On Atoms and KnowledgeThis started as a note to my daughter. Which means it is partly about physics. But mostly about how to learn.

Hey Darling,

Your young old man here again. Or, depending on when you read this, maybe just your old old man.

Today I want to talk to you about my favorite formula:

Estart=EendE_{start} = E_{end}Estart​=Eend​

The conservation of energy.

Not because I prefer being sloth-like, or because I love shortcuts, though to be fair both are true. I love it because it was my first favorite atom of knowledge . And atoms are one of the most important things you can learn to look for.

I’ll come back to this one. But let me start somewhere you have already been, with an atom you mastered before you could even read this letter.

What I mean by atoms

You can’t memorize how to say every word. There are too many of them, and people make up new ones all the time. So nobody tried. Instead, someone taught you the sounds of just twenty-six letters. And the first time you saw a word you had never seen before, something like galumphing or persimmon, you could already say it out loud.

Nobody taught you that word. You built it yourself, out of letters.

That is an atom of knowledge . The letters are the atoms. The words are the molecules. You learned twenty-six small things instead of a million big ones, and in return you can say any word there is, including the ones nobody has invented yet.

More generally, an atom is a small truth that helps you derive bigger truths. Atoms matter because they reduce the number of things you have to simply “remember.”

Knowledge works a bit like chemistry:

Level<br>Meaning

Atoms<br>Simple truths

Molecules<br>Combinations of those truths

Structures<br>Bigger systems built from those combinations

On their own, each atom is useful. But when you combine them, they become molecules of understanding.

Why bother? It is harder than memorizing

It is. But it is worth it.

Memorizing is easy today and exhausting forever: you collect molecule after molecule and have to carry every single one. Learning the atoms costs more up front, then pays you back for the rest of your life. Three reasons:

You carry less. A handful of atoms can rebuild dozens of molecules. You never memorized every word in the language, and you never had to: twenty-six letters carry all of them. Learn four ideas about energy and you can derive the speed of a roller coaster, a ramp, a pendulum, and a spring, formulas you would otherwise memorize one at a time.

You can rebuild what you forget. You will forget almost everything you memorize. You can never forget how to rebuild it. A molecule you lose is gone; an atom you keep makes the molecule again.

The atoms travel. Memorize physics and you know physics. Understand its atoms and you walk into chemistry, biology, even people already half-fluent. Molecules stay home; atoms come with you.

So memorizing is renting answers: cheap this week, and you pay again every week after. Learning the atoms is owning them.

Memorize the molecules and you have to carry all of them. Learn the atoms and you can build any of them.

The same trick, everywhere

The reason I love atoms so much is that this move, small truths combining into bigger ones, is not a physics thing. It shows up almost everywhere. So as you grow up, I hope you do not worry too much about whether you are a “math person,” a “science person,” a “creative person,” or a “people person.” Those categories are mostly fake. What matters is learning how to find the atoms in whatever world you are drawn to.

Below are a few of those worlds. Start with the one you love most, but peek at the others too, because the pattern is the point: different costumes, same atoms.

If you love physics

This is the world that taught me all of this. Here are four of its atoms:

Atom<br>Idea<br>Formula

Energy is conserved<br>Estart=EendE_{start} = E_{end}Estart​=Eend​

Height stores energy<br>PE=mghPE = mghPE=mgh

Motion stores energy<br>KE=12mv2KE = \tfrac{1}{2}mv^2KE=21​mv2

Springs store energy<br>SE=12kx2SE = \tfrac{1}{2}kx^2SE=21​kx2

Each one is useful on its own. The interesting part is what happens when you combine them.

Molecule 1: height becomes speed

Take a roller coaster rolling down a hill, a skier going down a slope, a ball falling from a window, or a pendulum swinging downward. At first those look like different problems. Say you just want the speed at the bottom, and you have forgotten there is an atom underneath.

So you do the honest, exhausting thing. You track the forces down the slope with Newton’s laws:

F=mgsin⁡θF=maa=gsin⁡θs=h/sin⁡θv2=u2+2asv2=2(gsin⁡θ)(h/sin⁡θ)v=2gh\begin{aligned}<br>F &= mg\sin\theta \\<br>F &= ma \\<br>a &= g\sin\theta \\<br>s &= h / \sin\theta \\<br>v^2 &= u^2 + 2as \\<br>v^2 &= 2(g\sin\theta)(h / \sin\theta) \\<br>v &= \sqrt{2gh}<br>\end{aligned}FFasv2v2v​=mgsinθ=ma=gsinθ=h/sinθ=u2+2as=2(gsinθ)(h/sinθ)=2gh​​

Look at everything that road made you carry: forces, angles, acceleration, distance, the equations of motion,...