Explanations simple enough that non-mathematical audiences can understand [closed]

Question

The following (debatable) quote is attributed to Einstein:

"You do not really understand something unless you can explain it to your grandmother."

I feel very enlightened when there is a simple explanation of an important idea in mathematics. Below are some of my favorite ones.

My Question: Are there other explanations like this?

• (Credit to my analysis professor, many years ago): The geometric series $\sum_{n=1}^{\infty} \frac{1}{2^n} = 1$ can be explained as follows: take a disc. Cut it in half. Now take half of the disc, and cut that in half. Repeat this process. Then we have $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +...$ disc. But we started out with a whole disc, so the total is a single disc!

Definitions, etc.

• By "explanation," I am asking for a proof or a heuristic argument which is simple (in the sense defined below).

• Ideally, the fact is a central part of some subfield of mathematics. (For instance, the geometric series is certainly an important example in calculus since it is the main idea behind many comparison tests. It is also the most basic example of a series that can be computed explicitly (besides perhaps telescoping series). ) The facts themselves must also be simple.

• By "simple", I mean you can explain it to someone without any mathematical background (say a child under 10 years old). In particular, words like "derivative," "group," and "Riemann curvature tensor," are considered to be "too hard," but expressions like "speed/velocity," "symmetry" and "how much a surface/curve curves" are acceptable. (In this regard, words from elementary physics (e.g. Newtonian mechanics, electromagnetism, wave mechanics) are great, but quantum mechanics and relativity are too hard. Notions from middle/high school (e.g. Euclidean geometry) are great too.)

• Simple pictures are okay too, although the picture cannot be too complicated. (Again, the main criteria here is that your average non-mathematical audience can understand it.)

Answer 1

Pythagoras theorem.

Albert Einstein wrote about two pivotal moments in his childhood. The first involved a compass that his father showed him when he was four or five. The second involved his early exposure to Euclidean plane geometry. He was impressed by the idea that a mathematical assertion could “be proved with such certainty that any doubt appeared to be out of the question”.

Steven Strogatz discusses a breathtakingly simple proof of the Pythagorean theorem whose provenance is traced to Einstein as a child. "Though we cannot be sure the following proof is Einstein’s, anyone who knows his work will recognize the lion by his claw."

Einstein's first proof.

_{The proof relies on the insight that a right triangle can be decomposed into two smaller copies of itself. That’s a peculiarity of right triangles. If you try instead, for example, to decompose an equilateral triangle into two smaller equilateral triangles, you’ll find that you can’t. So Einstein’s proof reveals why the Pythagorean theorem applies only to right triangles: they’re the only kind made up of smaller copies of themselves.}