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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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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:

  1. proper administrative support for the existing ideas to be correctly implemented,
  2. a positive change in leadership so that the employees will be treated with respect, and
  3. for the Board of trustees to take seriously the education standards there should be for a place bearing the name "National Museum of Mathematics."
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    $\begingroup$ @GerryMyerson : Glassdoor is a website where current and former employees anonymously review companies. $\endgroup$
    – Timothy Chow
    Commented Dec 30, 2019 at 16:04
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    $\begingroup$ @GerryMyerson I've added a link to a particularly insightful review. From there you can find a plethora of similar ones. $\endgroup$
    – j0equ1nn
    Commented Dec 31, 2019 at 6:48
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    $\begingroup$ It’s about time! Can also personally vouch that these concerns are well-founded. $\endgroup$
    – Noah Snyder
    Commented Apr 9, 2021 at 20:57
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    $\begingroup$ I just hate them so much, and it's such a shame that mathematicians don't know how terrible a workplace it is and keep working with them. Maybe the meta thread about news of interest to MO would be more appropriate? Joe have you considered writing a letter to the Notices? $\endgroup$
    – Noah Snyder
    Commented Apr 10, 2021 at 2:31
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    $\begingroup$ My wife was their first employee. It’s a Theranos-level disaster of an organization. The linked Glassdoor review is spot-on. The letter barely scratches the surface. It’s been like this for 10+ years and the employees all keep saying the same things. $\endgroup$
    – Noah Snyder
    Commented Apr 10, 2021 at 13:20
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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"...

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    $\begingroup$ It's a planimeter (en.wikipedia.org/wiki/Planimeter). $\endgroup$
    – Zev Chonoles
    Commented Dec 25, 2010 at 17:18
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    $\begingroup$ And it's an example of Green's theorem in action. $\endgroup$
    – Nate Eldredge
    Commented Dec 26, 2010 at 0:00
  • $\begingroup$ When I visited the Science Museum a few years ago I singled out this object as the most impressive thing I saw there! $\endgroup$
    – Dan Piponi
    Commented Dec 26, 2010 at 16:34
  • $\begingroup$ I love these things. My physicist colleague actually has one. $\endgroup$
    – Jon Bannon
    Commented Oct 14, 2015 at 23:14
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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)

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    $\begingroup$ I love the emphasis on interactivity. Perhaps also they could offer the possibility of playing asteroids on either a torus or a klein bottle? $\endgroup$
    – stankewicz
    Commented Dec 26, 2010 at 22:28
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    $\begingroup$ The "visualizing the tenth dimension" video is complete pseudoscientific nonsense. $\endgroup$
    – Harry Gindi
    Commented Dec 29, 2010 at 16:31
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    $\begingroup$ As far as walking on a Moebius strip is concerned, I favor the idea of a band that is moving and a walker that is stationary (similar to a hamster wheel). $\endgroup$
    – André Henriques
    Commented Dec 30, 2010 at 23:24
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    $\begingroup$ I've taken it upon myself to remove the offending link. $\endgroup$
    – Harry Gindi
    Commented Jan 1, 2011 at 7:04
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    $\begingroup$ I'm now completely convinced that a sphere eversion is what 3D movie technology was made for. $\endgroup$
    – Harrison Brown
    Commented May 8, 2011 at 12:30
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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.

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    $\begingroup$ For searchability, this device is called a Galton box. $\endgroup$
    – Zsbán Ambrus
    Commented Dec 25, 2010 at 22:54
  • $\begingroup$ Thanks for letting me know! I'll add a link to the answer. $\endgroup$
    – Sam Nead
    Commented Dec 26, 2010 at 11:36
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    $\begingroup$ They have (had?) a Galton box in the math section of the Museum of Science in Boston with an added feature which I found intriguing (and clever): while most of the balls in the box are white, only a handful were black. After operating the machine the balls would overall arrange themeselves in a bell curve, BUT the few black balls would be scattered here and there in a unstructured random way. This shows that the expected distribution is reached only after a large amount of trials (=balls) while the theory is ineffective for a small amount. Unfortunately no panel on the exhibit explained this! $\endgroup$
    – Andrea Mori
    Commented Dec 31, 2010 at 14:01
  • $\begingroup$ They have one of these at the nearby Queens Hall of Science. I wonder if MoMath can just purchase the Hall of Science's math exhibits. They're nice but completed ignored by most visitors. $\endgroup$
    – Dan Lee
    Commented Feb 4, 2011 at 20:40
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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.

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  • $\begingroup$ This could serve as a nice introduction to modular arithmetic as well! $\endgroup$
    – Kevin H. Lin
    Commented Dec 26, 2010 at 2:25
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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.

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  • $\begingroup$ You could also include material on knots conducive to nice visuals, e.g. knotting of molecules and DNA. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 26, 2010 at 5:29
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    $\begingroup$ Wire frame knots, that you can dip in a bubble table. Then you can compare your creations to pictures of Seifert surfaces. $\endgroup$
    – Sam Nead
    Commented Dec 31, 2010 at 18:56
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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).

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    $\begingroup$ Also you could let people play around with different Penrose tilings, as in, you could have a big set of plastic tiles and a big board for people to try and fail to form a periodic pattern. $\endgroup$
    – Dan Petersen
    Commented Dec 26, 2010 at 6:13
  • $\begingroup$ Ah, right, I was going to mention something about quasiperiodic tilings but forgot. Another good source of historical examples (in Islamic architecture, I think). $\endgroup$
    – Qiaochu Yuan
    Commented Dec 26, 2010 at 6:17
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    $\begingroup$ The Math Midway actually already has Penrose's kite and dart tiles on magnets. (As well as interlocking money tiles.) $\endgroup$
    – Dan Lee
    Commented Feb 4, 2011 at 20:50
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"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

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Sculptures of surfaces would be lovely.

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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.

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    $\begingroup$ +1! I strongly agree with this. $\endgroup$
    – Kevin H. Lin
    Commented Dec 26, 2010 at 1:23
  • $\begingroup$ I'm imagining 3 driving games: the first one gives you a target for where to be at each moment, the second one gives you a target for how fast to go at each moment, and the third one gives...the integral?!?! $\endgroup$
    – Kevin O'Bryant
    Commented Dec 26, 2010 at 3:52
  • $\begingroup$ I do agree with the handson side ( just google "Mathematics is not a spectator sport"). Yet I hope you did not think that dead science is implicit in the word Museum ( otherwise there is no more painting). $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Jan 8, 2011 at 23:59
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    $\begingroup$ My experience with museums is that it is a passive experience and, even when there is an exception and something for the visitor to do, it is rather superficial and does not convey at all the experience of, say, doing or using mathematics. Certainly, there are very few art museums that allow you to do the painting yourself. And there is a reason why most people view the word "museum" to mean "a rather dull place". $\endgroup$
    – Deane Yang
    Commented Jan 9, 2011 at 0:43
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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! ! :-)

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  • $\begingroup$ I wonder what the Oppenheimer brothers would have thought of such legal action. $\endgroup$
    – Tom Copeland
    Commented Sep 1, 2015 at 19:12
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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.

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    $\begingroup$ Vi Hart's videos are super great! Some of the things in her videos would make cool museum exhibits or activities, as well. $\endgroup$
    – Kevin H. Lin
    Commented Dec 26, 2010 at 4:30
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    $\begingroup$ License a copy of xkcd's drawing of winning tic-tac-toe. xkcd.com/832 $\endgroup$
    – user1504
    Commented Dec 30, 2010 at 16:23
  • $\begingroup$ Something tells me that the MoMath people are well aware of Vi Hart's amazing doodles... (Their chief of content happens to be a fellow named George Hart.) $\endgroup$
    – Dan Lee
    Commented Feb 4, 2011 at 20:47
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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.

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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.

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  • $\begingroup$ Hi George...thanks for posting. I notified Ken Fan and Glen Whitney by email when I first posted this question but I guess the word didn't circulate among the official MoMath people as quickly as I thought it would. $\endgroup$
    – Timothy Chow
    Commented Jan 4, 2011 at 15:48
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Mirrors exhibiting plane tilings: enter image description here

Conic section billiard tables (the reflection properties!): enter image description here enter image description here enter image description here

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.

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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.

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    $\begingroup$ Links to images are dead. $\endgroup$
    – Dag Oskar Madsen
    Commented Dec 3, 2016 at 22:20
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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

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  • $\begingroup$ Very nice. An interesting engineering challenge to make such a thing. $\endgroup$
    – Dylan Thurston
    Commented May 2, 2011 at 3:01
  • $\begingroup$ I'm skeptical about easily imagining holding it. What does it feel like to pull on it? When you're stretching it into a new shape, you can't simply pull it by the edges because a Riemann mapping has very rigid boundary behavior. $\endgroup$
    – Tom Goodwillie
    Commented May 2, 2011 at 4:16
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    $\begingroup$ Yes, unfortunately conformal fabrics would violate conservation of energy, as a small motion in one place may lead to a huge motion somewhere else. Perhaps I should settle for a crank-powered approximate conformal fabric, that would at least illustrate the difficulty in making the real thing. $\endgroup$
    – Dror Bar-Natan
    Commented May 11, 2011 at 19:28
  • $\begingroup$ @Dror Bar-Natan: Why do you say "conformal fabrics would violate conservation of energy, as a small motion in one place may lead to a huge motion somewhere else"? When you use a pair of scissors, a small motion in one place leads to a large motion somewhere else---do scissors violate conservation of energy too? $\endgroup$
    – Vectornaut
    Commented Jan 4, 2012 at 20:50
  • $\begingroup$ @Dylan Thurston: Although it wouldn't be as nice as a physical conformal fabric, you can build almost anything under the glass of a touch screen... $\endgroup$
    – Vectornaut
    Commented Jan 4, 2012 at 20:59
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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.

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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????)

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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.

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    $\begingroup$ It's high time the English-speaking world decided to drop the name "hairy ball" theorem and started using a more civilized term (like the hedgehog combing theorem). $\endgroup$
    – Thierry Zell
    Commented Jan 2, 2011 at 19:44
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    $\begingroup$ I prefer the term "windy planet" to "hairy ball". $\endgroup$
    – DavidLHarden
    Commented Mar 10, 2011 at 16:42
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    $\begingroup$ You could change "ball" to "coconut", and no one would be offended. $\endgroup$
    – Todd Trimble
    Commented Nov 30, 2012 at 21:33
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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...

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    $\begingroup$ "they will answer you immediately" They must still be on the first couple of drinks. $\endgroup$
    – Dan Piponi
    Commented Dec 26, 2010 at 16:31
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    $\begingroup$ Gelfand's example reminds me of the difference between the following two questions. Question 1: Shown 4 cards on a table, displaying respectively "25," "16," "B," and "C," what is the minimum number of cards you need to turn over to verify the statement, "every card with a B on it has a number > 20 on the reverse side"? Question 2: There are 4 people at a bar; the first is 25 years old, the second is 16 years old, the third has a beer, and the fourth has a coke. What is the minimum number of people you need more information about to verify that there is no underage drinking going on? $\endgroup$
    – Timothy Chow
    Commented Dec 27, 2010 at 17:33
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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.

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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.

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    $\begingroup$ 1. I'm not too sure the elegance of QR can be done justice in an exhibit that will be glanced at for at most a minute or two. 2. A (heavily modified) power method is used, not QR proper: mathworks.com/moler/exm/chapters/pagerank.pdf $\endgroup$
    – J. M. isn't a mathematician
    Commented Dec 28, 2010 at 0:39
  • $\begingroup$ 3. Can the Radon transform be talked about without mentioning integrals too much? This would be great if so; it would be great to show how these machines "slice" bodies. $\endgroup$
    – J. M. isn't a mathematician
    Commented Dec 28, 2010 at 0:40
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    $\begingroup$ @J. M.: the museum does not have to explain the math, just mention that it's there, and point out that we wouldn't know how to solve the same problem without that specific tool. Remember, it's easy for us to be jaded about applications of mathematics, but many (even educated) people don't even begin to suspect how much mathematics is involved in the devices they're so fond of. Not to mention that many people who should know better (e.g. Claude Allegre) seem to think that "we'll just let the computer do it" is an answer and an end to a problem rather than the beginning of it. $\endgroup$
    – Thierry Zell
    Commented Jan 2, 2011 at 20:35
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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.

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  • $\begingroup$ Dan Lee commented on Qiaochu Yuan's answer that the Math Midway already has some magnetic Penrose tiles. Hopefully these will end up in MoMath! (Just wanted to point this out here so nobody else wastes ten minutes writing a pitch for magnetic Penrose tiles, complete with estimated magnet prices.) $\endgroup$
    – Vectornaut
    Commented Oct 26, 2011 at 5:56
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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.

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    $\begingroup$ There is also the Arithmeum (arithmeum.uni-bonn.de/en/home) in Bonn, which features a history of calculational devices, many of which the visitors can touch and play with. Interestingly, the museum also houses a collection of modern art which has been inspired by mathematics. $\endgroup$
    – JCollins
    Commented Jan 2, 2011 at 19:09
  • $\begingroup$ There is also the ix-quadrat founded in 2002 by Jürgen Richter-Gebert. So the count of mathematics museum in Germany alone stands at 4. The ix-quadrat is probably the smallest in physical size, yet it's a gem. $\endgroup$
    – Peter Heinig
    Commented Aug 26, 2017 at 7:16
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A working differential analyzer and other early computers would be pretty cool.

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A bicycle with square wheels.

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    $\begingroup$ The Exploratorium already has one... exs.exploratorium.edu/exhibits/square-wheels maybe something different could be done for this new museum? $\endgroup$
    – J. M. isn't a mathematician
    Commented Dec 27, 2010 at 12:22
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    $\begingroup$ This was implemented by MoMath in the form of a square-wheeled tricycle. See mathmidway.org/math-midway-activities-pedal.php. $\endgroup$
    – Ken Fan
    Commented Dec 27, 2010 at 17:13
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    $\begingroup$ It might be novel to go beyond the nearly hackneyed square wheel to other shapes. See this MO question: mathoverflow.net/questions/29988/… $\endgroup$
    – Joseph O'Rourke
    Commented Dec 27, 2010 at 19:47
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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.

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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.

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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.

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