I spent the better part of November writing miniature essays in this. It's really quite addictive. My favourites:
Parallax and cepheid variables (Dead stars that flash in space)
Basic linear algebra (four-sided boxes of numbers that eat each other)
The Gold Standard (Should a bit of money be the same as a bit of sun-colored stuff that comes out of the ground?)
The Central Limit Theorem (The Middle Thing-It-Goes-To Idea-You-Can-Show-Is-True-With-Numbers - when you take lots of Middle Numbers of lots of groups, it looks like the Normal Line!)
Complex numbers ("I have just found out I can use the word 'set'. This makes me very happy.")
Utility, utilitarianism and the problems with interpersonal utility comparison ("If you can't put all your wants into this order, you have Not-Ordered Wants")
The triumvirate brain hypothesis ("when you lie down on the Mind Doctor's couch, you are lying down next to a horse, and a green water animal with a big smile")
Arrow's Impossibility Theorem ("If every person making their mark on a piece of paper wants the Cat Party more than the Dog Party, then the Dog Party can't come out higher in the order than the Cat Party.")
The concept of "degenerate case" ("If your boyfriend or girlfriend has a different meaning for 'box' than you do, and you give them a line, not only will they be cross with you, but you will be wrong, and that is almost as bad")
The word "sublimate" ("When Dry Ice goes into the air, it is beautiful, like white smoke. There is a word for this situation, and we also use that word to talk about things that are beautiful, because they are perfect, and become white smoke without being wet first")
Complex numbers
I don't know why that one caught my eye, but here I go.
You've probably seen the number line before, a straight line from left to right (or right to left, if you like) with a point on the line for every real number. A real number, before you ask, is just that: real. You can see it in the world. If I point to a finger on my hand and ask, "how many of these do I have?", the answer is a real number. So is the answer to "how tall am I?", and the answer to "How much money do I have?" The answer to that last questio...
xkcd's Up-Goer Five comic gave technical specifications for the Saturn V rocket using only the 1,000 most common words in the English language.
This seemed to me and Briénne to be a really fun exercise, both for tabooing one's words and for communicating difficult concepts to laypeople. So why not make a game out of it? Pick any tough, important, or interesting argument or idea, and use this text editor to try to describe what you have in mind with extremely common words only.
This is challenging, so if you almost succeed and want to share your results, you can mark words where you had to cheat in *italics*. Bonus points if your explanation is actually useful for gaining a deeper understanding of the idea, or for teaching it, in the spirit of Gödel's Second Incompleteness Theorem Explained in Words of One Syllable.
As an example, here's my attempt to capture the five theses using only top-thousand words:
If you make a really strong computer and it is not very nice, you will not go to space today.
Other ideas to start with: agent, akrasia, Bayes' theorem, Bayesianism, CFAR, cognitive bias, consequentialism, deontology, effective altruism, Everett-style ('Many Worlds') interpretations of quantum mechanics, entropy, evolution, the Great Reductionist Thesis, halting problem, humanism, law of nature, LessWrong, logic, mathematics, the measurement problem, MIRI, Newcomb's problem, Newton's laws of motion, optimization, Pascal's wager, philosophy, preference, proof, rationality, religion, science, Shannon information, signaling, the simulation argument, singularity, sociopathy, the supernatural, superposition, time, timeless decision theory, transfinite numbers, Turing machine, utilitarianism, validity and soundness, virtue ethics, VNM-utility