Hyperbolic Space
Watch Euclidean space transform into the Poincaré disk
The problem with flat space
When you visualize a tree in Euclidean space, you run into a fundamental limitation: there isn't enough room.
A binary tree at depth 20 has over a million leaf nodes. But the area of a circle grows as πr². Nodes grow exponentially; display area grows polynomially.
Euclidean
V(r) ~ rd
Polynomial growth. Not enough room.
Hyperbolic
V(r) ~ er
Exponential growth. Room for everything.
Why hyperbolic space works
Hyperbolic geometry has exponential volume growth—exactly matching tree structures.
The Poincaré disk maps infinite hyperbolic space onto a finite disk. The boundary represents infinity—you can fit arbitrarily many points without crowding.
What you just saw
As you scrolled, the uniform grid transformed. In the Poincaré disk, squares near the edge appear smaller—but they all have equal hyperbolic area.
This compression near the boundary gives hyperbolic space its power. Navigation (Möbius transformations) brings any region smoothly to the center.