Math

13 chapters in the order to read them. 34 equations, each with a line underneath saying what it says out loud. 10 of the chapters have something to press.

1. Everything pulls on everything
   Two things pull on each other. The pull goes up with the weight and down with the square of the distance. That one line is the whole engine.
2. Two is easy
   With two bodies you can write the answer down, once, and it is right forever. This is the only case that works that way.
3. Add one rock
   The equations for three bodies fit on one line and hide nothing. What breaks is not the writing down. It is the solving.
4. Ten things that never change
   Eighteen numbers, ten of them pinned down by conservation laws. The eight left over are where the trouble lives — and two theorems say no further law is coming.
5. A hair's difference, and a different sky
   Move one rock by the width of an atom and an hour later the whole arrangement is different. The equations are exact; the answer is still out of reach.
6. Two big, one small
   Let the third body be a pebble too light to pull back. Now the problem has a fence you can draw, five places where a pebble can sit still, and a number that says which side of the fence you are on.
7. The two ways three rocks can hold a shape
   Three bodies can keep the same shape while they spin, and there are exactly two shapes that allow it: a straight line and an equilateral triangle. Euler found the first in 1767, Lagrange the second in 1772.
8. The ones that come back around
   Some starts run a closed loop and repeat forever. Three equal weights chasing each other around a figure eight is the famous one, and it was found by looking, not by solving.
9. There is a formula. Nobody can add it up.
   In 1912 Karl Sundman wrote a series that converges to the answer for almost every three-body start. One estimate of how many terms you would need to use it runs to a 1 followed by eight million zeros.
10. Crashes, and the kind of trouble that is not a crash
    Solutions can stop existing. With three bodies the only way is a collision, and a three-way collision needs the whole system to have no spin. With four or more, something stranger is possible.
11. So you step it
    Nobody solves the three-body problem. Everybody steps it: work out the pulls, move everything a little, do it again. The skill is in knowing when your own arithmetic has started lying.
12. What usually happens
    Throw three bodies together and the usual ending is a fight two of them win: one gets thrown out, the other two are left closer than they started. You cannot predict which — but you can price it.
13. What it is not
    Five things the phrase 'the three-body problem' gets used for that it does not mean.

Nothing here needs mathematics past squaring a number and taking a square root. Where a symbol turns up it is named the first time, and again at Words.