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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