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Ok, hotshots. You're at a party, and you're chatting with some non-mathematicians. You tell them that you're a mathematician, and then they ask you to elaborate a bit on what you study, or they ask you to explain why you like math so much.

  1. What are some engaging ways to do this in general? What are some nice elementary results, accessible to people with any mathematical background or lack thereof, that can be used to illustrate why math is interesting, and its depth, breadth, and beauty? Note the scenario: you're at a party so it should be a relatively quick and snappy kind of thing that doesn't require a blackboard or paper to explain.

  2. Easy Mode: How do you explain what your particular sub-field of math is about in an accurate but still understandable and engaging way? Hard Mode: Assuming you work in a less applied area, how can you do this without mentioning any real-world applications? Again, note the scenario.

This question is inspired by this one, in particular Anton's answer.

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This is not a question: But apparently seven dimensional laser calculus is a good answer. –  Peter McNamara Nov 13 '09 at 22:47
    
Following the precedent of questions like mathoverflow.net/questions/5353/… and mathoverflow.net/questions/3559/… , I figured this one would be ok as well. –  Kevin H. Lin Nov 13 '09 at 23:04
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seems like quite a reasonable question to me, I don't see a difference between this and something like "Do good math jokes exist?" or "Mathematicians who were late learners" or the other two that Kevin mentions –  Michael Hoffman Nov 14 '09 at 6:20
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This is the problem with making decisions based on precedents. Once one undesirable question is allowed to slip through the net, it becomes a free for all. –  Peter McNamara Nov 14 '09 at 16:31
    
@Peter: I propose to take this discussion to the meta board –  Kevin H. Lin Nov 14 '09 at 17:00

25 Answers 25

In parties where people are eating pizza, it is quite nice to see people taking their slice and curving the edge so that the slice stays straight. Then you can tell them that this is effectively Gauss's "Theorema Egregium": the initial Gaussian curvature is zero, so by curving the slice in one direction they force the slice to stay straight in the perpendicular direction.

You can probably continue the discussion talking about rolling pieces of paper into cylinders but not into tori ("doughnuts"), or maybe talking about soap bubbles!

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I love the pizza illustration! –  Elisha Peterson Nov 14 '09 at 23:58
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Ohhhh maaaan! I've been curving pizza like that since I was twelve, and I never saw the differential geometry connection! :) –  Vectornaut Nov 15 '09 at 5:34
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I don't get it.. or maybe I'm too stupid and have not eaten enough pizzas.. what does it mean "curving pizza" or curving the edge of the slice? You mean picking up the sliced slice and moving it away from the pizza in a curved-like manner to cut off the cheese part? sorry I feel so silly here –  Jose Capco Nov 15 '09 at 22:10
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The idea is that, if you pick up a slice of pizza by the crust, gravity makes the tip of the slice droop downwards. This makes it difficult to eat, and the toppings can slide off. However, if you use your hand to fold it lengthwise a bit (i.e., perpendicular to the crust), then the pizza stays straight and no longer droops. (See slice.seriouseats.com/images/FoldHoldHalf-thumb.jpg for a picture of this.) Essentially, you make one of the principal curvatures positive -- but since the Gaussian curvature is zero, this forces the other principal curvature to be zero. –  Ari Nov 15 '09 at 22:38

I've always found that a brief description of the Stable Marriage Problem gets people interested. I frame it as a high school dance with n boys and n girls each with a preference relation defined on the members of the opposite sex. Of course I don't call it a preference relation, but that's what it is. I then explain the disaster and heartache waiting if an unstable matching occurs.

I'll pair up people standing around at the party to illustrate the theorem, and get people involved. I'll then close with the fact that the National Resident Matching Program, which places graduating medical students into residence positions, uses the Stable Marriage Theorem to determine those placements.

The example is interesting enough to keep people listening and just technical enough that the answers aren't clear from the outset. If people are still interested you can segway into bipartite graphs and other kinds of matchings or just to graphs in general. Graph Theory is a rich area for cocktail party discussion.

NRMP Site: http://www.nrmp.org/res_match/about_res/algorithms.html

Stable Marriage Problem can be found on Wikipedia. I can't post a second link as a new user.

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Yeah, this is an example I use a lot too. The sociologically-minded friends also find it really interesting that the male courtship method is male-optimal and female-pessimal. –  Jonah Ostroff Nov 13 '09 at 23:26
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Have you really paired people up? That's awesome. –  Qiaochu Yuan Nov 14 '09 at 1:59
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Yes, I have. And I always put myself with my first choice. :) See kids? Math DOES help you meet women. –  Hank Turowski Nov 14 '09 at 5:55

One of the things I like to mention, since I study topology, is the Brouwer fixed point theorem. The idea to explain is that if you pick up a piece of paper, DON'T RIP IT, but crumple it, turn it over, fold it, whatever, put it down on top of another one, then there will always be at least one point that will match up with the one below it on the other paper. It's very physical, very counterintuitive, and thoroughly math, though it's better demonstrated with a decorated napkin than plain.

Alternatively, in the same vein, one can talk about the hairy-sphere theorem (the idea that you can't comb a hairy sphere all the way around without a cowlick; i.e., you can't have a nonvanishing continuous vector field along the sphere).

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Borsuk-Ulam is nicely illustrated with deflated toy balloon: crumple it all you want, but once you lay the balloon flat, there will be a pair of antipodal points one on top of another. –  Boris Bukh Jan 29 '10 at 23:13

There's no reason to stick to your subfield of mathematics. Nonmathematicians aren't going to care about the difference between category theory and Fourier analysis.

I like to tell people about Arrow's theorem, that the only voting systems which are consistent are dictatorships. Consistency can be illustrated by this joke:

"Would you like tea or coffee?"

"Tea, please."

"Oh, we also have lemonade."

"In that case, I'd like coffee."

That seems silly, but it's what democratic systems do, e.g., in 1992 the American population preferred Bush over Clinton, but with the right wing 3rd party candidacy of Perot, Clinton was elected.

Most people get the joke, and see that consistency might be desirable, and then that mathematics can say some things which are meaningful to them.

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This is really interesting. And I guess your comment about category theory/Fourier analysis is true....... though I really wish it weren't..... –  Kevin H. Lin Apr 20 '10 at 8:57

I have considered some of the lack of knowledge and mistaken impressions of mathematics (and scientific research in general) that can be held by nonmathematicians. In mathematics, there's nothing to prove, just things to calculate. Or all of the important things to prove were established ages ago. Or if there are things still to prove, it's because the remaining questions are incredibly complicated, or incomprehensibly abstract. Or remaining questions could be mystical ones with no right answer, only opinions. Maybe contemporary mathematicians are much smarter than their predecessors because they have powerful computers. In any case, applications — technology, health, some other science that's actually interesting — could be the serious reason to do mathematics. Or if not that, it could be sheer ego and hero worship.

To be fair, a lot of nonmathematicians don't have any such depressing view of our profession. However, they can adopt it very quickly in response to bad explanations. Certainly most nonmathematicians have little sense of the basic coin of research in pure mathematics: theorems, proofs, definitions, conjectures, open problems. They also generally don't know that mathematics was already sophisticated in the 19th century, that a vast amount was accomplished in the first half of the 20th century, and that there are plenty of open problems left. (19th century mathematics is largely invisible in newspapers. On the one hand, very few readers or journalists know any of it; on the other hand, it certainly isn't news.)

To counter every side of that, I like to discuss questions that are not only accessible and fun, but also have a historical narrative. The narrative can go from an easy question, to some 19th or early 20th century result, to open problems. It can also cite great results from mathematicians other than the most famous heroes. I think that this can be done in lots of ways, but it is important to stick to clear explanations. Here is an example and a half:

  1. Knot theory. Is an overhand (a trefoil) different from a nothing (an unknot)? Is a right-handed overhand different from a left-handed one? Yes and yes, according to Heegaard, Tietze, and Dehn from a century ago. Are there knots that aren't handed? For example, the figure-eight. Are there non-invertible knots? Yes, but that's harder; it was only established in 1964 by Trotter. Is it possible to distinguish any pair of knots? Yes, as was first proved by Haken in the late 1960s. How would you do it? The current best way is with Thurston's ideas, using hyperbolic geometry. (Hyperbolic geometry is then another big topic.) How many crossings do you need to switch to convert one knot to another? That is a big open problem, although it has been solved in many interesting cases. What's the easiest solution to the first of this whole chain of questions? Reidemeister moves and 3-colorings. Etc.

  2. Real algebraic curves. Hyperbolas, parabolas, and ellipses are all quadratic curves. A hyperbola has two branches, but these are halves of one oval that passes through infinity. How many ovals can you have in higher degrees? An indirect argument tells you that you can't have an unbounded number in any fixed degree. In degree 4 you can have 4 ovals; in degree 6 you can have 11 ovals. Harnack discovered and proved these upper bounds in the 19th century. Can the ovals nest any way you like? No...

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There are some good ones in topology (or maybe I just know these examples because it's my field).

  1. People always seem to think the hairy ball theorem (already mentioned) is interesting because it explain why people have cowlicks on their head, not to mention the name itself usually gets a few giggles.

  2. The Meteorological Theorem (Borusk-Ulam) implies there are antipodal points on the surface of the Earth where the temperature and barometric pressure are the same.

  3. My favorite one is similar to the plate trick above. If you hold a coffee cup in your hand and rotate your wrist until the cup is oriented the original way--your arm is all tangled up, but if you rotate it again, your arm straightens out. It demonsrates that the fundamental group of the group of rotations in R^3 is Z_2.

  4. Vin de Silva, who does works in applied topology, has the best one though. Take a piece of paper draw a few dots on it and ask what the shape it. Draw more until it's clear that you're "sampling" points from a circle. Then tell them that math (persistent homology in this case) lets you find the shape of a sampling of points. Leads to simple discussions of using math to solve lots of applied problems.

Then you can start making jokes using the work "functor." ("Functor? I hardly know her!" or just randomly say "Clusterfunctors!")

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I'm a bit confused by #3. I tried to do what you suggested but my wrist doesn't bend that way. –  Michael Lugo Nov 14 '09 at 15:28
    
@Michael Lugo: Here's an okay illustration, along with a nice description (on the previous page) of what the plate trick has to do with SO(3): books.google.ca/… I can't believe there's no video of this on YouTube! –  Vectornaut Nov 14 '09 at 17:07
    
@Michael: My fault for posting before morning coffee. Vectornaut's link has a good picture. If it's done with a coffee cup it's easy to see when you've went around 360 degrees. Bredon's "Topology and Geometry" has a nice picture of this, but I can't find it online. –  jd.r Nov 14 '09 at 20:46

I researched some properties of words with a limited alphabet, which has applications for genetic modelings. So at a party, I tell people that I model genes.

They do tend to look at me funny after I say it, though.

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Noncommutativity for transformations: when dressing up, you first put on socks, then shoes, when undressing, you have to take off shoes first!

Percentages and interest rates: it sort of matters whether to put 2,000 into your savings account this year and 1,000 the next year or 1,000 this year and 2,000 the next year (not obvious to many people).

Abstract maths can go far from "reality" into the "world of ideas": Banach-Tarski paradox.

And as to the questions like "why do you like maths so much?" I usually prefer to draw parallels between maths and arts/languages (not denying the applications either).

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On computer role-playing forums, I have seen a lot of general strategy advice regarding difficult group battles ("attack the boss monster first", "start with the highest-damage enemies", etc.). I have decided to see for myself which the optimal tactics is. The answer requires some basic (school-level but nontrivial) mathematics: see http://www.mathlinks.ro/viewtopic.php?t=326811 , scroll down to the remark (the problem turned out to be more or less identical to an American Math. Monthly question which is older than CRPG).

Navigation in mazes.

Is it possible to brute-force a combination lock by repeatedly changing a digit without checking one and the same combination several times? (Yes, by induction, at least if you can cycle every digit all the way from 0, 1, 2, ... to 9 and back to 0. If you cannot move between 9 and 0 without going through the intermediate digits, then no, by a semi-invariant argument. The keyword here is Hamiltonian path. If none exists, the next natural question is how to find a path through every vertex of minimal length...) And as we are talking about Hamiltonian paths, Euler paths can be interesting as well...

Huffman coding. It's in your base compressing your files.

For some reason I never understood, many people not particularly close to mathematics seem to be fascinated by Rubik's cube. As opposed to some other popular riddles like Sudoku, this one becomes more or less trivial once one knows the maths behind it.

Huh, the four-color theorem has not been mentioned yet? This is the best example for the notion of beautiful vs. ugly in mathematics that comes into my mind. The five-color theorem is not difficult and rather nice to prove; the only proofs of the four-color theorem known go the "classify and solve for hundreds of particular cases" way. Mathematics is probably the only science where people care for the difference.

The isoperimetric inequality, with all of its, sorry for the pun, variations (such as the case of a curve in a half-plane with two given ends).

You want to encrypt a message in a way that each of $n$ persons gets a key such that any $k$ of them can, in a joint effort, unambiguously decrypt the it, while any $k-1$ will not have the slightest idea about the message (i. e., every possible message including gibberish will be equiprobable). It's called Shamir's Secret Sharing.

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Hah, your RPG strategy is more or less identical to one I figured out when I was 18 or so wondering about the same things. Although I approached it from a slightly different perspective, it is basically equivalent. It's also not too difficult to account for some basic additional factors, such as area-of-effect attacks, ranged attacks, who's nearest to you, etc. Similarly, before that, I'd come up with a nice approximate solution to what basically amounts to the knapsack problem in RPGs. I can carry x pounds, I have items worth d_i dollars, and want to maximize the value of stuff I can take. –  jeremy Feb 10 '10 at 7:57
    
The knapsack thing leads to the rearrangement inequality ( en.wikipedia.org/wiki/Rearrangement_inequality ) which I also have thought about mentioning but didn't because most people would do the right thing by intuition here (in RPG or in real life), and the mathematics would just give them a proof that their intuition is right - not particularly exciting. –  darij grinberg Feb 10 '10 at 13:09
    
I am a bit skeptical about area of effect tactics - while it is certainly a good idea to get the toughest enemies in one row and shoot some cone-shaped effect on them, it is usually a matter of luck and of enemy AI whether getting them in one row actually works. –  darij grinberg Feb 10 '10 at 13:10

In Ramsey theory, there is a theorem that states that if the edges of the complete graph on six points are divided into two sets one set will contain all the edges of a triangle. This could be illustrated by taking 6 people at the party and finding either three that know each or three that don't. This is easy to demonstrate and could be generalized in various ways.

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Here's a couple of examples from the Japanese TV show "Coma University of Mathematics". There have been some excellent episodes which are accessible to non-mathematicians yet I think really conveys the creativity in mathematical research, and how very different that is from doing complicated arithmetic.

  1. The moving sofa problem: explain that, to get to your room, you have to pass through two corridors, 1m in width, which meet at right angles. What shape of sofa has the largest area but can still get into your room? It's not immediately clear what shapes one should try, and it might surprise your audience that such a simple-looking question is an open problem.

  2. The art gallery theorem: in short, how many guards (or CCTV cameras) do you need to cover all of a polygonal art gallery? Again, a problem that's easy to understand, but the proof for the upper bound requires a clever idea.

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I'm in applied math so I have it easy: I use math to pick the cancer treatments with the highest probability of being safe and effective.

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I like to talk about planarity of graphs, because it's easy to introduce, easy to draw on a napkin, and on the "application" side of things you can mention computer chip design, transportation networks, and other things. The nonplanarity of $K_{3,3}$ can be introduced via the "three utilities" puzzle: you have three houses and you want to connect each one to each of three utilities (water, hydro, gas) -- can you do it without having the lines cross? Similarly the nonplanarity of $K_5$ can be posed as a puzzle, and Kuratowski's theorem basically asserts that these two examples give the only obstructions to planarity: a graph is planar if and only if it contains no subgraph homeomorphic to $K_5$ or $K_{3,3}$.

Going further, you can talk about embedding graphs on a torus (i.e. in a game of Asteroids), for which you can draw examples of embeddings of $K_5$ and $K_{3,3}$, and say that there's an analogy of Kuratowski's theorem (by corollary to the Robertson-Seymour theorem): there is a finite list of "forbidden graphs" which are the only obstruction to a graph being embeddable on the torus, in the sense that any nonembeddable graph contains one of those forbidden graphs as a minor.

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I found that playing planarity.net gave me the intuition for planar graphs before I thought of looking up what they actually were. Perhaps if you're near a computer you can introduce people to that? –  Elizabeth S. Q. Goodman Nov 18 '09 at 5:46

Especially when talking to engineers, when asked about math I tend to mention a fact I learned in Hilbert's book "Geometry and the Imagination": the two rulings on a quadric surface in 3-space can be used to design gear trains which allow for transferring motion between two arbitrary skew axes of rotation. (Basically, make gears shaped like real hyperboloids of one sheet. There's a picture in Hilbert's book, p. 287.) For some reason I find this a particularly snazzy (albeit very elementary) example of why anybody might care about algebraic geometry.

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One of the issues that came up during the challenger explosion investigation was whether some port had a circular opening. The engineers had taken two parallel planes, placed the port between them, measured the distance between the planes, and then rotated & repeated. Since all the measured distances were equal, they concluded the port was circular.

Anyone who's studied a bit of geometry knows this is an error. There are curves of constant width that are not circles -- a good counterexample is the 50p coin if you have some UK currency.

It's easy to extend this into a discussion of whether manhole covers need to be circular. Any manhole cover that's a curve of constant width cannot fall into its hole.

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I like the following description of the inverse galois problem I came up with at one point:

Do you remember polynomials? Maybe from high school math? Do you ever remember trying to factor polynomials in school? Like for example $x^2 -4$ factors as $(x+2)(x-2)$. So notice that you can get from one factor to the other just by switching from 2 to -2, so this polynomial is symmetric in the simplest way possible, you just hold up a mirror and switch roots from one to the other.

Well every polynomial is symmetric in some way or another, and there are very many ways a polynomial can be symmetric. We as mathematicians know how to take a polynomial and say in a very precise way "What kinds of symmetry does this polynomial have?"

Now, can we go the other way? Can we start with some symmetries and ask for a polynomial that is symmetric in that way? Well, we don't know. We should know, but we don't yet!

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I have successfully used examples involving covering chessboards with dominoes. I start with the question about whether you can tile a chessboard with opposite corner squares removed. If this gets any interest, I might go on from there. This leads people to the idea that mathematics is not just about numbers, but is actually about thinking logically about pretty much anything.

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Does that work with no chess board and dominoes around? –  Douglas Zare Jan 30 '10 at 6:47

I have a go-to math fact to bring out at parties when people want to hear one. The cute part is that they can do it themselves.

Take a pen and paper and draw a quadrilateral. There are no restrictions (it can be concave or self-intersecting), but don’t make it too close to the sides of the paper. Now, for each edge, draw the square containing that edge that is outside the quadrilateral. Put a dot in the center of each of the four squares, and draw a line connecting opposite dots, ie, those that came from opposite edges.

The Punchline: The lines you just drew are the same length, and perpendicular.

I wrote it up and drew up a pictoral proof on my (mostly-defunct) blog sometime ago.

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When people are asking for practical applications (which they inevitably do), I always explain how number theory (that's prime numbers for the layperson) was at some time studied by some people because it couldn't have any application ever (so it's pure and artistic), but now is the basis for cryptography, which is for example behind the banking system; implying that there can be a delay of 50-100 years between a mathematical discovery and a potential everyday application. I also mention that there are often immediate applications inside mathematics, so it's not totally useless :).

Topology has some interesting stuff too, as already mentioned.

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A bizarre consequence of the Crystallography Restriction is that one can't have a wallpaper pattern with with fivefold rotational symmetry but one can have 2-, 3-, 4-, and 6-fold rotational symmetry. It's tricky to explain "fivefold rotational symmetry" concisely and precisely, but I just tried this example out on a friend of mine, and I must say his reaction was fairly enjoyable. He found the result unbelievable.

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Starfish have fivefold rotational symmetry. That might help future explanations. –  Michael Lugo Dec 23 '09 at 15:02

Get piece of paper in the shape of long rectangle. Draw the line along long side exactly in the middle of the rectangle. Then make Moebius strip from piece of paper. Finally cut it in the half - You will obtain two pieces of paper. Now make another Moebius strip but now, cut in in say 1/3 from the side along long side of the rectangle ( it is hard to describe as English is not my primary language ) . See what happen. You may ask people to guess before;-)

Do You know Steinhaus http://en.wikipedia.org/wiki/Hugo_Steinhaus book "Mathematical Kaleidoscope"? Read it and You will be ready for any occasion;-)

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When trying to talk about specific results, I really like talking about Cantor's Theorem (or at least, the special case of $2^{\aleph_0}$), and then, if they're willing to accept that, talk about ordinals a bit. If the audience isn't taking it, I'll generally talk about some arbitrary graph problem that comes to my head.

But when asked, I typically try to approach it from the more philosophic perspective of "what mathematicians do"-- study abstract structure. (Or at least, that's the approach I take to math), and try to explain what that means, providing some examples here and there. The definition of group shows up a lot, since there are easy to understand examples of groups. I also view myself as a bit of an artist, so I tend to use analogies that deal with music and painting.

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If you work in Theoretical computer science (or related fields) you can say what Anup Rao says in this article.

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I just thought about this a bit more, and wonder whether, with a good set of people who can describe mathematics well, something in the realm of Bill Nye the Science Guy could be created for math. It might end up Geometry heavy, i'm not sure.

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