MMW

Mathematics in the Modern World

Instructor: Engr. Grant Phill S. delos SantosSchedule: Friday, 12:30 PM – 3:30 PM

Specific Objectives

  1. Understanding Mathematics

    • Grasp the role of mathematics in the modern world.

  2. Revisiting Mathematical Landscape

    • Appreciate the variety and depth of mathematical concepts.

  3. Mathematics as a Utility

    • Recognize the practical applications of mathematics.

  4. Awareness of Our Role

    • Understand the individual and collective roles in mathematics.

The Nature of Mathematics

  • Definition:

    • According to Stewart in "Nature’s Number," mathematics is a gradually developed formal system of thought.

  • Human Insight:

    • Early humans recognized measures and patterns in nature.

    • This led to the understanding that the natural world is rich in mathematical patterns that benefit the natural order.

  • Historical Significance:

    • Mathematics has been crucial in every era of human development.

Mathematics as a Tool

  • Utility of Mathematics:

    • Serves as a practical tool for problem-solving.

  • Perspective Change:

    • Encourages new perspectives to understand problems by merging insights and creativity.

  • Pattern Recognition:

    • Helps to decipher patterns, relationships, and logical connections.

  • Broad Discipline:

    • Encompasses shapes, quantities, arrangements, and more, functioning as a universal language.

Mathematics is Everywhere

  • Useful across various fields:

    • Sciences

    • Humanities

    • Literature

    • Medicine

    • Music and Arts

  • Mathematical Landscape:

    • Comprises numbers, symbols, equations, operations, calculations, abstractions, and proofs.

How Mathematics is Done

  • Building Concepts:

    • Complex concepts derive from simpler ones; strong foundational knowledge is essential.

  • Inclusivity of Mathematics:

    • Mathematics exists in pure and applied forms, serving various professions like engineering, commerce, healthcare, and energy utilization.

Patterns

  • Definition:

    • A pattern is a regular or consistent structure or design.

  • Occurrence of Patterns:

    • Found in nature, human-made designs, and abstract concepts.

  • Mathematical Modeling:

    • The regularities in patterns can be modeled mathematically through

      • Investigation

      • Observation of Relationships

      • Establishment of Logical Connections

      • Generalizations

Types of Patterns

Visual Patterns

  • Often unpredictable, featuring fractals and self-similar replication in nature (e.g., trees and plants).

Flow Patterns

  • Observed in the movement of liquids like water or the growth of trees.

Movement Patterns

  • Regular rhythms in walking; consistent strides in humans and complex patterns in animals.

Rhythm Patterns

  • Fundamental to many biological processes, resembling heartbeats and breathing patterns.

Texture Patterns

  • Various attributes felt upon touch, like rough or smooth surfaces.

Geometric Patterns

  • Regularly repeated shapes observable in nature (e.g., on cacti).

Patterns Found in Nature

Waves and Dunes

  • Waves are disturbances that carry energy; include mechanical waves in air and water.

Spots and Stripes

  • Result from reaction-diffusion systems in organisms.

Spirals

  • Found on cosmic scales and microscopic organisms; ubiquitous in nature.

Fractals

  • Infinitely complex patterns appearing consistently at different scales.

Symmetries

  • Definition of Symmetry:

    • A figure is symmetric if it can be divided into two identical halves.

  • Types of Symmetry:

    • Reflection Symmetry:

      • Left half equals right half.

    • Rotational Symmetry:

      • Remains unchanged when rotated.

    • Translational Symmetry:

      • Identical figures created by repeating units (e.g., honeycomb patterns).

Fibonacci Sequence

  • Presents a sequence of numbers (1, 1, 2, 3, 5, 8, ...) found in various natural forms.

  • Recognized as one of the mathematical mysteries embedded in natural patterns.

Golden Ratio

  • Known as Phi (φ), approximately 1.6180339887, and recognized for its occurrence in nature and art symbolism.