Shapes of spacesTopology and manifolds — the geometry that survives stretching, in three playable sections

The essays on this site keep saying things like the trajectory lives on a low-dimensional manifold, or the winding number is a topological invariant. Those sentences do real work, and none of it needs equations. It needs three ideas, and each gets a section of its own here: what kind of “shape” survives stretching, what it means for a space to look flat up close, and how to count the ways a loop can be trapped. Every idea arrives as something you drag, draw, or play before it gets a name.

This module assumes nothing beyond the rotations primer, and even that only lightly. Twelve chapters of the site lean on what's built here.