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.