What would you want to see at the Museum of Mathematics? EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no longer relevant".

As some of you may already know, there are plans in the making for a Museum of Mathematics in New York City.  Some of you may have already seen the Math Midway, a preview of the coming attractions at MoMath.
I've been involved in a small way, having an account at the Math Factory where I have made some suggestions for exhibits.  It occurred to me that it would be a good idea to solicit exhibit ideas from a wider community of mathematicians.

What would you like to see at MoMath?

There are already a lot of suggestions at the above Math Factory site; however, you need an account to view the details.  But never mind that; you should not hesitate to suggest something here even if you suspect that it has already been suggested by someone at the Math Factory, because part of the value of MO is that the voting system allows us to estimate the level of enthusiasm for various ideas.
Let me also mention that exhibit ideas showing the connections between mathematics and other fields are particularly welcome, particularly if the connection is not well-known or obvious.

A couple of the answers are announcements which may be better seen if they are included in the question.
Maria Droujkova: We are going to host an open online event with Cindy Lawrence, one of the organizers of MoMath, in the Math Future series. On January 12th 2011, at 9:30pm ET, follow this link to join the live session using Elluminate. 
George Hart: ...we at MoMath are looking for all kinds of input. If you’re at the Joint Math Meetings this week, come to our booth in the exhibit hall to meet us, learn more, and give us your ideas.
 A: Some Pixar/Dreamworks stuff might be good...a Pixar guy gave a cool talk at ICM a few years ago about the mathematics they use to do the 3d rendering, topping it with harmonic coordinates.
A: I've once made a 3D model of a contact structure. This is a remarkable object, and the feeling one gets by looking at it is difficult to describe in words. I've already spent a lot of time pondering at the idea of making a big sculpture out of it (including going and talking with someone whose job is to make metallic constructions).
Here is a mathematical description the object: take the Cayley graph of the Heisenberg group <a,b | [a,b] is central > and embed it in 3D. This Cayley graph is infinite, and I'm of course imagining taking a finite portion of it (5x5x5 nodes works pretty well). The vertices are 4-valent, and at each vertex, the directions of the four incident edges are coplanar. On those four edges, you then position a small piece of plane: that's the contact structure!

If you want to make this object interactive, you could imagine little cars moving on it, their  x-y-coordinates could be somehow specified by the user...

Here's a model I once made:

and here's the same model upside down, before it was completely finished:

A: I would like someone to make hyperbolic glass. I'm not sure that the technology exists to make it though - fiber optics cables use glass of varying index of refraction, but I don't know if it is isotropic.  
A: A bubble table, like the one at the Exploratorium. I couldn't quickly find a good link at the Exploratorium website, so check out the list of Google images instead.  It would be particularly cool to connect this with a discussion of minimal surfaces.
Edit: Just to give more details -- The bubble table at the Exploratorium is a large (4 feet?) and shallow (4 inches?) bath filled with bubble solution, at waist height of a 6 year old.  The museum provides metal loops which visitors use to make large tubular bubbles.  It is particularly amazing to lift the hoop up and the pull it down over your head: you get a moment of looking out of a bubble. 
I don't remember if they provide other wire frames.  It would be cool to have the standard ones to play with (one-skeleta of Platonic solids) and various saddle inducing frames (say, subgraphs of the one-skeleton of the cube).  Also interesting: wire frames of knots (interesting unknots, trefoil, figure eight) shaped to allow seeing their Seifert surfaces.  Another suggestion: parallel plates of clear plastic connected by rods, to allow the creation of Steiner networks (or at least their approximation). 
A: I would like to see a clock which illustrates the Chinese Remainder Theorem. Since $3600 = 16 * 9 * 25$, have wheels which spin once every $16$, $9$, and $25$ seconds, with marked points which align once every hour. Of course, I would like there to be an explanation of the CRT with this, but I think the clock should be visible from a distance.
A: Build a fundamental region of a Platonic solid out of mirrors facing inward, e.g., $1/48$ of a cube, omitting the side of the tetrahedron which is part of the exterior of the solid. When you look into those three mirrors, you see copies of yourself looking into a Platonic solid from each of the other fundamental regions. 
If you truncate the vertex corresponding to the center of the regular polyhedron appropriately with an opaque triangle, the mirror images of the triangle form the polyhedron or the dual. I think a few of these, made by another math major in my year, might still be in the math lounge at New College.
This is a striking visual effect which can be observed by nonmathematicians in passing. Similarly, two large vertical mirrors set at an angle of $\pi/n$ show the viewer as one of $2n$ copies.
A: Working mathematicians live!!
Movies showing sessions of working mathematicians, with some comments and explanations along it.  
So at last the general public (and sadly the not so general one as well) will be aware that mathematics has more to do with art and understanding than with formulas and logic.  
Five to ten "movies" would do, it is not easy but neither hard nor expensive to produce, 
Some people are very good at producing documentaries. Those professionals should be asked/used of course. 
A: A slide rule! The physical embodiment of the isomorphism between $\left(\mathbb{R},\cdot\right)$ and $\left(\mathbb{R},+\right)$. There are pretty pieces of history here too - Napier's bones and so forth. A giant one (maybe >1m long) mounted on a wall so that people can make it work - now that would plant the idea of the isomorphism making + and * "the same" operation deeply in the mind of anyone who played around with it seriously.
A: A gömböc (or gomboc, as I would prefer it to be spelt in English) 3 metres across was exibited at the World Expo 2010 in Shanghai, China. I don't know where it is now; but it deserves a permanent place in a major museum of mathematics.
A: A wonderful interactive mathematics exhibition called Beyond Numbers was designed by the Maryland Science Center and the George Washington University Department of Mathematics, especially the co-director Rodica Simion. See http://www.gjbgraphics.com/usefulstring/BNTofC.html. It was displayed during the period 1994-1999. Many of the ideas for this exhibition could be carried over to MoMath.
A: Flexible polyhedron.
A: This hyperbolic tesselation applet:
http://www.plunk.org/~hatch/HyperbolicApplet,
maybe enhanced so as to also include the Eucledian and spherical cases.


(Picture taken from http://aleph0.clarku.edu/~djoyce/poincare/PoincareApplet.html)
A: Some mathematics was motivated by astronomy in ways which are hard to notice now due to light pollution and alternatives to staring at the sky at night. I would like to see an exhibit which shows the motions of the planets, Sun, and Moon, sped up and made easier to see, along with a presentation of mathematical results and techniques developed for astronomy, from numerical methods to mechanics to Kepler's laws. Newton and Euler contributed extensively to the mathematics of astronomy, and astronomy influenced many of their mathematical works. 
A: A room dedicated to waves, waterwaves, soundwaves and lightwaves illustrating interference, refraction, Fourier transform and so on with the help of concrete (and playful)
devices, and explaining that waves are as much mathematics (trigonometric functions,
differential equation, complex numbers) as physics (optics, acoustics, quantum mechanics).
Perhaps one could also do something around the heat equation?
A: At the science museum in London they have this very cute little gadget used by mapmakers 150 years ago: an axle with a rubber ring around it, and the ring pressing against a cone. The whole lot is attached to a metal stylus; you trace around an area on a map with the stylus and a little reader tells you the area of what you've traced around. I always found that ingenious. The exhibit in London then goes on to show how you can use the same idea to integrate and hence solve differential equations, and finishes with a monster machine that can solve ordinary 4th order ODEs using basically the same trick; you set the coefficients with dials and then the machine draws a graph of the output. I'm afraid I know neither the name of the cute gadget nor the machine :-( but it strikes me as being appropriate for a "math museum"...
A: Vi Hart's Doodling in Math Class series on YouTube seems to be quite popular.  You could either incorporate ideas from the videos or ask for her permission to use them.  
A: A few years ago there was an exhibition devoted to mathematics which took place at the Science Museum near the Hebrew University of Jerusalem and also at the Abu Dis Al Kuds University. This was an Italian-Isreali-Palestinian joint endeavor. There were many exhibits (and some were mentioned already among the answers) like: The decimal number system, exponential growth, Konisberg bridges, Tilings periodic and non periodic, knots, The Tower of Hanoi Game, Soap bubbles, Reuleaux triangle, models for graphs of polyhedra, demonstration of Buffon's needle problem, and many more. Some movies (in Hebrew, but still easy to understand) can be found here  http://www.cet.ac.il/math/mada.asp See also here 
A: I would like to see RSA encryption included somehow, ideally in a hands-on way (letting them do some arithmetic with aid of calculators which are part of the exhibit) so that people can get the sense that whenever there is an https:// in their browser, a lot of simple but remarkable arithmetic is happening in the background.
A: If you want to give the audience some sense for what mathematical argument is about, I like the topic of divisibility rules (by 2, 3, 4, 9, etc).  Most people have seen these but take them completely for granted - indeed, some people take "ends in an even digit" as a suitable definition for even number.  One main characteristic which separates mathematicians from the rest of the world is seeing such a rule and asking "does that always work, and if so why?"  So perhaps one could first put some plausible false rules out there to create some doubt and the arguments that these rules work - both with algebra and if possible avoiding algebra.  I found that emphasizing this material worked well in a class I taught for future elementary school teachers.  I told them that even most/ all of their science major friends who passed AP calculus didn't really know why these rules work, so they had learned something special.
A: What about some large-number phenomena? This seems to be something the general public would appreciate and could relate to the "Computers in Modern Mathematics" booth others have suggested.
What I have in mind is not really Ackerman function/Graham's number business (which I don't think I could wrap my head around any more easily at a museum), but facts that involve small-ish large numbers. For instance:

The smallest positive integer $n$ for which $n$ divides $2^n-3$ is $4,700,063,447$.

There are many other great examples (though not all interesting or accessible to non-mathematicians) in answers to this MO question. It also might be nice to see comparisons of smallest counterexamples like this to 'real-world' numbers like the population of China (~$1.34$ billion), or the number of cells in the human body (~$10^{14}$), or the number of elementary particles in the observable universe (~$10^{80(\pm10?)}$). 
To me, the goal of such an exhibit should be (1) to provide a few examples (like the one above) illustrating the importance of proof over verification of the first $10^{10}$ cases, and (2) to help museum-goers conceptualize the small-ish large numbers that come up in analyzing real-world phenomena.
A: *

*A transparent model of Cayley's cubic surface, with the 27 real lines marked on them. I've only seen plaster models of this. (Actually, I want this for my birthday. Ahem...)

*A transparent plastic cone and a laser light to cut it into conics and, more fun, similarly transparent models of quadrics and lights to check the theorem that the shadow lines on cuadrics are plane curves.
A: I'm not sure if this fits with the type of "museum" they have in mind, but I'd love to see Fermat's copy of Diophantus' Arithmetica.  (Ignoring the fact that noone knows what happened to it)
A: I'd love to see large and detailed historical montages centered around specific developments or results that took significant time and evolution from conception or conjecture to actual proof. For example, we could see a large montage of the development of the proof of Fermat's theorum from Fermat's cryptic anecdote through 2 centuries of developments in number theory,algebra and elliptic curve theory concluding with Wiles' proof of the Taniyama–Shimura conjecture for semistable elliptic curves and Ribet's proof of the epsilon conjecture. 
The level of detail could be modular-several levels of explaination could be present from general audience to PHD level. 
A: I think holographs are a compelling technology that seems like magic except in the light of some pretty cool mathematics. If some kind of learning module could get these ideas across, it'd be neat. 
A: I come across "mind reading" games based on elementary number theory from time to time; e.g. http://www.digicc.com/fido/.  It bugs me a bit when people are wowed by such tricks, but not enough to sit down and figure out the mechanics of the thing. But the surprise factor may make a good museum activity -- where the second part of the activity is teaching why the trick works the way it does.
In general, math-based magic tricks would be good for an interactive exhibit: Magic trick based on deep mathematics 
A: Klein bottle (with a description containing at least 15 characters)
A: I'd love to see an exhibit devoted to beautiful and intuitive proofs. Most of us mere mortals will never be able to understand Wiles' proof of Fermat's Last Theorem, but there are some phenomenally interesting and important proofs out there that the average person might be excited to learn about. For instance, using Cantor Diagonalization to prove the uncountability of real numbers. Fascinating and accessible!
A: A history of Pythagoras' theorem - from Egypt and Babylon through to the proof, then higher dimensional versions, and then a jump from that to non-Euclidean geometry (surfaces of positive and negative curvature), and then introducing the idea of a metric space, with $\mathbb{R}^2$ as an example. 
A: The trammel of Archimedes.
 See http://blog.makezine.com/archive/2010/06/my_10_favorite_mechanical_animation.html
A: I think an exhibit on sangaku, geometry puzzles offered to shrines and temples in Japan, would work well because there are such interesting physical objects to look at.
A: 1) A high quality 3D movie (with glasses!) of sphere eversion, like this one
2) A Let's Make a Deal game show room, where people can play the game to death on a computer until they believe that they should switch doors. Offer candy prizes.
3) A scaled down Bridges of Koinsberg room, where you can try to walk across each bridge only once.
4) A large transparent (working!) replica of an Enigma machine.
5) A Velcro covered life-size Mobius strip which you can walk on with Velcro shoes (I hope you have good insurance)
A: How about leading them through an interesting problem, like a geometry IMO problem or, if that is asking too much, a Mathcounts problem? It could be set up on square tiles, the left most of which would contain the problem, with the following tiles showing the steps of the solution. It should be a problem that can be written such that viewers see a surprise toward the end, thereby possibly giving a glimpse into why mathematicians enjoy so much what they do. Although a Mathcounts problem would no doubt be accessible, a very beautiful IMO problem could be inspiring. Very likely, it would be entertaining for both children and their parents.
One might also include multiple solutions to a problem to dispel the notion that for each problem only one solution exists.
One can see examples of interesting presentations and ideas for problems at Rusczyk's Mathcounts channel at 
http://www.youtube.com/user/mathcountsfdn
Using the same format from above, one could present a suitable Putnam problem and show its connection to research. This is discussed in Kedlaya, Poonen, and Vakil's book.
Finally, this response might be related to Kevin Lin's. and ein's.
A: An exhibit on the role of computers in pure (and applied) mathematics. It would especially be nice to see something about experimental mathematics and viewing math, at times, as not purely deductive, but even empirical. I think this would give an idea as to how some mathematicians work and think, and also emphasize the growing importance of computers in verifying or finding new theorems.
A: In an Italian museum (probably the Leonardo da Vinci Museum in Florence) I saw a compass for drawing arbitrary conical sections. I believe the legend mentioned only ellipses, but it could in principle draw the others too.
The basic principle is that the "central" arm (in general, the focal arm) of the compass is held at a fixed (per drawing) angle to the desk, while the pencil arm adjusts in length (so that it is shorter at the perigee and longer at the apogee).
A: An exhibit - with both individual workstations and one whose results are projected (to draw people in) - where users use parallelograms to specify affine transformations which in turn define Iterated Function Systems and their associated fractals.  See here for background.  With a few moves/ clicks a user can start making fractal ferns, clouds, spirals and starfish as well as classical objects such as the Sierpinski gasket.  It is great because you don't have to know anything to start making pretty pictures, but you immediately get a sense that there is something significant going on.  For those who want to see beyond the pretty pictures, one could explain the contraction mapping theorem  (terrific fun in its own right) and develop affine-linear transformations starting with rotations, scalings and translations.  Trying to find the transformations which define a particular fractal by "finding enough smaller copies of the fractal to cover the fractal" is also great fun, which is compelling even for children.  
A: In such a museum, I would like to see how mathematics are used in real life, not just for their internal beauty (well, beauty, simplicity and usability are certainly related). I mentioned above in a comment how Thales' theorem has been the tool to measure the height of pyramids. This can make a nice mathematics experiment: a lamp (the sun) a small pyramid and a stick. And suddenly math comes alive. Another kind of living mathematics: put salt into a thin aquarium such that the density vary, top to bottom, from zero to (almost) infinity. Send a light beam to the aquarium and the light will follow a geodesic of Poincaré's half plane (this experiment has been actually presented at the Paris "Palais de la découverte"). These are just two examples of "math in real life", I'm confident in mathematician's skills to find a lot more of such examples (not just in geometry: prime number and securing communications, statistics and controlling epidemics, etc...). I'm sure that understanding with our eyes how mathematics are used in real life makes mathematics even more sexy.
A: A moving sculpture approximating Smale's turning of the sphere inside out. (but what material would you use?)
A sphere made out of elastic plastic with fotoreactive proteins. The proteins are laid down so that they only react when antipodal points touch.
A similar thing for Brower's theorem instead of Borusk-Ulam, with two discs.
A sphere made out of some flexible but not so flexible material, the spectator gets to sculpt the sphere into some shape, then he plays the shape with a stick. There are (say) seven choices of sticks in front of him each corresponding to one eigenvalue of the shape. When the visitor plays the sphere a computer computes the corresponding eigenvalue and translates it into a sound (maybe the sound of a drum).
A family of itouch-made-material 2d-surfaces, and a big sphere in the middle of the room. When one traces a curve (with the finger) on the surfaces. The Gauss map on the big sphere in the middle is animated.
A linkage that has the earth at the center of the solar system and traces the moving planets perfectly (this can be done by universality of Thurston and Kapovich-Millson right?) The spectator stands in the middle and sees the planets move around him. At the end he gets to wear an Inquisition hat and burn a paper sculpture of Galileo.
A metalic model to see percolation: The vertices will be represented by magnetized vertical thin tubes coming from the floor. The spectator stands in the second floor and from above (not above the spectator but above the magnetized tubes) a bunch of small tubes (edges) fall. Depending on the strength of the magnets (controlled by the user) some stay sticking to the tubes and some go to the ground, the spectator is asked to repeat the experiment many times and conjecture with what probability this random graph percolates.
In a dark room. A hospital bed with a set of cards is lighten like in a noir film. The visitor is supposed to play solitaire lying on the hospital bed, if he gets to the end in one round a screening of the H-bomb appears on a big screen in front of him. The sound of the bomb is heard in very loud speakers so that everybody in the museum hears this.
A 3d animation of a contorted 2-sphere Ricci flowing to a round sphere. Again the visitor gets to choose the starting sphere. At the end he is given a phone number. He tries to call Perleman.
A huge fountain that doesn't work.
An observatory with stars at random positions in which suddenly log n of them turn out to form a convex polygon. It should look like an astrological map.
Many microscopes looking at cells growing. In the first one like f(t)=t, then f(t)=t^2 and a few polynomials more. Then f(t)=2^t. (Can this be done with unlimited resources? I'm not joking, this is a honest experimental biology question.)
A mechanically transformable translucent skate park. Each configuration of the skate park corresponds to a link, an element in pi_1(C_n(R^3)). A tube of say two meters of radious traces the curve followed by one of the points in configuration space. So there are say 5 base points and 40=10*4 switches (like in a wiring diagram but going around 5 circles) The user gets to select what switches (adjecent transpositions are on).
The skaters are encouraged to use a helmet.
A liquid based model explaining Kepler's laws with a very eccentric ellipse.
A finance millionaire looking at the numbers of the stock market, very focused.
Three hallways that meet at the center. One with triangles on the walls, other with curves of functions, the third one with polynomials. At the center (e^i\pi=-1). Like that is a bit cheesy but if there was a computer app illustrating the geometry of elementary operations (+ and *) that foced the visitor to define what product by i does in the complex plane this would be quite cool.
A room with two walls closing on the visitor like in that famous star wars scene. But without the intergalactic trash. With ungraded calc exams instead.
A performance-theatre show. An artificial beach with Newton (with a wig) looking at a shell and at a pebble, figuring out which one is smoother. We hear a loud applouse and two giants carry him out of the building. The two giants come back to find a seminaked Archimedes computing an integral on the sand, they slaughter him.
This one will save mathematics from the financial crisis: A chair in which the visitor sits and his brain activity is monitored as he does some easy mathematics. Whenever his brain activity seams to resemble math thinking he is injected endorphins. 
sorry, this got a bit out of control....
Im trying to think of something good for some basics about Galois theory or Covering Space theory, but this is harder....
Oh and in the store you get to draw your favorite planar graph and you leave with a clay model of the corresponding circle packing.
A: Stereographic projection lamps, please!
I'd love to see a bunch of clear plastic spheres with colored patterns (triangle tilings, Escher pieces, and so forth), lit from a pole to project the patterns onto whatever's nearby. Or stick models of polyhedra with bright lights suspended at their centers. When I was a kid, I had a toy that worked on a similar principle---it was reasonably effective, and apparently considered safe.
A: A cool gadget I've seen in a few science museums:  There is a vertical board with a lattice of nails in it.  You drop balls in from the top, at the center.  After dropping enough balls, you always see a Bell curve, "proving" the central limit theorem.  Then a catch releases the balls, they are transported back to the top, and you start again.  The cooler versions of this have the Gaussian predrawn in the background (which displays a certain level of confidence!  And a willingness to replace missing balls). 
Edit - This is sometimes called a Galton box.
A: The standard orientation on $S^1$, if you can borrow it from NIST.
A: Rather than providing concrete examples, I would like to make a suggestion about two possible guiding lines.
If you agree that mathematics is about solving problems of a certain kind, then two sensible goals for a museum of mathematics could be showing:


*

*what kind of problems mathematics deals with, and 

*how it deals with them.


The first goal cannot be exhaustive for obvious reasons, but I think it should be broad enough to give the visitor an idea of the diversity of mathematics (applied or not), i. e. examples from as many various branches as possible should be given.
For the second goal, Polya's views about how to solve mathematical problems could be helpful, or modern works about "visual" mathematical thinking. I think visitors should be able to feel some connection between mathematical problem solving and applying "common sense" strategies (here is a hidden goal: to demistify a bit the work of mathematicians).
To focus on one or the other goal in different degree can result in very different kinds of museum, but I think any of them would deserve the name "Museum of mathematics". Not so one that didn't meet any of both goals.
A: I'm a Mathematics & Theoretical Physics undergrad, so I though I'd share what got me hooked on Maths. As a teenager it first dawned on me that Maths wasn't the "dry", boring subject taught at school when, on my own at home, I first started playing around with nD geometry and generalised Euler's Theorem to nD by noticing the patterns. The thing that really made me fall for Maths though was reading about the Riemann Hypothesis in New Scientist when I was 16. [It's hard to believe, but none of the Maths teachers I'd had had even mentioned primes, and I was hooked.
The thing I'd put in MoMath is the Mandelbrot set across a HUGE wall using a projector that gradually zoomed in and in. After a set time it could start again but zoom in on a different area. You could include higher-order Mandelbrot sets and other infinite fractals. The beauty of fractals is their beauty appeals to everyone [Importantly including NON-Mathematicians!] and gets an important point across: "MATHS IS BEAUTIFUL!"; in the blink of an eye. It would show clearly & quickly both the richness & beauty that lies within Mathematics and that it's NOT the dead, dull, dry subject many people think.
I would suggest screening a range of pre-recorded maths "lectures", catering for the diverse spectrum of Mathematical-understanding of the visitors.
MoMath, if done right, seems like a brilliant idea. Good luck with the project, and I wish you every success! :-)
A: I'd love to see a wall sized picture of the matrix of a random element of the monster group acting in a faithful representation.
A: We are going to host an open online event with Cindy Lawrence, one of the organizers of MoMath, in the Math Future series: http://mathfuture.wikispaces.com/events
On January 12th, at 9:30pm ET, follow this link to join the live session using Elluminate: http://tinyurl.com/math20event
A: Crystallography, illustrated by optical diffraction.  See the repeated pattern under a magnifier, then project with different wavelengths (would a prism work or do you need different laser pointers).   Introduction to Fourier analysis, optical transforms,etc.
A: Something about the "Quaternion Demonstrator" (the belt trick demonstrating that $\mathbf{SO}\left(3\right)$ has a double cover). An exhibit could centre on three distinct, accessible and interesting to the general public (like myself) mathematical topics:
1) As a model for spin 1/2 particles - I recall being enthralled as a young teen by the idea that some objects might not come back to their same state after a $2 \pi$ rotation. At that age, on seeing the belt trick, I recall one of my first reactions was "Interesting, but might not we build something with a fancier arrangement of ribbons and strings that would need, say $6 \pi$ rotations to bring it back to the "same state"?" I think it would be interesting to say in the exhibit that there is sound mathematics behind the assertion that, no, there is no such fancier arrangement, so that, unless the topology of our Universe is radically different from what we can imagine, there is very strong evidence, grounded on mathematics alone, that half-integer spin is the only possibility - and we don't need billion dollar particle accelerators to know this.
2) The quaternions and the idea of number systems beyond "everyday" rational and real numbers. That only restricted systems can be built if one wants to preserve "real world" properties like continuity of the "multiplication" - that mathematics isn't just postulating arbitrary axiom systems and playing games with them. History of complex numbers could be included, maybe even a feel for Hatcher's Algebraic Topology proof thereof (something like the YouTube clip http://www.youtube.com/watch?v=nRO_4IYOdq8).
3) Thinking about the belt trick itself (the physical thingie, rather than the mathemetics of the $\mathbf{SO}\left(3\right)$  double cover) for me stridently raises the question of what the distinction between mathematics and physics really is, or even whether there is one. The belt trick is compelling to even small children - I showed it to my five year old recently and was astonished to find that she seemed to understand many of the ideas of symmetry involved and played around with different numbers of twists and untangling them for quite some time. Of course, most serious mathematicians will say that the belt trick is not a proof, but when you begin to look at it hard, and see that the ribbon is directly encoding a "history" of rotations of a Frenet-Serret frame, you realise that the contraption is a pretty spot on analogue of the mathematical construction of a universal cover - so much so that you begin to wonder whether the mathematical construction isn't part of the subconscious visual processing and understanding of the physical contraption in almost anyone - mathematician or layperson.
A: A Foucault pendulum, explaining the concept of parallel transport in a manifold.
A: Look at the 'surfer video' which among other things shows how visualizations of algebraic geometry can be presented in real-time in an exhibition.
A: Something from Erik Demaine on linkages would have wide appeal.
A: I think that there should be an exhibit on fractals and a history of mathematics exhibit. Fractals are very beautiful and mathematically interesting, and the concept of self-iteration is fairly easy for an general audience to understand. You could even discuss the motivating problem of determining the length of the coast of Britain and how making measurements of non-smooth curves on smaller and smaller scales eventually limits to infinity. As for the history of mathematics, I think that's pretty obvious. There are a lot of interesting stories that make up the history of mathematics. Museums also love to show cultural diversity, so the development of mathematics around the world could be a possibility.
A: I'd like to see a picture of a person throwing a ball (eg Michael Jordan) and next to it, the corresponding parabola in a Cartesian plane.   
A: An example of the Ricci flow equation used to solve the Poincare conjecture on different surfaces. I think an animation, and a brief explanation and demonstration as the changes were applied would be incredible to watch.
A: An exhibit on how cryptography works, and how it keeps online payments and transactions secure. Perhaps a demo or game where kids get to code a message, and other kids have to try to decode it.
A: A knot table, with the knots in it made out of a nice (pretty and pliable) material. It's aesthetic, and people might have fun playing with them.
One might include also the Perko pair! They come with a story, and it's a lovely (terribly difficult, but tremendously fun) challenge to figure out how to change onto into the other.
A: Tiling and symmetry!  You could start with the wallpaper groups, maybe have a station where people learn to recognize and name them (I guess using Conway's orbifold notation or something similar).  The great thing about this is that there are beautiful examples throughout history to use.  Then move on to the crystallographic groups and explain the application to chemistry; again a lot of nice pictures here.  Finally maybe something about hyperbolic tilings, explaining all those Escher drawings.  
Related: a guided tour through the proof of the classification of Platonic solids.  Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things might be a good place to look for inspiration, as well as Mumford, Series, and Wright's Indra's Pearls for branching out to more exotic groups (although I hesitate to suggest that you do anything about fractals because they already have a disproportionate grip on the public imagination).
A: "The Forbidden Forest"
Mathematical objects, the existence of which was once forbidden:
More than one parallel to a given line
Square roots of 2, -1
etc etc [so many examples from different fields]
To show how mathematical development has required real courage against the status quo
A: Sculptures of surfaces would be lovely.
A: First, I don't like using the term "Museum", which has too many undesirable implications for me. I have to say I like the word "Factory".
Second, it seems to me that most exhibits give only an impressionistic, usually visual view-from-the-outside of mathematics. For me mathematics is a powerful tool combining deductive logic and abstraction, and I'd like to see exhibits or "labs", where ordinary people are allowed to experience the power of mathematics firsthand by showing them how to use deductive logic and abstraction themselves to gain new knowledge or insight. This, of course, means making the visitor work or think harder than usual, but I think it would be well worth having some exhibits like this, because I think it would create a deeper level of both understanding and excitement about mathematics.
I can't claim to have many concrete examples to offer, but one that comes from my experience teaching precalculus and calculus is to have an exhibit that introduces people to what a function is and then showing them in very concrete terms what a derivative is (i.e., the sensitivity of the output to changes in the input) and also the definite integral (if the function is measuring a rate of change then the definite integral recovers the total or net change). The important here is avoid an exhibit that just shows this visually but to actually make visitors work through a series of exercises (almost as if they were calculus students themselves) where they learn through firsthand experience. The analogy for me is sports or crafts (like, say, knitting). Instead of having visitors just watch someone else do things or look at the finished product, let them actually have the experience of doing the craft of mathematics (I like thinking of math as a craft rather than a science or body of knowledge or whatever) themselves.
A: I have been involved with an online Mathematical Museum (called unsurprisingly The Virtual Math Museum, and located at http://VirtualMathMuseum.org). There is also an interactive version in the form of an application called 3D-XplorMath that is freely available at http://3D-XplorMath.org. In both you will find many "Galleries" of different types of mathematical objects (curves, surfaces, ODEs, Fractals,...) and in each gallery we have attempted to put all the interesting objects of that type that we could find and that had names. Some years ago I also wrote an article called "The Visualization of Mathematics: Towards a Mathematical Exploratorium" that appeared in the Notices of the AMS and that is now freely available online, and you may find that of interest. By the way, be careful with the use of the word "Exploratorium"... the San Francisco Exploratorium feel they own that word and got very mad at me for using it in the title---even got their lawyers after me to emphasize their displeasure! ! :-)
A: A game section for kids with good strategy games where the player can win if they figure out how and makes no mistakes (nim, pursuit on a lattice, etc.) but not otherwise would be nice (with some prizes for really hard games). Some puzzles will be nice too.
Also, look at this. I would really love those to be played in the museum theater.
A: Hendrik Lenstra and others worked out the mathematics behind Escher's "Print Gallery" print, and filled in the hole in the center. Their website is here. Since then many people have used the same technique on photographs, a google search shows many examples. What I haven't seen, and would be an excellent exhibit, is a real-time video implementation of this.
Perhaps a good setup would involve a video camera pointed at a picture frame. The inside of the frame would be green or blue, so that green/blue screen technology could be used to detect the inside of the frame and distinguish it from objects or people overlapping it. The rest of the calculations are not mathematically difficult, but it would need a fast processor to get it to be real time.
A: I just saw this question and its many answers today.  As Chief of Content at MoMath, there is much I could tell you about in response.  Most important: we at MoMath are looking for all kinds of input.  If you’re at the Joint Math Meetings this week, come to our booth in the exhibit hall to meet us, learn more, and give us your ideas.  We have many activities scheduled there.  With your help, we'll be the coolest museum of any kind anywhere, because mathematics is so rich with engaging concepts.
As to the comment about Persi Diaconis, he certainly is involved.  MoMath will be inaugurating a free public lecture series on recreational mathematics in NY City later this year, and Perci is one of the wonderful speakers you can come hear.  Check momath.org for an announcement or go there to add yourself to our email list.
Many of the exhibit concepts suggested in these answers are already on our drawing boards, including the walk-on Mobius strip, but this isn’t the place to delve into the details of individual exhibits. A couple of answers mention Vi Hart’s Math Doodles.  She is already involved with MoMath and you can meet her at our JMM booth, along with MoMath's executive director, Glen Whitney, our chief of operations, Cindy Lawrence, and me.
Finally, a big thank you to Timothy, for posting this question, and to the many people who contributed interesting answers.
A: Mirrors exhibiting plane tilings:

Conic section billiard tables (the reflection properties!):



These are taken from the exhibitions documented at http://atractor.pt
Update: Elliptical pool tables can now be bought from http://loop-the-game.com.
A: https://www.scribd.com/document/479581247/Letter-to-MoMath-Board
Update: many of us got together to take a stand against unethical practices at the museum. See the above open letter to the Board of Trustees which recommends the replacement of the CEO Cindy Lawrence.
The concerns raised therein are at the intersection of the problems observed by the cosigners and are not comprehensive.

After serving as Chief of Mathematics at MoMath for the better part of 2 years, I'd like to shed some new light on this.
Firstly, this thread was a beautiful idea, to prompt the community for ideas before the Museum opened. However, at this point, the reality is: the last thing the Museum needs is more math ideas. What they need is proper implementation, and support for education. There is just a huge amount of work that one must do to get from a concept in a mathematician's brain to an interactive exhibit/lesson/activity that will work with kids. That is an ambitious thing to take on even if you don't have any other problems ... which the Museum does (see, for instance, the long history of complaints on Glassdoor -- they are a bit emotional, but having been there, I can say the complaints are well founded). I feel I did some great work there that I'm very proud of, but it was an uphill battle.
So here's what I'd like to see at the Museum:

*

*proper administrative support for the existing ideas to be correctly
implemented,

*a positive change in leadership so that the employees
will be treated with respect, and

*for the Board of trustees to take seriously the education standards there should be for a place bearing the name "National Museum of Mathematics."

A: A book scanner.
A: Maybe it will be good to place in such a museum some of the famous original papers or copies (like the Poincare's article on topology or homology or the Grothendieck's papers on schemes or maybe the famous page where Fermat had declared his Great Theorem).
Also it will be good to place here some of the drawings of (for example) mobile telephones with mathematical calculations.
A: How to form your own math circle or some other teaching movies, especially with kids solving problems and having fun at it.   I mentioned this in a comment, but it could use elaboration and advertisement.   Museums are for people, especially kids.  One goal is to empower kids to feel they can think.   Some sort of exhibits on what really good mathematics teaching feels like could be tremendously inspiring.  The museum could be a living organizing center for this.
A: The cosmic distance ladder: the mathematics used in (and very often developed for) measuring astronomical distances: the radius the Earth, the distance from the Earth to nearby bodies (the Moon and Sun), their radii, and on up to the shape and size of the universe. The first of these were first done with good accuracy in antiquity; the latter are still being worked on (indeed, so are the former, to amazing levels of precision).
I like this topic for several reasons. It shows applications of mathematics to physics. I would guess that much of the mathematics used was developed for this purpose, and if some of it was developed independently, well both of these are important aspects in mathematics. It shows different mathematics and different non-mathematical ideas all intertwined in a single endeavour.
I have seen a recording of an excellent lecture by Terence Tao on this topic.
A: There are some cool little formulas you can "prove" by putting together a 3d block puzzle.
For example, the formula for the sum of the first n squares can be seen by putting together 6 puzzle pieces to form a square prism with sides n, n+1, and 2n+1.  Here, each piece is a "staggered square pyramid" of volume 1+ ...+ n^2.  (Suggestion:  Take n=5) There are other such puzzle-ready formulas like the sum of triangular numbers. I believe "The Book of Numbers" by Conway and Guy has some. You could build nice big soft ones that schoolchildren can play with and grownups can appreciate.
A: Car parking by Lie group techniques!
A: As a late adapter to smart phones, I just recently started idling away some time playing the popular app Flow Free (Big Duck Games).  The goal is to connect pairs of dots of the same color with grid paths that do not intersect and cover the entire grid.  It makes me think of the Gessel-Viennot lemma about a determinant counting nonintersecting sets of paths (although there often only moves right and up are allowed).
One could make a computer display where the pairs of dots have a unique set of nonintersecting paths.  An initial step could lead visitors through the number of paths between two points being counted by binomial coefficients.  Then finding a set of nonintersecting paths is similar to the game (same sort of touchscreen interface), with the bonus that your work shows that a particular determinant is 1 (without all the arithmetic and plus / minus signs).
The same interface could have an exploration of paths strictly below the diagonal that lead to Catalan numbers, which connects to a whole host of visually engaging things such as triangulating regular polygons and making "penny piles" (Richard Stanley is up to 202 things counted by these numbers -- that could be a whole special exhibit).
A: 
I am sure a museum of Mathematics could not miss a selection of beautiful pictures of fractals.
A: Finding Tumors with Linear Algebra (and a lending hand by logarithms):
Example of the use of linear algebra (and with less emphasis, logarithms) in a toy example of computerized tomography (CT scana). 
Computer (Bob) chooses a transparent plexiglass box among four such boxes in a horizontal square grid in which to place an opaque box/tumor. The top of the grid must be covered so the investigator (Alice) can't see where the opaque box is. She can, however, rotate a laser penlight to shine a laser beam horizontally through the boxes and observe any shadow on the opposing side (complete attenuation). Bob displays a diagram of the four empty boxes and asks where the tumor/opaque box is. After Alice figures out how to locate the box, Bob asks what the minimum number of positions for the penlight for effectively locating the opaque box is. Next a second opaque box is placed randomly in the grid and the process iterated.
Placement of a third opaque box results in failure, but now a light meter is placed opposite the penlight, the three opaque blocks are replaced by semi-transparent boxes with their attenuation coefficients (ACs) given (nice integers), and Bob explains that for a box the AC is log[intensity of light in / intensity out] so that the total attenuation displayed by the light meter is log[intensity at penlight / intensity measured by the light meter] = AC of row or column of grid = sum of ACs of boxes in the row or column. Can Alice figure out the placement of each box? Must Bob demonstrate the linear algebra required to solve the problem? Finally, four boxes with differing ACs (nice integers) are configured in the grid and Alice asked if she can determine the ACs of each box. Bob gives a grey level display for each result--hands-on tomography.  
In my experience, very young kids with decent attention spans like such interactive challenges (no shock and awe approach, just simple reasoning). 
A: The Duel: Who Fired First?
Projections, Minkowski space, special relativity, and a legal paradox
A computer displays a line of observers outside a speeding train and a parallel line inside a train, all with respectively synchronized stop watches. Between the two lines and displaced along the length of the train are Galois and his frenemy with pistols poised. An animation captures the interest of the museum guests.
Two coordinate systems are superimposed with the origins of both systems coinciding at time $t=0$ well before the pistols are fired with respect to the observations of the ground crew.
A spacetime graph is displayed depicting the two events of the pistols being fired. In the classical Newtonian world, the two spatial coordinate lines are superimposed and depicted parallel to the two lines of observers with their origins displaced by the motion of the train by the time the shots are fired. The single time axis is depicted vertically perpendicular to both coordinate axes. The spatial displacement between the duelers is determined by drawing lines parallel to the single time axis through the point-events and down to the spatial axes, and analogously for the temporal displacement between the firings of the pistols. Let's have the two events happen simultaneously as recorded by the ground crew. Then the events will occur simultaneously according to the observers on the train also with exactly the same spatial displacement between the adversaries. These facts are easily demonstrated by the projections, and, indeed, are equivalent to the projections. 
In the world of special relativity, the facts change. While keeping the time and space axes for observations by the ground observers unchanged, the time and space axes for the observers on the train must be displayed pivoted about the origin towards each other. Projections parallel to the skewed time axis though the events to the spatial axis reveal that the train observers will record a smaller spatial displacement between the duelers, and projections parallel to the skewed spatial axis reveal a non-zero temporal displacement in the events, i.e., the times the pistols are observed to have been fired are not equal as measured by the train observers. 
A legal paradox! The ground observers might conclude the duelers are equally guilty of premeditated  murderous intent whereas the train observers might accuse one of premeditated murderous intent and the other of reacting only in retaliation.      
A: This:
alt text http://dl.dropbox.com/u/5390048/genus_2.gif alt text http://dl.dropbox.com/u/5390048/orange_torus.gif
see this MO post, by Bill Thurston.
A: A piece of conformal fabric. A conformal fabric is some membrane-like material that can stretch and unstretch, yet locally at any given point, only by equal amounts along a direction and a perpendicular to it. Such fabric you could stretch to any planar shape; if you stretch it from a circle to a square, say, you'd have found the Riemann mapping that maps a circle to a square! So holding a piece of conformal fabric and playing with it you'd at last get some "feel" for what the Riemann mapping theorem is all about.
Unfortunately, basic as it is, I could not find where one could get a piece of this valuable material. I'm not quite sure why - I'm not asking for something that depends on the axiom of choice, or that may live only in 4D, or for the moon. I can easily imagine holding a piece of conformal fabric, yet I have no clue how to make one.
[Edit by A. Henriques]: I think that this link might be showing exactly that material: http://www.bbc.co.uk/news/science-environment-35818924
A: There are many interesting films at the site http://www.etudes.ru/ (not in English): curves of constant width, Pick's theorem, geometry of polyhedra, an infinite staircase with the harmonic series, mechanisms of Chebyshev, etc. They can provide some interesting ideas for exhibits, and the people who are putting together the math museum in NY should consider contacting the folks behind this website (click on 6th link on the left, with the envelope icon). I saw a presentation of several of these films by the "main" person on the contact page, Nikolai Andreev, and it was quite impressive. 
A: I think graph theory is a good source for nice "labs" (see Deane Yang's post)...
There are nice activities you can do involving:


*

*The Königsberg Bridges, Eulerian paths, Hamiltonian paths

*The non-planarity of $K_5$ and $K_{3,3}$ 

*Map coloring and graph coloring, leading up to a discussion of the four color theorem

*Euler characteristics of graphs, leading up to a discussion of topology

*Traveling salesman problems (... leading up to a discussion of NP-completeness????)
A: The hairy ball theorem demonstrated with a ball with hair on it and a comb. 
What happens if we deform the ball a little, so that it is shaped like a banana? 
What happens on a torus? 
(I'm not so sure that it's a good idea to emphasize the name "hairy ball".) 
Euler characteristics of polyhedra and possibly of manifolds.
I would like to see something about manifolds and the shape of the universe. Maybe something about string theory as well.
A: I think it would be nice to have exhibits (or "labs" -- see Deane Yang's post) on various probability "paradoxes", such as the Monty Hall problem, the false positive paradox, the birthday paradox...
Like the central limit theorem (see Sam Nead's post), many of these "paradoxes" can be experimentally demonstrated. The birthday paradox can be quite impressive when you have a group of around 30 to 40 people -- assuming it works out, that is ;-)
The Monty Hall problem can also be demonstrated experimentally. Once, at a party with non-mathematicians, I played 20 instances of "the Monty Hall game", and already one could see that the "switch doors" strategy was usually more successful. Happily, my audience was actually rather unsatisfied with my experimental demonstration, and wanted a more conceptual explanation. (I actually found this to be somewhat curious -- for me personally at least, the experimental demonstration is very satisfying!) This lead into a long and fun discussion.
I like the following quote by Israel Gelfand:

Mathematics is a way of thinking in everyday life. It is important not to separate mathematics from life. You can explain fractions even to heavy drinkers. If you ask them, ‘Which is larger, 2/3 or 3/5?’ it is likely they will not know. But if you ask, ‘Which is better, two bottles of vodka for three people, or three bottles of vodka for five people?’ they will answer you immediately. They will say two for three, of course.

I think it can be difficult for many people to appreciate math "for its own sake". We mathematicians usually find, for example, the infinitude of primes, and the proof thereof, to be pretty awesome. But I don't think that you can expect most people to react to such things the same way that we do. I think the reason is because, as in the Gelfand quote, it is often not apparent how these things connect to "the real world", and it is often not apparent that these kinds of considerations can arise very naturally. So to get people excited about math, I think that it can be useful to first get them to care about a problem or arouse their curiosity in something, and then demonstrate that math can be used to solve that problem. This nice TED talk also argues for this point.
Sorry for rambling...
A: The Antikythera Mechanism, a stone encrusted mechanical computer from 150-100 BCE designed to calculate astronomical positions. It has a degree of mechanical sophistication is comparable to a 19th century Swiss clock. Nothing as complex is known for the next thousand years.
In addition it'd be nice to have an explanation of its workings along with a modern functional copy that one can directly manipulate.
A: If a Museum is a place where mathematics meets people of every kind, it is important to let them think that our discipline is useful, and is not just a game. I advocate to display applications of good mathematics in the everyday life. Cryptography has been evoqued before and I voted +1 for this answer. Let me add a few others :


*

*Radon transform, with application to tomography, and therefore to medical diagnosis.

*QR algorithm, with application to searching on the web (Google page rank algorithm).

*Dynamical systems, saddle points and their application to the launch of spacecrafts away from the ecliptic.


I have not been involved in the elaboration of any mathematical exhibition, but I am convinced that if these topics have been successfully used by non-mathematician, they can be explained to a non-scientific audience. I except that they contribute to a positive judgement of mathematics by the population.
A: Various aspects of Symmetry have been mentioned, but one aspect which could be explored is "near symmetry", for example:
A very large set of Penrose Tiles to play with.
A: There are two such hands-on Math museums in Germany, and they are tremendously successful. Number one is Beutelspacher's Mathematikum, which had over a million visitors since 2002. More recent is the Math Adventure Land in Dresden, which also attracts a high number of visitors.  
A: A working differential analyzer and other early computers would be pretty cool. 
A: I would like to see an exhibit on the mathematics of perspective drawings. This is an old application of mathematics that has lead to some interesting theory. It is also an application in an area that most people don't think of as mathematical. 
Related to this there should be 3D-models of 4D-objects. (I can not believe nobody has mentioned this.) One should point out that they can be seen as the analogue of 2D-drawings of 3D-objects. This is an excellent illustration of mathematicians tendency for abstraction and generalization. 
A: A bicycle with square wheels.
A: Dynkin diagrams and regular polyhedra!!! And the list of all possible finite simple groups!!!
I mean, cute applications listed here are cute, but you should also put a tangible, direct representation of human achievement in mathematics. It might be a bit hard to properly explain them, but still ... 
As a person with background in elementary particle physics, the fact that human beings have classified possible symmetries themselves (not just the symmetry realized) strikes my inner cord quite strongly.
A: I would enjoy a "hall of infinities", listing countable ordinals... not all of them, but enough to get the idea across.  It's possible to draw nice pictures of some them, at least up to $\omega^3$ or so, and even kids know how to count, so they might enjoy knowing what comes after the numbers they learned about in school.
I tried to present this information in story form in "week236" of This Week's Finds.
Actually, now that I think about it, there should be a "hall of numbers" that starts by listing lots of interesting natural numbers and then moves on to countable ordinals.
MIT has an "infinite corridor" that would do well for this, but I guess a shorter version would still be okay.
A: See the water demonstration of the Pythagorean theorem:
[]1
https://www.youtube.com/watch?v=CAkMUdeB06o
A: I'd suggest an interactive exhibit where people can tweak the parameters of a population model with 3 species in it.  Have an information panel which explains what the parameters represent.  Suggest goals such as (1) keep the rabbits from going extinct, (2) create a stable equilibrium where all three species survive, (3) find a cyclic solution, etc.  Experts could probably come up with a good system that exhibited lots of interesting behavior.  Let people visualize their solutions both graphically (3-D graph of all three populations as well as 2-D graphs of any two populations of their choosing).  There are probably other clever graphical representations that others could come up with as well.
Overall, I think this is a great endeavor and I hope that you will focus on making the exhibits interactive, with pathways to learning.  Ideally, the same person could visit an exhibit a half dozen times and learn something new each time.  Of course, making things fun and interesting is really important too - but I think that should naturally emerge from the design of interactive ways to explore a beautiful piece of mathematics.
A: The Body Counter: Making Math Matter in Human Rights 
An exhibit explaining the use of multiple systems estimation (MSE, a type of statistical inference) in human rights investigations and trials, based on Tina Rosenberg's article The Body Counter.
A: An original S.I. metre bar.
Who wouldn't love to see one of those in person?
A: *

*Models of the sphere eversion that emulate the chicken wire ones. I think someone made some for Morin at one time. 

*Interactive computer graphics of Seirpinski $n$-simplices,  interactive computer graphics of hypercubes, the 120 cell, and the  24 cell. 

*Transparent models of knotted surfaces.
A: Something like the clickable math atlas:
http://www.math.niu.edu/~rusin/known-math/index/mathmap.html
but the clicks should probably lead to places showing for laypeople what these different fields are!
A: Nothing. In a museum you put thing that are obsolete now. In mathematics nothing is obsolete (yet).
