Mathematics in the Modern World Notes

The Nature of Mathematics

  • Mathematics is the study of numbers.
  • It evolves into the study of numbers and shapes.
  • Then becomes the study of numbers, shape, motion, change and space.
  • Expands to include the mathematical tools used in the study of numbers, shape, motion, change and space.
  • Mathematics is the science of patterns.

Mathematics as the Science of Patterns

  • Arithmetic and Number Theory study patterns of number and counting.
  • Geometry studies patterns of shape.
  • Calculus allows us to handle patterns of motion.
  • Logic studies patterns of reasoning.
  • Probability theory deals with patterns of chance.
  • Topology studies patterns of closeness and position.

Where Mathematics is Found

  • Hints or clues in nature.
  • Daily routines.
  • Work environments.
  • People and communities.
  • Events.

Purpose of Mathematics

  • Helps unravel the puzzles of nature.
  • Provides tools to understand everything around us.

How Mathematics is Done

  • With curiosity.
  • With a penchant for seeking patterns and generalities.
  • With a desire to know the truth.
  • With trial and error.
  • Without fear of facing more questions and problems to solve.

Mathematics in the World: Patterns and Numbers in Nature

  • Patterns in nature are visible regularities of form found in the natural world and can also be seen in nature.
  • These patterns recurring in different contexts can sometimes be modeled mathematically.
  • Humans have developed a formal system of thought (mathematics) for recognizing, classifying, and exploiting patterns.
  • Applying mathematics helps organize and systematize ideas about patterns, leading to the discovery of patterns in nature.
  • Nature patterns are vital clues to the rules that govern natural processes, not just for admiration.
  • Patterns are observed in:
    • Stars moving in circles across the sky.
    • Weather seasons cycling each year (winter, spring, summer, fall).
    • Snowflakes, which exhibit six-fold symmetry with no two being exactly the same.
    • Fish patterns (e.g., spotted trunkfish, spotted puffer, blue spotted stingray, spotted moray eel, coral grouper, red lion fish, yellow box fish and angel fish).
    • Animal patterns: stripes on zebras, tigers, cats, and snakes; spots on leopards and hyenas; blotches on giraffes.
    • Natural patterns: intricate waves across the oceans, sand dunes on deserts, formation of typhoons, and water drop ripples.
    • Other patterns: schools of mackerel, V-formation of geese, tornado formation of starlings.
    • Locomotion: scuttling of insects, flight of birds, pulsations of jellyfish, wavelike movements of fish, worms, and snakes.

Patterns in Everyday Events

  • Daily routines.
  • Traffic.
  • Spending patterns.
  • Weather patterns.
  • Sleeping patterns.

Elements of Pattern

  • Repetition – involves a collection of distinct objects of some sort.
  • Regularity – elements repeat in a predictable manner.
  • Rule – order and structure/homogeneity/transformation.
  • Generator – element that is repeated following a specific rule.

Geometric Transformations

  • Translations
  • Reflection
  • Rotation
  • Dilation

Mathematical Patterns - Pattern Recognition

Fibonacci Sequence

  • Discovered by Leonardo of Pisa (Fibonacci).
  • The sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
  • Each number is obtained by adding the last two numbers of the sequence.
  • Rabbit problem: A man puts a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits are produced from that pair in a year, if it is supposed that every month each pair produces a new pair, which from the second month onwards becomes productive?
  • The number of pairs of mature rabbits living each month determines the Fibonacci sequence.
  • Formula:
    Fn = 1 if n = 0;
    Fn = 1 if n = 1;
    Fn = Fn-1+ Fn-2 if n>1.
  • Flower petals often exhibit Fibonacci numbers:
    • White calla lily: 1 petal
    • Euphorbia: 2 petals
    • Trillium: 3 petals
    • Columbine: 5 petals
    • Bloodroot: 8 petals
    • Black-eyed Susan: 13 petals
    • Shasta daisy: 21 petals
    • Field daisies: 34 petals
    • Other daisies: 55 and 89 petals
  • Sunflower seeds convey the Fibonacci sequence through spirals in opposing directions (clockwise and counterclockwise).
    • The number of clockwise and counterclockwise spirals are consecutive Fibonacci numbers, usually 34 and 55.
  • Pineapples have spirals formed by their hexagonal nubs, often forming 5 and 8 spirals or 8 and 13 spirals that rotate diagonally upward to the right.
  • Pine cones also exhibit spirals from the center, having 5 and 8 arms or 8 and 13 arms, depending on the size.

Golden Ratio

  • Two quantities are in the Golden Ratio if their ratio is the same as the ratio of their sum to the larger of two quantities.
  • Formula:
    \frac{a+b}{a} = \frac{a}{b} = \varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887…

Golden Rectangle

  • The numbers in a Fibonacci sequence can be applied to the proportions of a rectangle, called the Golden Rectangle.
  • The Golden Ratio is visually satisfying.

Golden Spiral

  • The Golden Rectangle is related to the Golden Spiral, which is created by making adjacent squares of Fibonacci dimensions.
  • A Fibonacci spiral approximates the Golden Spiral using Fibonacci sequence squares.

Golden Ratio in Nature

  • The Golden Ratio relates to human beauty, aesthetics, and growth patterns in plants and animals.
  • Positions and proportions of key dimensions of many animals are based on Phi.
  • Examples: horn of ram, wing dimensions and location of eye-like spots on moths, body sections of ants and other insects, body features of animals, spirals of sea shells.
  • The growth pattern on branches of trees is Fibonacci.
  • The human face contains spirals, and human DNA contains phi proportions.