# What would you want to see at the Museum of Mathematics?

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
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
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
@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
@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

Mirrors exhibiting plane tilings:

Conic section billiard tables (the reflection properties!):

These are taken from the exhibitions documented at http://atractor.pt

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See the water demonstration of the Pythagorean theorem:

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I am sure a museum of Mathematics could not miss a selection of beautiful pictures of fractals.

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As a late adapter to smart phones, I just recently started idling away some time playing the popular app Flow Free (Big Duck Games). The goal is to connect pairs of dots of the same color with grid paths that do not intersect and cover the entire grid. It makes me think of the Gessel-Viennot lemma about a determinant counting nonintersecting sets of paths (although there often only moves right and up are allowed).

One could make a computer display where the pairs of dots have a unique set of nonintersecting paths. An initial step could lead visitors through the number of paths between two points being counted by binomial coefficients. Then finding a set of nonintersecting paths is similar to the game (same sort of touchscreen interface), with the bonus that your work shows that a particular determinant is 1 (without all the arithmetic and plus / minus signs).

The same interface could have an exploration of paths strictly below the diagonal that lead to Catalan numbers, which connects to a whole host of visually engaging things such as triangulating regular polygons and making "penny piles" (Richard Stanley is up to 202 things counted by these numbers -- that could be a whole special exhibit).

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Car parking by Lie group techniques!

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Something like the clickable math atlas: http://www.math.niu.edu/~rusin/known-math/index/mathmap.html

but the clicks should probably lead to places showing for laypeople what these different fields are!

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I'd suggest an interactive exhibit where people can tweak the parameters of a population model with 3 species in it. Have an information panel which explains what the parameters represent. Suggest goals such as (1) keep the rabbits from going extinct, (2) create a stable equilibrium where all three species survive, (3) find a cyclic solution, etc. Experts could probably come up with a good system that exhibited lots of interesting behavior. Let people visualize their solutions both graphically (3-D graph of all three populations as well as 2-D graphs of any two populations of their choosing). There are probably other clever graphical representations that others could come up with as well.

Overall, I think this is a great endeavor and I hope that you will focus on making the exhibits interactive, with pathways to learning. Ideally, the same person could visit an exhibit a half dozen times and learn something new each time. Of course, making things fun and interesting is really important too - but I think that should naturally emerge from the design of interactive ways to explore a beautiful piece of mathematics.

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1. Models of the sphere eversion that emulate the chicken wire ones. I think someone made some for Morin at one time.

2. Interactive computer graphics of Seirpinski $n$-simplices, interactive computer graphics of hypercubes, the 120 cell, and the 24 cell.

3. Transparent models of knotted surfaces.

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Hypercubical arrays of Hasse diagrams for divisibilities. –  Scott Carter Oct 27 '11 at 3:32

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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I'd like to see a picture of a man throwing a ball (eg Michael Jordan) and next to it, the corresponding parabola in a Cartesian plane.

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A Foucault pendulum, explaining the concept of parallel transport in a manifold.

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Stereographic projection lamps, please!

I'd love to see a bunch of clear plastic spheres with colored patterns (triangle tilings, Escher pieces, and so forth), lit from a pole to project the patterns onto whatever's nearby. Or stick models of polyhedra with bright lights suspended at their centers. When I was a kid, I had a toy that worked on a similar principle---it was reasonably effective, and apparently considered safe.

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• A transparent model of Cayley's cubic surface, with the 27 real lines marked on them. I've only seen plaster models of this. (Actually, I want this for my birthday. Ahem...)

• A transparent plastic cone and a laser light to cut it into conics and, more fun, similarly transparent models of quadrics and lights to check the theorem that the shadow lines on cuadrics are plane curves.

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I think that there should be an exhibit on fractals and a history of mathematics exhibit. Fractals are very beautiful and mathematically interesting, and the concept of self-iteration is fairly easy for an general audience to understand. You could even discuss the motivating problem of determining the length of the coast of Britain and how making measurements of non-smooth curves on smaller and smaller scales eventually limits to infinity. As for the history of mathematics, I think that's pretty obvious. There are a lot of interesting stories that make up the history of mathematics. Museums also love to show cultural diversity, so the development of mathematics around the world could be a possibility.

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There are some cool little formulas you can "prove" by putting together a 3d block puzzle. For example, the formula for the sum of the first n squares can be seen by putting together 6 puzzle pieces to form a square prism with sides n, n+1, and 2n+1. Here, each piece is a "staggered square pyramid" of volume 1+ ...+ n^2. (Suggestion: Take n=5) There are other such puzzle-ready formulas like the sum of triangular numbers. I believe "The Book of Numbers" by Conway and Guy has some. You could build nice big soft ones that schoolchildren can play with and grownups can appreciate.

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A moving sculpture approximating Smale's turning of the sphere inside out. (but what material would you use?)

A sphere made out of elastic plastic with fotoreactive proteins. The proteins are laid down so that they only react when antipodal points touch. A similar thing for Brower's theorem instead of Borusk-Ulam, with two discs.

A sphere made out of some flexible but not so flexible material, the spectator gets to sculpt the sphere into some shape, then he plays the shape with a stick. There are (say) seven choices of sticks in front of him each corresponding to one eigenvalue of the shape. When the visitor plays the sphere a computer computes the corresponding eigenvalue and translates it into a sound (maybe the sound of a drum).

A family of itouch-made-material 2d-surfaces, and a big sphere in the middle of the room. When one traces a curve (with the finger) on the surfaces. The Gauss map on the big sphere in the middle is animated.

A linkage that has the earth at the center of the solar system and traces the moving planets perfectly (this can be done by universality of Thurston and Kapovich-Millson right?) The spectator stands in the middle and sees the planets move around him. At the end he gets to wear an Inquisition hat and burn a paper sculpture of Galileo.

A metalic model to see percolation: The vertices will be represented by magnetized vertical thin tubes coming from the floor. The spectator stands in the second floor and from above (not above the spectator but above the magnetized tubes) a bunch of small tubes (edges) fall. Depending on the strength of the magnets (controlled by the user) some stay sticking to the tubes and some go to the ground, the spectator is asked to repeat the experiment many times and conjecture with what probability this random graph percolates.

In a dark room. A hospital bed with a set of cards is lighten like in a noir film. The visitor is supposed to play solitaire lying on the hospital bed, if he gets to the end in one round a screening of the H-bomb appears on a big screen in front of him. The sound of the bomb is heard in very loud speakers so that everybody in the museum hears this.

A 3d animation of a contorted 2-sphere Ricci flowing to a round sphere. Again the visitor gets to choose the starting sphere. At the end he is given a phone number. He tries to call Perleman.

A huge fountain that doesn't work.

An observatory with stars at random positions in which suddenly log n of them turn out to form a convex polygon. It should look like an astrological map.

Many microscopes looking at cells growing. In the first one like f(t)=t, then f(t)=t^2 and a few polynomials more. Then f(t)=2^t. (Can this be done with unlimited resources? I'm not joking, this is a honest experimental biology question.)

A mechanically transformable translucent skate park. Each configuration of the skate park corresponds to a link, an element in pi_1(C_n(R^3)). A tube of say two meters of radious traces the curve followed by one of the points in configuration space. So there are say 5 base points and 40=10*4 switches (like in a wiring diagram but going around 5 circles) The user gets to select what switches (adjecent transpositions are on). The skaters are encouraged to use a helmet.

A liquid based model explaining Kepler's laws with a very eccentric ellipse.

A finance millionaire looking at the numbers of the stock market, very focused.

Three hallways that meet at the center. One with triangles on the walls, other with curves of functions, the third one with polynomials. At the center (e^i\pi=-1). Like that is a bit cheesy but if there was a computer app illustrating the geometry of elementary operations (+ and *) that foced the visitor to define what product by i does in the complex plane this would be quite cool.

A room with two walls closing on the visitor like in that famous star wars scene. But without the intergalactic trash. With ungraded calc exams instead.

A performance-theatre show. An artificial beach with Newton (with a wig) looking at a shell and at a pebble, figuring out which one is smoother. We hear a loud applouse and two giants carry him out of the building. The two giants come back to find a seminaked Archimedes computing an integral on the sand, they slaughter him.

This one will save mathematics from the financial crisis: A chair in which the visitor sits and his brain activity is monitored as he does some easy mathematics. Whenever his brain activity seams to resemble math thinking he is injected endorphins.

sorry, this got a bit out of control....

Im trying to think of something good for some basics about Galois theory or Covering Space theory, but this is harder....

Oh and in the store you get to draw your favorite planar graph and you leave with a clay model of the corresponding circle packing.

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A slide rule! The physical embodiment of the isomorphism between $\left(\mathbb{R},\cdot\right)$ and $\left(\mathbb{R},+\right)$. There are pretty pieces of history here too - Napier's bones and so forth. A giant one (maybe >1m long) mounted on a wall so that people can make it work - now that would plant the idea of the isomorphism making + and * "the same" operation deeply in the mind of anyone who played around with it seriously.

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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 '14 at 18:01

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

Something about the "Quaternion Demonstrator" (the belt trick demonstrating that $\mathbf{SO}\left(3\right)$ has a double cover). An exhibit could centre on three distinct, accessible and interesting to the general public (like myself) mathematical topics:

1) As a model for spin 1/2 particles - I recall being enthralled as a young teen by the idea that some objects might not come back to their same state after a $2 \pi$ rotation. At that age, on seeing the belt trick, I recall one of my first reactions was "Interesting, but might not we build something with a fancier arrangement of ribbons and strings that would need, say $6 \pi$ rotations to bring it back to the "same state"?" I think it would be interesting to say in the exhibit that there is sound mathematics behind the assertion that, no, there is no such fancier arrangement, so that, unless the topology of our Universe is radically different from what we can imagine, there is very strong evidence, grounded on mathematics alone, that half-integer spin is the only possibility - and we don't need billion dollar particle accelerators to know this.

2) The quaternions and the idea of number systems beyond "everyday" rational and real numbers. That only restricted systems can be built if one wants to preserve "real world" properties like continuity of the "multiplication" - that mathematics isn't just postulating arbitrary axiom systems and playing games with them. History of complex numbers could be included, maybe even a feel for Hatcher's Algebraic Topology proof thereof (something like the YouTube clip http://www.youtube.com/watch?v=nRO_4IYOdq8).

3) Thinking about the belt trick itself (the physical thingie, rather than the mathemetics of the $\mathbf{SO}\left(3\right)$ double cover) for me stridently raises the question of what the distinction between mathematics and physics really is, or even whether there is one. The belt trick is compelling to even small children - I showed it to my five year old recently and was astonished to find that she seemed to understand many of the ideas of symmetry involved and played around with different numbers of twists and untangling them for quite some time. Of course, most serious mathematicians will say that the belt trick is not a proof, but when you begin to look at it hard, and see that the ribbon is directly encoding a "history" of rotations of a Frenet-Serret frame, you realise that the contraption is a pretty spot on analogue of the mathematical construction of a universal cover - so much so that you begin to wonder whether the mathematical construction isn't part of the subconscious visual processing and understanding of the physical contraption in almost anyone - mathematician or layperson.

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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.

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The cosmic distance ladder: the mathematics used in (and very often developed for) measuring astronomical distances: the radius the Earth, the distance from the Earth to nearby bodies (the Moon and Sun), their radii, and on up to the shape and size of the universe. The first of these were first done with good accuracy in antiquity; the latter are still being worked on (indeed, so are the former, to amazing levels of precision).

I like this topic for several reasons. It shows applications of mathematics to physics. I would guess that much of the mathematics used was developed for this purpose, and if some of it was developed independently, well both of these are important aspects in mathematics. It shows different mathematics and different non-mathematical ideas all intertwined in a single endeavour.

I have seen a recording of an excellent lecture by Terence Tao on this topic.

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In an Italian museum (probably the Leonardo da Vinci Museum in Florence) I saw a compass for drawing arbitrary conical sections. I believe the legend mentioned only ellipses, but it could in principle draw the others too.

The basic principle is that the "central" arm (in general, the focal arm) of the compass is held at a fixed (per drawing) angle to the desk, while the pencil arm adjusts in length (so that it is shorter at the perigee and longer at the apogee).

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What about some large-number phenomena? This seems to be something the general public would appreciate and could relate to the "Computers in Modern Mathematics" booth others have suggested.

What I have in mind is not really Ackerman function/Graham's number business (which I don't think I could wrap my head around any more easily at a museum), but facts that involve small-ish large numbers. For instance:

The smallest positive integer $n$ for which $n$ divides $2^n-3$ is $4,700,063,447$.

There are many other great examples (though not all interesting or accessible to non-mathematicians) in answers to this MO question. It also might be nice to see comparisons of smallest counterexamples like this to 'real-world' numbers like the population of China (~$1.34$ billion), or the number of cells in the human body (~$10^{14}$), or the number of elementary particles in the observable universe (~$10^{80(\pm10?)}$).

To me, the goal of such an exhibit should be (1) to provide a few examples (like the one above) illustrating the importance of proof over verification of the first $10^{10}$ cases, and (2) to help museum-goers conceptualize the small-ish large numbers that come up in analyzing real-world phenomena.

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How to form your own math circle or some other teaching movies, especially with kids solving problems and having fun at it. I mentioned this in a comment, but it could use elaboration and advertisement. Museums are for people, especially kids. One goal is to empower kids to feel they can think. Some sort of exhibits on what really good mathematics teaching feels like could be tremendously inspiring. The museum could be a living organizing center for this.

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Crystallography, illustrated by optical diffraction. See the repeated pattern under a magnifier, then project with different wavelengths (would a prism work or do you need different laser pointers). Introduction to Fourier analysis, optical transforms,etc.

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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 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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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
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

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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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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