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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
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: – 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 : 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

94 Answers 94

An exhibit - with both individual workstations and one whose results are projected (to draw people in) - where users use parallelograms to specify affine transformations which in turn define Iterated Function Systems and their associated fractals. See here for background. With a few moves/ clicks a user can start making fractal ferns, clouds, spirals and starfish as well as classical objects such as the Sierpinski gasket. It is great because you don't have to know anything to start making pretty pictures, but you immediately get a sense that there is something significant going on. For those who want to see beyond the pretty pictures, one could explain the contraction mapping theorem (terrific fun in its own right) and develop affine-linear transformations starting with rotations, scalings and translations. Trying to find the transformations which define a particular fractal by "finding enough smaller copies of the fractal to cover the fractal" is also great fun, which is compelling even for children.

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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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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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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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@kjetilbhalvorsen, if you get it working, I'd love to see pictures! – Vectornaut Mar 21 at 21:52

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

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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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I'm a Mathematics & Theoretical Physics undergrad, so I though I'd share what got me hooked on Maths. As a teenager it first dawned on me that Maths wasn't the "dry", boring subject taught at school when, on my own at home, I first started playing around with nD geometry and generalised Euler's Theorem to nD by noticing the patterns. The thing that really made me fall for Maths though was reading about the Riemann Hypothesis in New Scientist when I was 16. [It's hard to believe, but none of the Maths teachers I'd had had even mentioned primes, and I was hooked.

The thing I'd put in MoMath is the Mandelbrot set across a HUGE wall using a projector that gradually zoomed in and in. After a set time it could start again but zoom in on a different area. You could include higher-order Mandelbrot sets and other infinite fractals. The beauty of fractals is their beauty appeals to everyone [Importantly including NON-Mathematicians!] and gets an important point across: "MATHS IS BEAUTIFUL!"; in the blink of an eye. It would show clearly & quickly both the richness & beauty that lies within Mathematics and that it's NOT the dead, dull, dry subject many people think.

I would suggest screening a range of pre-recorded maths "lectures", catering for the diverse spectrum of Mathematical-understanding of the visitors.

MoMath, if done right, seems like a brilliant idea. Good luck with the project, and I wish you every success! :-)

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Downvoted because it's a rant; also, Dick Palais has already suggested fractals. – Daniel Moskovich Dec 30 '10 at 20:53

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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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

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 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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Indeed, it is somewhat out of control. Rather than flag and state my objections or alert the moderators or start a meta thread listing objections, I will wait and see if you bring it back under control. Your unedited opinions may be OK for your blog; I think your posting (at this writing) needs pants. Gerhard "Ask Me About System Design" Paseman, 2011.10.21 – Gerhard Paseman Oct 22 '11 at 3:53

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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Something from Erik Demaine on linkages would have wide appeal.

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I'd love to see a wall sized picture of the matrix of a random element of the monster group acting in a faithful representation.

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S. Carnahan---I think that's a really good argument in favor of the proposal! When I read your comment, my first thought was, "Come on... 196882 can't be that big." Then I multiplied it by a millimeter, and---WHOA. I would love to see an auditorium-sized wall filled with a corner of one of the monster matrices, with the entries in 4-millimeter type, and a little plaque saying that two write out the whole matrix, you'd need a wall two Empire State Buildings high! – Vectornaut Oct 26 '11 at 6:25

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, at 9:30pm ET, follow this link to join the live session using Elluminate:

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Instead of an answer, I think this should be edited into the original question. Otherwise it will be lost in the long list of suggestions. – Douglas Zare Jan 3 '11 at 16:19

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

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A book scanner.

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Please elaborate? – Kevin H. Lin Dec 25 '10 at 23:41
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

Maybe it will be good to place in such a museum some of the famous original papers or copies (like the Poincare's article on topology or homology or the Grothendieck's papers on schemes or maybe the famous page where Fermat had declared his Great Theorem).

Also it will be good to place here some of the drawings of (for example) mobile telephones with mathematical calculations.

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I honestly can't imagine that the general public would get anything out of this. – Qiaochu Yuan Dec 31 '10 at 16:54

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

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Finding Tumors with Linear Algebra (and a lending hand by logarithms):

Example of the use of linear algebra (and with less emphasis, logarithms) in a toy example of computerized tomography (CT scana).

Computer (Bob) chooses a transparent plexiglass box among four such boxes in a horizontal square grid in which to place an opaque box/tumor. The top of the grid must be covered so the investigator (Alice) can't see where the opaque box is. She can, however, rotate a laser penlight to shine a laser beam horizontally through the boxes and observe any shadow on the opposing side (complete attenuation). Bob displays a diagram of the four empty boxes and asks where the tumor/opaque box is. After Alice figures out how to locate the box, Bob asks what the minimum number of positions for the penlight for effectively locating the opaque box is. Next a second opaque box is placed randomly in the grid and the process iterated.

Placement of a third opaque box results in failure, but now a light meter is placed opposite the penlight, the three opaque blocks are replaced by semi-transparent boxes with their attenuation coefficients (ACs) given (nice integers), and Bob explains that for a box the AC is log[intensity of light in / intensity out] so that the total attenuation displayed by the light meter is log[intensity at penlight / intensity measured by the light meter] = AC of row or column of grid = sum of ACs of boxes in the row or column. Can Alice figure out the placement of each box? Must Bob demonstrate the linear algebra required to solve the problem? Finally, four boxes with differing ACs (nice integers) are configured in the grid and Alice asked if she can determine the ACs of each box. Bob gives a grey level display for each result--hands-on tomography.

In my experience, very young kids with decent attention spans like such interactive challenges (no shock and awe approach, just simple reasoning).

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The Duel: Who Fired First?

Projections, Minkowski space, special relativity, and a legal paradox

A computer displays a line of observers outside a speeding train and a parallel line inside a train, all with respectively synchronized stop watches. Between the two lines and displaced along the length of the train are Galois and his frenemy with pistols poised. An animation captures the interest of the museum guests.

Two coordinate systems are superimposed with the origins of both systems coinciding at time $t=0$ well before the pistols are fired with respect to the observations of the ground crew.

A spacetime graph is displayed depicting the two events of the pistols being fired. In the classical Newtonian world, the two spatial coordinate lines are superimposed and depicted parallel to the two lines of observers with their origins displaced by the motion of the train by the time the shots are fired. The single time axis is depicted vertically perpendicular to both coordinate axes. The spatial displacement between the duelers is determined by drawing lines parallel to the single time axis through the point-events and down to the spatial axes, and analogously for the temporal displacement between the firings of the pistols. Let's have the two events happen simultaneously as recorded by the ground crew. Then the events will occur simultaneously according to the observers on the train also with exactly the same spatial displacement between the adversaries. These facts are easily demonstrated by the projections, and, indeed, are equivalent to the projections.

In the world of special relativity, the facts change. While keeping the time and space axes for observations by the ground observers unchanged, the time and space axes for the observers on the train must be displayed pivoted about the origin towards each other. Projections parallel to the skewed time axis though the events to the spatial axis reveal that the train observers will record a smaller spatial displacement between the duelers, and projections parallel to the skewed spatial axis reveal a non-zero temporal displacement in the events, i.e., the times the pistols are observed to have been fired are not equal as measured by the train observers.

A legal paradox! The ground observers might conclude the duelers are equally guilty of premeditated murderous intent whereas the train observers might accuse one of premeditated murderous intent and the other of reacting only in retaliation.

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

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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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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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Car parking by Lie group techniques!

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The Body Counter: Making Math Matter in Human Rights

An exhibit explaining the use of multiple systems estimation (MSE, a type of statistical inference) in human rights investigations and trials, based on Tina Rosenberg's article The Body Counter.

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