Notes on Patterns and Numbers in Nature

Introduction

• In the 21st century, problems in science and technology are the main focus for scientists and mathematicians. They use mathematical concepts in diverse fields to obtain significant results (e.g., forecasting relationships between variables, developing vaccines, locating earthquake epicenters and estimating damages, creating programs/algorithms).
• Questions posed: Is mathematics a science or an art? What is the nature of the world without mathematics?

What is Mathematics?

• The word “mathematics” comes from the Greek word Mathema, meaning “knowledge” or “learning.”
• Mathematics is the science of patterns, structures, relationships, and logical reasoning.
• It allows us to analyze data, predict trends, solve problems, and describe the world in precise ways.
• Mathematics as a Way of Thinking
• Mathematics as a Universal Language
• Mathematics as a Tool

Mathematics as a Science, Mathematics as an Art

• Mathematics as a Science: a science of numbers and magnitudes.
• Mathematics as an Art: can deduce any data or set of information into forms (graphs, diagrams, statements, figures) without losing informative details.

Mathematics in Nature

• The existence of mathematics in our environment is undeniable.
• Mathematics helps predict the behavior of nature and phenomena in the world.
• Mathematics helps control nature and occurrences in the world for our own ends.

Mathematics in the Modern World

• It is about mathematics as a system of knowing or understanding our surroundings.
• Deals with the nature of mathematics, appreciation of its practical, intellectual, and aesthetic dimensions, and the application of mathematical tools in daily life.

Pattern are regular, repeated, or recurring forms or designs

• Types of Pattern
  • A. Symmetry – an exact correspondence of form and constituent configuration on opposite sides of a dividing line, plane, or about a center/axis.
  • Bilateral or Reflection
• B. Spiral – curved pattern that focuses on a point and a series of circular shapes that revolve around it.
• C. Fractal – built from simple repeated shapes that are reduced in size when repeated.
• D. Tessellations – created with identical shapes that fit together with no gaps.

Pattern and Symmetry: Key concepts

• Patterns indicate structure and organization; humans often perceive intelligent design in nature’s patterns.
• Example prompts from the section:
  • What number comes next in 1, 3, 5, 7, 9, ?
  • What comes next in A, C, E, G, I, ?

Leonardo da Vinci’s Vitruvian Man

• Example illustrating symmetry and proportion in the human body.

Rotational Symmetry and Related Concepts

• Radial Symmetry / Angle of Rotation – the smallest measure of angle that a figure can be rotated while preserving its original position.
• Order of Rotation – a common way of describing rotational symmetry.

Spiral, Fractal, and Tessellations (Continued)

• Spiral – a curved pattern revolving around a center.
• Fractal – a structure built from simple repeated shapes that scale down.
• Tessellations – tilings of the plane with no gaps using identical shapes.

Vedic Math: An Example Calculation

• Example: 23 × 32 = ?
• Direction (conceptual steps):
  • Draw 2 horizontal lines and, below them, draw the multiplicand lines: 23.
  • Draw 3 vertical lines across the horizontal lines and, to the right, draw 2 vertical lines for the multiplier 32.
  • Count the dots/vertices for each group.
  • Group 1 (unit digits): the lower-right group gives 6.
  • Group 2 (diagonal sums): sum diagonally as 4 + 9 = 13; since it exceeds 10, retain the unit digit 3 and carry over 1 to the next group.
  • Group 3 (hundreds/thousands): 6 + 1 = 7.
• Therefore, 23 × 32 = 736.
• Representation from the page suggests the following digits were considered: 6, 4, 9, 6, with the final result 736.

Pascal’s Triangle Explanation

• The digits highlighted (red) serve as the numerical coefficients of the answer.
• Coefficients correspond to the expansion of a binomial expression (e.g., (a + b)^n).
• The digits can appear in different forms corresponding to binomial expansions; coefficients determine the final numeric result.

Fibonacci Sequence

• Overview:
  • Leonard o Fibonacci: Italian mathematician; nickname “Fibonacci” meaning “Son of Bonacci.”
  • Born about 1170; died after 1240 (as per the records shown in the transcript).
  • November 23 is celebrated as Fibonacci Day.
• Sequence terms (starting from 0): $0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ 89,\ 144,\ 233,\ 377,\ 610,\ \dots$
• Page references show petal patterns: 3 petals, 5 petals, 8 petals, 13 petals, 21 petals, illustrating Fibonacci in nature.

Golden Ratio

• The golden ratio φ (phi) = $\phi = \frac{1+\sqrt{5}}{2} \approx 1.618…$
• It is a mathematical proportion that appears in nature, art, and architecture; found in seashell spirals, hurricanes, and the proportions of human faces.

Table: The First 30 Fibonacci Numbers and Approximations to φ

• A table is presented (TABLE II) showing pairs of consecutive Fibonacci numbers Fn and Fn+1 and their ratios Fn+1/Fn converging to φ.
• Key takeaway: as n grows, Fn+1/Fn → φ.
• Example values (illustrative, not exhaustive):
  • F1 = 1, F2 = 1, ratio F2/F1 = 1.0
  • F3 = 2, ratio F3/F2 = 2.0
  • F4 = 3, ratio F4/F3 = 1.5
  • F5 = 5, ratio F5/F4 = 1.6667
  • F6 = 8, ratio F6/F5 = 1.6
• The table in the source shows precise floating approximations of φ as the sequence progresses.

Fibonacci Sequence Formulae

• Recursive definition (sequence formula):
  • $F<em>n = F</em>{n-1} + F<em>{n-2}, \quad n \ge 3,$ with initial conditions $F</em>1 = 1,\ F_2 = 1.$
• Binet’s formula (closed form):
  • Let $\phi = \frac{1+\sqrt{5}}{2},\quad \psi = \frac{1-\sqrt{5}}{2}.$ Then
  • $F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}.$

Example: Computing a Fibonacci Term

• Given the sequence starting with F1 = 1, F2 = 1 and Fn = F{n-1} + F_{n-2},
  • The sequence proceeds as: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
  • Find F9: F9 = 34.
  • Validation via recurrence: F3 = 2, F4 = 3, F5 = 5, F6 = 8, F7 = 13, F8 = 21, F9 = F8 + F7 = 21 + 13 = 34.

Connections and Implications

• Connections to previous lectures/founding principles:
  • Pattern recognition and the formalization of patterns via symmetry, spirals, fractals, and tessellations.
  • The relationship between simple rules (recurrence relations) and complex structures (Fibonacci, natural spirals, golden ratio).
• Real-world relevance:
  • Fibonacci numbers appear in biological settings (arrangements of leaves, seeds, petals).
  • The golden ratio appears in aesthetics and design, as well as natural phenomena.
• Ethical/philosophical considerations:
  • The discussion of intelligent design in patterns prompts reflection on the nature of mathematical order in the universe.
  • Mathematics as a universal language raises questions about its universality across cultures and its role in scientific inquiry.

Key Formulas to Remember

• Pattern/Structure concepts:
  • Angle of rotation: the smallest angle for which a figure maps onto itself under rotation.
  • Order of rotation: the number of times the figure maps onto itself during a full rotation.
• Fibonacci recurrence:
  • $F<em>n = F</em>{n-1} + F_{n-2}, \quad n \ge 3,$
  • $F<em>1 = 1, \ F</em>2 = 1.$
• Binet’s formula:
  • $F_n = \frac{\phi^n - \psi^n}{\sqrt{5}},$
  • $\phi = \frac{1+\sqrt{5}}{2}, \quad \psi = \frac{1-\sqrt{5}}{2}.$
• Golden ratio:
  • $\phi = \frac{1+\sqrt{5}}{2} \approx 1.6180339….$