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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.

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I'm reminded of the following quote, which perhaps would be good to include in the museum: "Numbers exist only in our minds. There is no physical entity that is the number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe." - Linear Algebra by Fraleigh + Beauregard –  Zev Chonoles Dec 25 '10 at 20:38
What an opportunity! Clearly, the fact that many of us mathematicians ourselves don't even know about this project (or related ones mentioned in other responses) means, above all, we need to hire marketing professionals! And designers should build the exhibits. (But as for content, I've always liked the Borromean rings: –  Eric Zaslow Dec 26 '10 at 3:09
I'm wary of both marketing professionals and designers. We are interested neither in selling junk people do not really need, nor in trying to beautify something that is ugly by its nature. If anything, we should get a few high level math. people with good taste and some knowledge of the outside world to make decisions about what to do. But I doubt it'll be done. I bet Percy Diaconis, say, has been neither invited as a consultant, nor even told of the project. –  fedja Dec 26 '10 at 15:34
@fedja : While Diaconis doesn't seem to be involved, the advisory board (listed here : includes a lot of very good mathematicians, for example Bjorn Poonen. That being said, I'm still pretty skeptical that a "museum of mathematics" is possible... –  Andy Putman Dec 26 '10 at 20:34
@Timothy : My skepticism comes precisely from the math sections of a number of science museums I have been too. They've all been pretty lame (and that's not just Andy the math-snob talking -- my wife and kids haven't enjoyed them either). We just don't have cool things like robots or spaceships or dinosaur bones or life-size models of the human heart to show off! –  Andy Putman Dec 26 '10 at 23:18

94 Answers 94

Something like the clickable math atlas:

but the clicks should probably lead to places showing for laypeople what these different fields are!

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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.

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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.

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Nothing. In a museum you put thing that are obsolete now. In mathematics nothing is obsolete (yet).

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Actually, putting the empty set in the museum, would not be a bad idea at all. –  Lucas K. Dec 26 '10 at 16:48

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