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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

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

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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I love the emphasis on interactivity. Perhaps also they could offer the possibility of playing asteroids on either a torus or a klein bottle? –  stankewicz Dec 26 '10 at 22:28
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The "visualizing the tenth dimension" video is complete pseudoscientific nonsense. –  Harry Gindi Dec 29 '10 at 16:31
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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). –  André Henriques Dec 30 '10 at 23:24
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I've taken it upon myself to remove the offending link. –  Harry Gindi Jan 1 '11 at 7:04
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I'm now completely convinced that a sphere eversion is what 3D movie technology was made for. –  Harrison Brown May 8 '11 at 12:30

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

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

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

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 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

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

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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"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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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

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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This:

alt text alt text

see this MO post, by Bill Thurston.

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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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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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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

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.

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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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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 –  J. M. Dec 28 '10 at 0:39
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@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. –  Thierry Zell Jan 2 '11 at 20:35

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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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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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. –  JCollins Jan 2 '11 at 19:09

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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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 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

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 bicycle with square wheels.

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This was implemented by MoMath in the form of a square-wheeled tricycle. See mathmidway.org/math-midway-activities-pedal.php. –  Ken Fan Dec 27 '10 at 17:13
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It might be novel to go beyond the nearly hackneyed square wheel to other shapes. See this MO question: mathoverflow.net/questions/29988/… –  Joseph O'Rourke Dec 27 '10 at 19:47

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 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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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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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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