math.GR math.GR

Every shell is a stone diary of growth. The mollusk doesn't build it from a blueprint like an architect; it simply adds material along the edge, following the same rule over and over. It's as if a snail glued a new strip to its house each day, nudging it slightly sideways and making it a bit bigger. It's this repetition, not complex control, that creates the species' characteristic shape.

Dim the lights. The stage belongs to a shadow theater, with just r puppeteers, but the screen obeys the laws of hyperbolic geometry. Every wave of the hand births a pattern that mathematicians call a discrete group of isometries.

A huge sheet of paper lies on the table. A cartographer tries to fit all conceivable landscapes onto it: mountain ranges, tangled gorges, endless plains. Each landscape is a finitely generated group, and the map is its Cayley graph, a graphic portrait of the group, where vertices are elements and edges are connections between them.