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  Museums are traditionally associated with history, but there's no reason to exclude other fields. There are already museums of science in general  some specialize in ideas from physics, chemistry, or biology. But the scale and scope of museums focusing on math is relatively miniscule. Why?
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  Inspired by some brainstorms of Daryl Williams on this idea, I'm adding this page to the tangent space to collect a few thoughts for possible exhibits for a more serious and largescale math museum. There can be many museums for different audiences. I feel that there is a great deal of mathematics which can be expressed intuitively without a laborious amount of rigorous introduction, and that a wide audience could appreciate even deep ideas from math without a deep background. So, personally, I would be most interested in exhibits which combine ideas from serious and modern mathematics with a very intuitive and enjoyable presentation. (Of course, old math is still good!)
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  A few possible ideas:
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  == Geometric constructions ==
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  See firsthand how to build compass and straightedge constructions. Perhaps some puzzles to go along with it, and some explanation of the limitations of the process  hints about why we can't work with transcendental numbers this way, but ways to cheat if we can use more than just a straightedge.
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  == Games from number theory ==
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  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.
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  == Card tricks from order theory and number theory ==
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  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.
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  == Surprising ideas from statistics ==
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  The birthday paradox can be demonstrated to any group of the right size (say 2540 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 pseudorandom number generator to mix up some numbers and show how pseudorandomness 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(lifedeath processes), poisson distributions (soccer goals, light bulb lifetimes and frequency of floods). Simpson's Paradox.
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  == Games from game theory ==
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  We could play the prisoner's dilemma in real life and see who wins (among the audience). We could teach people certain strategies that they must stick to, and see which one does best. There is nice history here, as John Nash is somewhat famous. There are also cool related ideas, like a dollar auction (perform one with the audience for fake money), or other unusual auction types with nice theory behind them.
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  == Tessellations ==
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  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 nonperiodic tiles (something for the girls).
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  == The game of life and cellular automata ==
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  Wolfram's book is filled with nifty pictures relating to this. We could show how a simple pattern evolves to resemble a nice and individualistic snowflake design. We could give audience members simple patterns to execute and see what happens, or allow them to tweak the initial conditions and watch things evolve. We could explain some of the many known patterns in Conway's game of life, and exhibit how they might interact. We can intuitively show how a designed system can be considered Turing complete.
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  == Computational complexity ==
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  We could try to show the ideas that tetris and minesweeper are NPcomplete, by showing some very hard situations to figure out. We could explain the ideas of graph isomorphism, or 3colorability, or of factoring large numbers, or of solving 3satisfiability (aka 3SAT), which are all problems considered difficult to solve via algorithm, though unsolved as of yet.
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  == Unsolvable or Unsolved problems ==
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  We could mention a few easytounderstand 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 warmup problems so they at least feel better when they can't solve the hard ones.
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  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.
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  == Other possible exhibits ==
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  * The game of nim
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  * Towers of Hanoi
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  * Tangrams
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  * General brain teasers, explained with the related math ideas
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  * General facts about triangles (with animations / interactive)
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  * Nice solids (like the Platonics, Archimedean, etc)
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  * Fourier series and music
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  * Vector fields, calculus and application through models (eg aerodynamics)
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  * Group theory and Rubik's cube
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  * Number theory and cryptography
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  * Set theory and different sizes of infinity
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  * Graph theory, maps, and networks
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  * Relativity and curved spacetime
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  * Statistics, fun conclusions (eg Tufte's explanation of investigating the plague in London), and easy mistakes
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  * Financial math and gambling strategies
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  * Fibonacci number and the golden ratio
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  * Some history (major characters, their thoughts)
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  * Bell curve generator (quincunx? see wikipedia)
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  * Harmonic analysers
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  * Early computers
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  * History of "Circle Squarers"
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  * Special numbers: pi, e, phi, etc
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