Hyperbolic PDEs on the surface of a sphere
by James A. Rossmanith

Surface with constant Gaussian curvature

The Dedekind Tessellation: a tessellation of the half-plane in hyperbolic triangles.

Hyperbolic geometry differs greatly from what we are used to in that the longest path between two points is a straight line. You can think of hyperbolic space like Euclidean space, but with a different way of defining the distance between two points (we call this a metric). Above is the hyperbolic equivalent of an icosidodecahedron. Check out more hyperbolic tilings here.