Representation Collapse
What you just saw
At the top, all tree nodes were clustered together—children overlapping parents, grandchildren invisible. This is representation collapse.
As you scrolled, the same tree expanded into hyperbolic space. Every node got its own room. The hierarchy became visible and navigable.
Collapsed
Preserved
Why does this happen?
Trees have exponentially many nodes at each depth. A tree with branching factor 3 has 3 nodes at depth 1, 9 at depth 2, 27 at depth 3...
But Euclidean space has polynomial volume growth. You run out of room. Embedding algorithms sacrifice rare relationships to preserve common ones.
Hyperbolic space has exponential volume growth—matching the tree. There's always room for more nodes at the boundary.
Real-world impact
Single-cell biology: Rare cell types become visible instead of being absorbed into dominant clusters.
NLP: Word hierarchies embed naturally with preserved relationships.
Networks: Social graphs and phylogenies maintain their hierarchical structure.