# Math Museum

 Revision as of 23:24, 31 March 2009 (view source)Neylon (Talk | contribs) (first draft)← Older edit Latest revision as of 10:01, 21 June 2011 (view source) (category) Line 15: Line 15: == Complex functions == == Complex functions == - Visualizations and interactive graphing of certain functions in the complex plane, along with how they relate to regular functions in the reals.  In particular, it would be cool to try to demonstrate Euler's formula $e^{ix}=\cos(x) + i\sin(x)$ and possibly even relate this to Fourier series.  For example, we could project the shadow of a slinky and explain that each orthogonal projection can be thought of as the real/imaginary part of $e^{ix}$. + Visualizations and interactive graphing of certain functions in the complex plane, along with how they relate to regular functions in the reals.  In particular, it would be cool to try to demonstrate Euler's formula $e^{ix}=\cos(x) + i\sin(x)$ and possibly even relate this to Fourier series.  For example, we could project the shadow of a slinky and explain that each orthogonal projection can be thought of as the real/imaginary part of $e^{ix}$. And something on Riemann's zeta function and Hypothesis,and Conformal mapping (air flow simulations). == Games from number theory == == Games from number theory == - Show how some tricks may be used to arrive at surprising conclusions, such as using easy trick for divisibility by 9, or other mind tricks.  The key here is to explain the math behind the tricks, so that they become real math instead of just tricks, and so the audience (with a good memory) will be able to repeat the trick on their own. + Show how some tricks may be used to arrive at surprising conclusions, such as using the easy trick for divisibility by 9, or other mind tricks.  The key here is to explain the math behind the tricks, so that they become real math instead of just tricks, and so the audience (with a good memory) will be able to repeat the trick on their own. == Card tricks from order theory and number theory == == Card tricks from order theory and number theory == - How many shuffles does it take to get the most random deck?  If I remember correctly, I think that 8 perfect shuffles gives you your original deck back exactly (assuming you shuffle it one particular way each time).  The number theory behind that is not too hard.  There are also some great card tricks that have very math ideas behind them - for example Gilbreath's principle. + How many shuffles does it take to get the most random deck?  If I remember correctly, I think that 8 perfect shuffles gives you your original deck back exactly (assuming you shuffle it one particular way each time).  The number theory behind that is not too hard.  There are also some great card tricks that have nice math ideas behind them - for example Gilbreath's principle. == Surprising ideas from statistics == == Surprising ideas from statistics == - The birthday paradox can be demonstrated to any group of the right size (say 25-40 people is usually enough to give a shared birthday and seem surprising).  The Monte Carlo problem might surprise people.  We could also play games of chance with unexpected outcomes, and explain what happened.  Maybe use a pseudo-random number generator to mix up some numbers and show how pseudo-randomness can fail.  We could teach people how to do coin tosses in their head with extremely high accuracy (that is, very close to 50/50 choices). + The birthday paradox can be demonstrated to any group of the right size (say 25-40 people is usually enough to give a shared birthday and seem surprising).  The Monte Carlo problem might surprise people.  We could also play games of chance with unexpected outcomes, and explain what happened.  Maybe use a pseudo-random number generator to mix up some numbers and show how pseudo-randomness can fail.  We could teach people how to do coin tosses in their head with extremely high accuracy (that is, very close to 50/50 choices). And something on Poisson processes(life-death processes), poisson distributions (soccer goals, light bulb lifetimes and frequency of floods). Simpson's Paradox. == Games from game theory == == Games from game theory == Line 35: Line 35: == Tessellations == == Tessellations == - We could show an exhibit of various tessellations, both of the usual plane, of hyperbolic space, or of standard 3 dimensions (or others?).  We could showcase the works of M.C. Escher here.  We could also allow audience members to build their own tessellations by modifying existing patterns.  Some interesting history of Penrose tilings.  We could relate these to crystal structures and maybe demonstrate how crystal sugar or quartz patterns might arise from basic tiling building blocks. + We could show an exhibit of various tessellations, both of the usual plane, of hyperbolic space, or of standard 3 dimensions (or others?).  We could showcase the works of M.C. Escher here.  We could also allow audience members to build their own tessellations by modifying existing patterns.  Some interesting history of Penrose tilings.  We could relate these to crystal structures and maybe demonstrate how crystal sugar or quartz patterns might arise from basic tiling building blocks. Also "Islamic" tilings, the story of Majorie Rice and her non-periodic tiles (something for the girls). == The game of life and cellular automata == == The game of life and cellular automata == Line 48: Line 48: We could mention a few easy-to-understand problems (like the 3x+1 problem, or the Goldbach conjecture) and a few harder problems, like the Riemann hypothesis.  We could also mention things that are impossible to do in math, such as the implications of Godel's incompleteness theorem or solving the quintic.  It might be nice to leave the audience with an idea of some problems they can actually think about, but they are known to be still unsolved (of course, give them fair warning!).  And to not kill their brains include some warm-up problems so they at least feel better when they can't solve the hard ones. We could mention a few easy-to-understand problems (like the 3x+1 problem, or the Goldbach conjecture) and a few harder problems, like the Riemann hypothesis.  We could also mention things that are impossible to do in math, such as the implications of Godel's incompleteness theorem or solving the quintic.  It might be nice to leave the audience with an idea of some problems they can actually think about, but they are known to be still unsolved (of course, give them fair warning!).  And to not kill their brains include some warm-up problems so they at least feel better when they can't solve the hard ones. + Plus the connection between Hilbert's 10th problem, renumerable sets (work of Davis, Putnam and Robinson) and Diophantine equations and primes (Matajasevic, Jones, Wada et al)See wikipedia "Hilbert's 10th problem". The Busy Beaver Problem. The Word Problem. The connection between some tessellations and decision problems. == Other possible exhibits == == Other possible exhibits == Line 55: Line 56: * Tangrams * Tangrams * General brain teasers, explained with the related math ideas * General brain teasers, explained with the related math ideas + * General facts about triangles (with animations / interactive) + * Nice solids (like the Platonics, Archimedean, etc) + * Fourier series and music + * Vector fields, calculus and application through models (eg aerodynamics) + * Group theory and Rubik's cube + * Number theory and cryptography + * Set theory and different sizes of infinity + * Graph theory, maps, and networks + * Relativity and curved spacetime + * Statistics, fun conclusions (eg Tufte's explanation of investigating the plague in London), and easy mistakes + * Financial math and gambling strategies + * Fibonacci number and the golden ratio + * Some history (major characters, their thoughts) + * Bell curve generator (quincunx? see wikipedia) + * Harmonic analysers + * Early computers + * History of "Circle Squarers" + * Special numbers: pi, e, phi, etc + + [[Category: Math]]