ALGEBRA AND NUMBER THEORY:
Modular arithmetic
Application complex numbers
General solution of a cubic equation
Pattern in Pascal’s triangle
Pythagorean triples
Loci and complex numbers
Egyptian fractions
Chinese remainder theorem
Twin primes numbers
Factorable sets of integers of the form ak+b
Combinatorics – art of counting
Root of unity
Recurrence expressions for phi (golden ratio)
Goldbach’s congeture
Diophantine equations
Applications of logarithms
Finding prime numbers
Mersenne primes
Matrices and Cramer’s rule
Complex numbers and transformations
Fermat’s last theorem
Hypercomplex numbers
Euclidean algorithm for GCF
Algebraic congruences
Boolean algebra
Fermat’s little theorem
Probabilistic number theory
Continued fractions
Polar equations
Random numbers
Magic squares and cubes
Divisibility test
Euler’s identity
Natural logarithms of complex numbers
Diophantine applications: Cole numbers
Palindrome numbers
Inequalities related to Fibonacci numbers
Graphical representation of roots of complex numbers
Prime number sieves
Geometry:
Non-Euclidean geometries
Ptolemy’s theorem
Geodesic domes
Tesseract – a 4D cube
Penrose tiles
Symmetries of spider webs
Fermat point for polygons and polyhedral
Conic sections
Regular polyhedra
Stacking cannon balls
Conic sections as loci of points
Mandelbrot set and fractal shapes
Squaring the circle
Architecture and trigonometry
Geometric structure of the universe
Fractal tiling
Pick’s theorem and lattices
Nine-point circle
Euler’s formula for polyhedra
Ceva’s theorem for triangles
Curves of constant width
Polyominoes
Spherical geometry
Rigid and non-rigid geometric structures
Euler line of a triangle
Properties of a regular pentagon
Geometry of the catenary curve
Eratosthenes – measuring earth’s circumference
Constructing a cone from a circle
Area of an ellipse
Spierpinksi triangle
Reuleaux triangle
Gyroid – a minimal surface
Tangrams
CALCULUS/ANALYSIS AND FUNCTIONS:
Mean value theorem
Applications of power series
Brachistochrone (minimum time) problem
Hyperbolic functions
Projectile notion
Torricelli’s trumpet (Gabriel’s horn)
Newton’s law of cooling
Second order differential equations
The harmonic series
Why e is base of natural logarithm function
Integrating to infinity
Fundamental theorem of calculus
L’Hopital’s rule and evaluating limits
Torus – Solid of revolution
STATISTICS AND MODELING:
Traffic flow
Modelling epidemics/spread of a virus
Central limit theorem
Modelling radioactive decay
Regression to the mean
Logistic function and constrained growth
Modelling the shape of a bird’s egg
Modelling change in record performances for a sport
Least squares regression
Modelling growth of computer power past few decades
Modelling growth of tumours
Correlation coefficients
Hypothesis testing
Modelling the carrying capacity of the earth
PROBABILITY AND PROBABILITY DISTRIBUTION:
The Monty Hall problem
Insurrance and calculating risks
Lotteries
Normal distribution and natural phenomena
Monte Carlo simulation
Poisson distribution and queues
Bayes’s theorem
Medical test and probability
Random walks
Determnation of pi by probability
Birthday paradox
Probability and expectation
GAMES AND GAME THEORY:
The prisoner’s dilemma
Poker and other card games
Zero sum games
Sudoku
Knight’s tour in chess
Gamble’s fallacy
Billiards and snooker
TOPOLOGY AND NETWORK:
Knots
Travelling salesman problem
Möbius strip
Steiner problem Königsberg bridge problem
Klein bottle
Chinese postman problem
Handshake problem
LOGIC AND SETS:
Codes and ciphers
Proof by contradiction
Set theory and different ‘size’ infinities
Zeno’s paradox of Achilles and the tortoise
Four colour map theorem
NUMERICAL ANALYSIS:
Linear programming Applications of iteration
Generating the number e
Fixed-point itereation
Newton’s Method
Descartes’ rule of signs
Methods of approximating pi
Estimating size of large crowds
Methods for solving differential equations
PHYSICAL, BIOLOGICAL AND SOCIAL SCIENCES:
Radiocarbon dating
Biostatics
Computing centers of mass
Fibonacci sequence and spirals in nature
Concepts of equilibrium in economics
Column buckling – Euler theory
Gravity, orbits and escape velocity
Genetics
Elliptical orbits
Predicting an eclipse
Mathematics of the ‘credit crunch’
Mathematical methods in economics
Crystallography
Logarithmic scales – decibel, Richter, etc…
Change in a person’s BMI over time
Branching patterns of plants
OTHERS/MISCELLANEOUS:
Paper folding
Mathematical card tricks
Applications of parabolas
Flatland by Edwin Abott
Photography
Sundials
Construction of calendars
Mathematics of juggling
Origami
Design of product packaging
Designing bridges
Curry’s paradox – ‘missing’ square
Music – notes, pitches, scales, …
Terminal velocity
Art of M.C. Escher
Navigational systems
Slide rules
Global positioning system (GPS)
Napier’s bones
Mathematical of weaving
Mathematics of rotating gears
Bar codes
Voting systems
Tower of Hanoi puzzle
Harmonic mean
The abacus
Different number systems
Optical illusions
Celtics designs/knotwork
…