I'm working on an educational project about the applications of reasonably 'lofty', high-ish-level mathematical concepts in the real world. I've already scoured these links (1) (2) (3) after exhausting my own knowledge, but I was wondering if anyone here can come up with some additional fun examples very much grounded in everyday life. The intended audience is laypeople - "imaginary numbers are used in electronic engineering" wouldn't impress them! "Gaussian curvature describes the best way to eat pizza" is more the level I'm going for.

I'm also interested in examples where mathematics doesn't necessarily provide the perfect solution to a real-world decision-making problem: for instance, where human intuition arrives at a good solution much more efficiently than a rigorous mathematical analysis. NP-hard problems, the fact that there is no 'equation' to solving the travelling salesman problem, that sort of thing.

closed as off-topic by Qiaochu Yuan, András Bátkai, Steven Landsburg, paul garrett, Alexandre Eremenko Jan 19 '16 at 23:41

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  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Qiaochu Yuan, András Bátkai, Steven Landsburg, paul garrett, Alexandre Eremenko
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    This looks like a very worthwhile question, but I wonder if it might belong better on matheducators.stackexchange.com (I am genuinely undecided right now) – Yemon Choi Jan 19 '16 at 18:59
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    How about the Monty Hall Problem: en.wikipedia.org/wiki/Monty_Hall_problem :-) – M.G. Jan 19 '16 at 19:07
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    I would be very cautious with your second example. Avoid caricatures and oversimplifications of mathematics. Your example with the travelling salesman problem seems wrong. Mathematicians are interested in finding approximate solutions to hard problems like the travelling salesman problem, and indeed we prove theorems about problems like that frequently. The idea that mathematicians can only look for some "equation to solve the travelling salesman problem" is ridiculous. That's not what an equation is for, and we don't say we want a complete solution or nothing. – Douglas Zare Jan 19 '16 at 19:08
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    By the way, make sure to check this question: mathoverflow.net/questions/5450/cocktail-party-math – Douglas Zare Jan 19 '16 at 19:12
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    Would be better to delete the second paragraph entirely, I think: "human intuition" and "rigorous mathematical analysis" is not a legitimate or genuine dichotomy, in the first place, and, as @DouglasZare comments, is a caricature. – paul garrett Jan 19 '16 at 20:02

Here are a few, in no particular order.

Hanging chains have the form of a catenary.

The chaotic mixing of coffee and milk is a rather common phenomenon of non-linear differential equations.

More abstract: simplicity sometimes requires complexity. All degree n polynomials have exactly n roots, but only if you accept complex roots.

If you want to walk around the bridges of some town without retracing your steps, you should know about Eulerian graphs.

Spherical surface maximizes volume for a given area; that explains soap bubbles.

When the referee puts the football back in the center after half-time, in general two points will be in exactly the same position they were at in the beginning of the match.

Your average monthly expense has a (approximately) Gaussian distribution.

In a party with 23 people, there is more than 50% chance two of them share a birthday.

When you walk towards a point, even if you only cover half the remaining distance in every step, you will still get there.

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    You will still get there … only if your steps get faster as well as shorter. I'm not sure that this counts as "everyday, real-life". – LSpice Jan 19 '16 at 23:41
  • Ok, maybe that's not the best one – Marcel Jan 20 '16 at 0:20
  • Several of these seem wrong. (You have to count roots with multiplicity. For most people, expenses are not Gaussian because surprisingly large expenses are much more common than surprisingly small expenses. The CLT does not apply. You need a lot more to explain soap bubbles. Was that supposed to be Zeno's paradox?) Others might be correct in a technical model that won't and shouldn't impress many lay people if you say enough to be correct. Few of these involve what I would call higher mathematics. – Douglas Zare Jan 20 '16 at 14:26
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    When you try to popularize mathematics, the statement to flesh out is not, "I find mathematics interesting!" but rather, "You (should) find mathematics interesting!" If you haven't heard of a catenary or hung many chains, do you care if the shape is a catenary or a parabola? How many people set up or solve nonlinear differential equations, or even $n$th degree polynomials? Many people do eat pizza, and can appreciate bending the pizza one way so it doesn't droop the other. Many people play games, or follow elections, or deal with health risks... – Douglas Zare Jan 20 '16 at 14:31

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