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An ordinary donut surface, but not smooth — instead, pierced by invisible layers, like puff pastry where each layer is slightly rotated relative to its neighbor. That's how a contact structure on a three-dimensional manifold works: at each point, a plane is defined, and these planes twist smoothly, touching each other according to a special law.

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.

Two passengers step into an empty car on a ring subway line and take their seats. After one full loop, they are surprised to find themselves in each other’s seats — even though nobody moved. That’s pretty much how monodromy plays out in vineyards — families of persistence diagrams, a method rooted in the work of Henri Poincaré.