is $E5$ ).
  - Squares: One, Four, Nine, Sixteen, … ( $1^2, 2^2, 3^2, 4^2$ , next is $25$ ).
  - Arithmetic Decession: $100, 90, 80, 70, ...$ (Next is $60$ ).
  - Increasing Intervals: $2, 6, 12, 20, ...$ (Differences are $+4, +6, +8$ , next is $+10$ which is $30$ ).