Download Algebra and Number Theory Modular arithmetic Goldbach`s

LIST OF 200 IDEAS/TOPICS FOR A MATHEMATICAL EXPLORATION
The topics listed here range from fairly broad to quite narrow in scope. It is possible that some of these 200 could
be the title or focus of a Mathematical Exploration, while others will require you to investigate further to identify a
narrower focus to explore. Do not restrict yourself only to the topics listed below. This list is only the “tip of the
iceberg” with regard to potential topics for your Mathematical Exploration. Reading through this list may
stimulate you to think of some other topic in which you would be interested in exploring. Many of the items listed
below may be unfamiliar to you. A quick search on the internet should give you a better idea of what each is
about and help you determine if you’re interested enough to investigate further – and see if it might be a suitable
topic for your Mathematical Exploration.
Modular arithmetic
Applications of complex numbers
General solution of a cubic equation
Patterns in Pascal’s triangle
Pythagorean triples
Loci and complex numbers
Egyptian fractions
Chinese remainder theorem
Twin primes problem
Odd perfect numbers
Factorable sets of integers of the form
ak + b
Combinatorics – art of counting
Roots of unity
Recurrence expressions for phi
(golden ratio)
Non-Euclidean geometries
Ptolemy’s theorem
Geodesic domes
Tesseract – a 4D cube
Penrose tiles
Symmetries of spider webs
Fermat point for polygons and
polyhedra
Conic sections
Regular polyhedra
tacking cannon balls
Conic sections as loci of points
Mandelbrot set and fractal shapes
Squaring the circle
Architecture and trigonometry
Geometric structure of the universe
Mean Value Theorem
Applications of power series
Brachistochrone (minimum time)
problem
Hyperbolic functions
Projectile motion
Algebra and Number Theory
Goldbach’s conjecture
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
Probabilistic number theory
Continued fractions
Polar equations
Random numbers
Magic squares and cubes
Divisibility tests
Euler’s identity:
Natural logarithms of complex numbers
Diophantine application: Cole numbers
Palindrome numbers
Inequalities related to Fibonacci numbers
Fermat’s little theorem
Graphical representation of roots of
complex numbers
Prime number sieves
Geometry
Cavalieri’s principle
Hexaflexagons
Proofs of Pythagorean theorem
Map projections
Morley’s theorem
Fractal tilings
Pick’s theorem and lattices
Packing 2D and 3D shapes
Heron’s formula
Minimal surfaces and soap bubbles
Tiling the plane – tessellations
Cycloid curve
Euler line of a triangle
Properties of a regular pentagon
Nine-point circle
Euler’s formula for polyhedra
Ceva’s theorem for triangles
Consecutive integral triangles
Curves of constant width
Polyominoes
Spherical geometry
Rigid and non-rigid geometric
structures
Calculus/Analysis and Functions
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
Geometry of the catenary curve
Eratosthenes – measuring earth’s
circumference
Constructing a cone from a circle
Area of an ellipse
Sierpinksi triangle
Reuleaux triangle
Gyroid – a minimal surface
Tangrams
Integrating to infinity
Fundamental theorem of calculus
L’Hôpital’s rule and evaluating limits
Torus – solid of revolution
Traffic flow
Modeling epidemics/spread of a virus
Central limit theorem
Modeling radioactive decay
Regression to the mean
The Monty Hall problem
Insurance and calculating risks
Lotteries
Normal distribution and natural
phenomena
The prisoner’s dilemma
Poker and other card games
Zero sum games
Statistics and Modeling
Logistic function and constrained
growth
Modeling the shape of a bird’s egg
Modeling change in record
performances for a sport
Least squares regression
Modeling growth of tumors
correlation coefficients
Hypothesis testing
Modeling the carrying capacity of the earth
Modeling growth of computer power
past few decades
Probability and Probability Distributions
Monte Carlo simulations
Random walks
Poisson distribution and queues
Determination of π by probability
Bayes’ theorem
Birthday paradox
Medical tests and probability
Probability and expection
Games and Game Theory
Sudoku
Knight’s tour in chess
Gambler’s fallacy
Biliiards and snooker
Topology and Networks
Knots
Traveling salesman problem
Möbius strip
Linear programming
Applications of iteration
Generating the number e
Steiner problem
Königsberg bridge problem
Klein bottle
Logic and Sets
Set theory and different ‘size’
infinities
Zeno’s paradox of Achilles and the
tortoise
Numerical Analysis
Fixed-point iteration
Newton’s method
Descartes’ rule of signs
Radiocarbon dating
Biostatistics
Computing centers of mass
Physical, Biological, and Social Science
Gravity, orbits, and escape velocity
Mathematical methods in economics
Genetics
Crystallography
Elliptical orbits
Logarithmic scales – decibel, Richter, etc.
Codes and ciphers
Proof by contradiction
Fibonacci sequence and spirals in
nature
Concepts of equilibrium in economics
Column buckling – Euler theory
Paper folding
Mathematical card tricks
Applications of parabolas
Flatland by Edwin Abbot
Photography
Sundials
Construction of calendars
Mathematics of juggling
Origami
Design of product packaging
Chinese postman problem
Handshake problem
Mathematical induction (strong)
Four color map theorem
Methods of approximating π
Estimating size of large crowds
Methods for solving differential equations
Predicting an eclipse
Change in a person’s BMI over time
Mathematics of the ‘credit crunch’
Branching patterns of plants
Miscellaneous
Designing bridges
Curry’s paradox – ‘missing’ square
Music – notes, pitches, scales, etc.
Terminal velocity
Art of M.C. Escher
Navigational systems
Slide rules
Global positioning system (GPS)
Napier’s bones
Mathematics of weaving
Mathematics of rotating gears
Bar codes
Voting systems
Towers of Hanoi puzzle
Harmonic mean
The abacus
Different number systems
Optical illusions
Celtic designs/knot work