High-dimensional space
A space with thousands of directions, where random things almost never point the same way.
Like you're twelve
On a line, two random points are often close by accident. On a page there is more room, so accidents get rarer. In a room, rarer still. Keep adding directions and the space becomes so roomy that two things picked at random are essentially never confusable — there is always another direction they differ in.
That is the whole trick. A big space is not a bigger filing cabinet; it is a space where you can throw things in blindly and they still land apart.
Precisely
In d dimensions, the angle between two randomly drawn vectors concentrates around 90° as d grows, and the spread shrinks like 1/√d. By a few thousand dimensions, near-orthogonality is not a tendency but a practical guarantee.
Two consequences follow, and every other idea in Heather leans on them. Unrelated items do not collide, so distinct things stay distinct without anyone assigning them ids. And superposed vectors stay readable — adding several vectors together does not smear them into an average, because each term stays recoverable along its own direction.
Dimensionality is the resource here in the way memory is the resource elsewhere. Spend more of it and you get more room to keep things apart.