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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

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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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? –  Deane Yang Jan 9 '11 at 0:56
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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 –  sigoldberg1 Jan 11 '11 at 11:00

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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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. –  David Walker Jan 22 at 18:01

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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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 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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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

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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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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 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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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

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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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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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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שלום גיל, ממש יפה, כל הכבוד, אבל זה בעברית –  Patrick I-Z Dec 30 '10 at 23:26
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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. –  Gil Kalai Dec 31 '10 at 11:02

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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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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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

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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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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An exhibit on the role of computers in pure (and applied) mathematics. It would especially be nice to see something about experimental mathematics and viewing math, at times, as not purely deductive, but even empirical. I think this would give an idea as to how some mathematicians work and think, and also emphasize the growing importance of computers in verifying or finding new theorems.

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I don't think enough people know about the deductive part of mathematics. Too many people believe calculations and mathematics are the same thing. An exhibit on experimental mathematics could help, but this would have to be done carefully. –  Douglas Zare Dec 31 '10 at 10:26

How about leading them through an interesting problem, like a geometry IMO problem or, if that is asking too much, a Mathcounts problem? It could be set up on square tiles, the left most of which would contain the problem, with the following tiles showing the steps of the solution. It should be a problem that can be written such that viewers see a surprise toward the end, thereby possibly giving a glimpse into why mathematicians enjoy so much what they do. Although a Mathcounts problem would no doubt be accessible, a very beautiful IMO problem could be inspiring. Very likely, it would be entertaining for both children and their parents.

One might also include multiple solutions to a problem to dispel the notion that for each problem only one solution exists.

One can see examples of interesting presentations and ideas for problems at Rusczyk's Mathcounts channel at

http://www.youtube.com/user/mathcountsfdn

Using the same format from above, one could present a suitable Putnam problem and show its connection to research. This is discussed in Kedlaya, Poonen, and Vakil's book.

Finally, this response might be related to Kevin Lin's. and ein's.

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In such a museum, I would like to see how mathematics are used in real life, not just for their internal beauty (well, beauty, simplicity and usability are certainly related). I mentioned above in a comment how Thales' theorem has been the tool to measure the height of pyramids. This can make a nice mathematics experiment: a lamp (the sun) a small pyramid and a stick. And suddenly math comes alive. Another kind of living mathematics: put salt into a thin aquarium such that the density vary, top to bottom, from zero to (almost) infinity. Send a light beam to the aquarium and the light will follow a geodesic of Poincaré's half plane (this experiment has been actually presented at the Paris "Palais de la découverte"). These are just two examples of "math in real life", I'm confident in mathematician's skills to find a lot more of such examples (not just in geometry: prime number and securing communications, statistics and controlling epidemics, etc...). I'm sure that understanding with our eyes how mathematics are used in real life makes mathematics even more sexy.

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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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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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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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A shallow-water-wave soliton demonstration would be nice. –  S. Carnahan May 2 '11 at 9:20

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

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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...

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Rather than providing concrete examples, I would like to make a suggestion about two possible guiding lines.

If you agree that mathematics is about solving problems of a certain kind, then two sensible goals for a museum of mathematics could be showing:

  1. what kind of problems mathematics deals with, and
  2. how it deals with them.

The first goal cannot be exhaustive for obvious reasons, but I think it should be broad enough to give the visitor an idea of the diversity of mathematics (applied or not), i. e. examples from as many various branches as possible should be given.

For the second goal, Polya's views about how to solve mathematical problems could be helpful, or modern works about "visual" mathematical thinking. I think visitors should be able to feel some connection between mathematical problem solving and applying "common sense" strategies (here is a hidden goal: to demistify a bit the work of mathematicians).

To focus on one or the other goal in different degree can result in very different kinds of museum, but I think any of them would deserve the name "Museum of mathematics". Not so one that didn't meet any of both goals.

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