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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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    $\begingroup$ 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 $\endgroup$
    – Zev Chonoles
    Commented Dec 25, 2010 at 20:38
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    $\begingroup$ 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: en.wikipedia.org/wiki/Borromean_rings) $\endgroup$
    – Eric Zaslow
    Commented Dec 26, 2010 at 3:09
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    $\begingroup$ 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. $\endgroup$
    – fedja
    Commented Dec 26, 2010 at 15:34
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    $\begingroup$ @fedja : While Diaconis doesn't seem to be involved, the advisory board (listed here : momath.org/about/advisory-council) 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... $\endgroup$
    – Andy Putman
    Commented Dec 26, 2010 at 20:34
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    $\begingroup$ @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! $\endgroup$
    – Andy Putman
    Commented Dec 26, 2010 at 23:18

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See the water demonstration of the Pythagorean theorem:

[Water demonstration of the Pythagorean theorem]1 https://www.youtube.com/watch?v=CAkMUdeB06o

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    $\begingroup$ Could someone help me either embed this video, or extract an image of the video for linking? I couldn't see how to do this. $\endgroup$
    – Joel David Hamkins
    Commented Jan 24, 2014 at 3:43
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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.

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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.

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  • $\begingroup$ The truncating triangle could be flat (in which case you see a Platonic solid at the center) or it could be a spherical triangle, in which case you get to see a triangle group on the sphere. If you make the truncating triangle mirrored, as well, then you get a nice object. However, it might be nice to have a regular mirrored sphere nearby, and a discussion of spherical reflection (not quite the same as spherical inversion....) $\endgroup$
    – Sam Nead
    Commented Jan 10, 2011 at 15:50
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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.

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    $\begingroup$ Heartily second the slide rule but a wide variety showing the many forms that these were implemented. The linear, spiral, cylindrical to generate scales multi feet long for five figure accuracy. Especially relevant in 2014 as the 400th anniversary of Napier's publication of logarithms in 1614. $\endgroup$
    – David
    Commented Jan 22, 2014 at 18:01
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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.

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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.

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  • $\begingroup$ Poincare, on page 66 of Science and Hypothesis (see books.google.com/books?id=2QXqHaVbkgoC&q=refraction) suggests using the ball model. This might be harder to build (as you have to layer material in concentric spheres instead of parallel planes) but could have other advantages. $\endgroup$
    – Sam Nead
    Commented Dec 29, 2010 at 14:19
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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).

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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.

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  • $\begingroup$ Involve marked cog wheels with 9, 16 and 25 teeth (and for that matter, with 9*16, 9*25 and 16*25 teeth). $\endgroup$
    – Henry Segerman
    Commented Jan 3, 2011 at 0:16
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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.

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    $\begingroup$ I don't believe that watching working sessions of mathematicians, even with commentary, would be particularly inspiring or interesting to non-mathematicians. What we do is far too foreign. Why would they want to watch us struggle through something they don't understand and have no a priori interest in? $\endgroup$
    – Deane Yang
    Commented Jan 9, 2011 at 0:56
  • $\begingroup$ I am not a musician , I like music yet had not follow any king of schooling about it ( I cannot read the key) , same for dancing (though I dance rock and roll). Now in both cases I have seen several movies / shows about professional musicians and dancers the way they work the questions they ask and it was very informative for me as for other people in the same situation as I. $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Jan 9, 2011 at 2:48
  • $\begingroup$ @Dean again you comment is interesting as you say "foreign" : it is precisely this strangeness that must be removed. $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Jan 9, 2011 at 2:51
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    $\begingroup$ How about movies of kids solving math (or other practical geometry) problems cooperatively, in classrooms even. If you were a kid in a boring school, you might be very gratified to see how a good problem solving session in school might operate. If it were done in the math circle fashion, kids could be motivated to join something like them. They could be arranged by grade level, or you could choose easier or harder ones. Grown up mathematicians would only be one of a series. Come to think of it, math circle organizing could be a major activity of the museum, like glee clubs $\endgroup$
    – sigoldberg1
    Commented Jan 11, 2011 at 11:00
  • $\begingroup$ YES! Very nice siglodberg. In France ( Maths en Jeans AT fr.wikipedia.org/wiki/MATh_en_JEANS ) gives research problem to kids ( from 10 years old on) most of the time the problem is actually an open one. The kids work in group under the aegis of a teacher whose role is solely to help them being clear and to communicate. They come out with a presentation of their work ( findings) at the end of the year. $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Jan 12, 2011 at 0:08
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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: Contact structure
and here's the same model upside down, before it was completely finished: Contact structure

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  • $\begingroup$ I would love to get involved in the actual construction of such an object. If there is actual interest, please take contact with me. $\endgroup$
    – André Henriques
    Commented Dec 27, 2010 at 23:25
  • $\begingroup$ You might consider shapeways.com or other 3D printing companies for constructing (a small-scale version of) your object. $\endgroup$
    – Timothy Chow
    Commented Dec 28, 2010 at 2:08
  • $\begingroup$ Thanks for the link. But I already made one in wood and cardboard (unfortunately, my mother threw it accidentally to the trash!). I'm thinking of making a new, bigger one, out of wood. $\endgroup$
    – André Henriques
    Commented Dec 28, 2010 at 22:36
  • $\begingroup$ My approach to making contact structures interactive would be to use the natural contact structure on the orientations of a ball, and let people roll the ball along one of several paths on the ball, perhaps to try to get the ball into a particular orientation. I have used this when trying to explain contact structures to nonmathematicians. $\endgroup$
    – Douglas Zare
    Commented Dec 31, 2010 at 10:11
  • $\begingroup$ @Douglas Zare: The fact that there is a "space of possible orientations of a ball" is probably a quite foreign concept to the visitors of the museum. So you can't take that as the starting point of an explanation of something else. I was just imagining a contact structure on a piece of the $\mathbb R^3$ in which we live. $\endgroup$
    – André Henriques
    Commented Dec 31, 2010 at 22:14
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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.

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Flexible polyhedron.

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  • $\begingroup$ Why not make it general: a collection of models of polyhedra (Platonic, Archimedean...)? $\endgroup$
    – J. M. isn't a mathematician
    Commented Dec 26, 2010 at 11:36
  • $\begingroup$ youtube.com/watch?v=reO7Qx3HiIg One could demonstrate the bellows conjecture by blowing smoke into a flexible polyhedron. $\endgroup$
    – Ian Agol
    Commented Dec 28, 2010 at 6:16
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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.

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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?

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    $\begingroup$ A shallow-water-wave soliton demonstration would be nice. $\endgroup$
    – S. Carnahan ♦
    Commented May 2, 2011 at 9:20
  • $\begingroup$ Addition and subtraction as constructive and destructive interference and Young's double slit experiment with connection to quantum mysteries--wave-particle duality. Interplay of complex analysis, Huyghen's principle, and quantum physics. $\endgroup$
    – Tom Copeland
    Commented Sep 3, 2015 at 19:04
  • $\begingroup$ Maybe showing the heat equation uised in image analysis? $\endgroup$
    – kjetil b halvorsen
    Commented Mar 16, 2016 at 14:29
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This hyperbolic tesselation applet: http://www.plunk.org/~hatch/HyperbolicApplet,
maybe enhanced so as to also include the Eucledian and spherical cases.

alt text

(Picture taken from http://aleph0.clarku.edu/~djoyce/poincare/PoincareApplet.html)

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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.

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    $\begingroup$ Her father, George Hart, is "Chief of Content" at the museum, and I think she has already contributed some things to the museum to do with music. $\endgroup$
    – Henry Segerman
    Commented Dec 26, 2010 at 21:32
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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.

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  • $\begingroup$ Not to be confused with: whenever there is an https:// in their browser, their data is secure... $\endgroup$
    – darij grinberg
    Commented Dec 31, 2010 at 15:57
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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.

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  • $\begingroup$ "some people take "ends in an even digit" as a suitable definition for even number". That is of course ok for the usual 10-base system. To dispel it, write out the first integers in base 3: 0 1 2 10 11 12 20 21 etc. Five in base three ends with an even digit. $\endgroup$
    – kjetil b halvorsen
    Commented Dec 1, 2012 at 3:23
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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

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    $\begingroup$ שלום גיל, ממש יפה, כל הכבוד, אבל זה בעברית $\endgroup$
    – Patrick I-Z
    Commented Dec 30, 2010 at 23:26
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    $\begingroup$ Dear Patrick, Yes it is in Hebrew, (but you can see what the exhibits are and guess what is said). In any case I think the mathematics museum exist now as a permanent exhibition in Abu Dis University. If I will fine more material/pictures I will add them. $\endgroup$
    – Gil Kalai
    Commented Dec 31, 2010 at 11:02
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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.

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  • 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.

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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)

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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.

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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.

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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

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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!

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  • $\begingroup$ I question your choice of "intuitive" for Cantor's proof (adopting for a moment the perspective of a layperson) although beautiful is more than apt! $\endgroup$
    – dvitek
    Commented Jan 3, 2011 at 22:57
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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.

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  • $\begingroup$ Rich Schwartz once made a physical model of a proof, based on the proof given in this applet: math.brown.edu/~res/Java/App42/test1.html $\endgroup$
    – Ian Agol
    Commented Dec 28, 2010 at 6:43
  • $\begingroup$ Yes, but I'd start with the ancient Chinese diagram, where you take 4 copies of the right triangle, build a square with sides of length a+b, and consider the inside "square" with sides of length c. Of course seeing that the inside rhombus really is a square requires the parallel postulate, but any 7 year-year old can see it is a square, and it would be very easy to come back to after you "jump to non-Euclidean geometry." $\endgroup$
    – Airymouse
    Commented Feb 16, 2017 at 16:30
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The trammel of Archimedes. See http://blog.makezine.com/archive/2010/06/my_10_favorite_mechanical_animation.html

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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.

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Klein bottle (with a description containing at least 15 characters)

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    $\begingroup$ Downvoted because any sensible interpretation of your answer is subsumed in J.M.'s answer. $\endgroup$
    – Daniel Moskovich
    Commented Dec 26, 2010 at 10:14
  • $\begingroup$ Reverted the purely cosmetic effort to a very old answer $\endgroup$
    – Yemon Choi
    Commented Sep 4, 2015 at 1:33
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