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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
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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) –  Eric Zaslow Dec 26 '10 at 3:09
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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
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@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... –  Andy Putman Dec 26 '10 at 20:34
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@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

90 Answers 90

A book scanner.

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Please elaborate? –  Kevin H. Lin Dec 25 '10 at 23:41
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It is full of mathematical technology and, out of all proposals in this thread, gave the most positive feedback to mathematics. –  darij grinberg Dec 26 '10 at 14:15

Sculptures of surfaces would be lovely.

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Do you know about bathsheba.com . The items there are really great. Also look up George Hart and Carlo Sequin. –  Dick Palais Dec 25 '10 at 16:59
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As many of these (nytimes.com/slideshow/2004/12/02/magazine/…) as you can find. They are beyond beautiful. –  Sam Nead Dec 25 '10 at 19:19
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A related MO question: mathoverflow.net/questions/32479/… –  Timothy Chow Dec 27 '10 at 3:23
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John Hempel has a nice sculpture of the pseudosphere in his office, for which he had made a small rubber mould of a patch of it. One can move it around and see that it always fits, demonstrating that it is constant curvature. –  Ian Agol Dec 28 '10 at 6:42

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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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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A game section for kids with good strategy games where the player can win if he figures 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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Vi Hart's videos are super great! Some of the things in her videos would make cool museum exhibits or activities, as well. –  Kevin H. Lin Dec 26 '10 at 4:30
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License a copy of xkcd's drawing of winning tic-tac-toe. xkcd.com/832 –  userN Dec 30 '10 at 16:23

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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It's a planimeter (en.wikipedia.org/wiki/Planimeter). –  Zev Chonoles Dec 25 '10 at 17:18
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And it's an example of Green's theorem in action. –  Nate Eldredge Dec 26 '10 at 0:00

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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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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+1! I strongly agree with this. –  Kevin H. Lin Dec 26 '10 at 1:23
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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". –  Deane Yang Jan 9 '11 at 0:43

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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For searchability, this device is called a Galton box. –  Zsbán Ambrus Dec 25 '10 at 22:54
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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! –  Andrea Mori Dec 31 '10 at 14:01

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'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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A working differential analyzer and other early computers would be pretty cool.

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The Curta would be nice: curta.org –  J. M. Dec 26 '10 at 11:40

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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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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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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Wire frame knots, that you can dip in a bubble table. Then you can compare your creations to pictures of Seifert surfaces. –  Sam Nead Dec 31 '10 at 18:56

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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"they will answer you immediately" They must still be on the first couple of drinks. –  Dan Piponi Dec 26 '10 at 16:31
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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? –  Timothy Chow Dec 27 '10 at 17:33

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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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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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. –  Dan Petersen Dec 26 '10 at 6:13
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The Math Midway actually already has Penrose's kite and dart tiles on magnets. (As well as interlocking money tiles.) –  Dan Lee Feb 4 '11 at 20:50

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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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. –  Henry Segerman Dec 26 '10 at 21:32

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

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Downvoted because any sensible interpretation of your answer is subsumed in J.M.'s answer. –  Daniel Moskovich Dec 26 '10 at 10:14

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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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). –  Thierry Zell Jan 2 '11 at 19:44
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I prefer the term "windy planet" to "hairy ball". –  DavidLHarden Mar 10 '11 at 16:42
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You could change "ball" to "coconut", and no one would be offended. –  Todd Trimble Nov 30 '12 at 21:33

Look at the 'surfer video' which among other things shows how visualizations of algebraic geometry can be presented in real-time in an exhibition.

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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

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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Tony deRose, most likely. maa.org/news/101509derose.html –  Allen Knutson Dec 26 '10 at 18:39

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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The standard orientation on $S^1$, if you can borrow it from NIST.

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An original S.I. metre bar.

Who wouldn't love to see one of those in person?

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I wouldn't. Talk about non-canonical choices. –  Qiaochu Yuan Dec 26 '10 at 20:59
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I don't understand how it's math history. It seems almost like engineering history. –  Daniel Moskovich Dec 28 '10 at 21:24

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