Equal-sized squares or hexagons can be arranged to fully tile a flat, two-dimensional plane, which has zero curvature. Pentagons can’t tile a plane, but they can be wrapped into a 3D dodecahedron and cover a sphere. Heptagons can’t be tiled at all, at least in familiar Euclidean geometry. A regular tiling of heptagons would require a hyperbolic surface with negative curvature—every point is a saddle point where space curves away from itself. Unlike a sphere, which has positive curvature, a hyperbolic surface cannot be realized in Euclidean space without distorting it. The top panel shows an example: a 2D projection (orange) of a regular heptagonal tiling.
Alicia Kollár (now at the University of Maryland) and colleagues at Princeton University have demonstrated a novel way of fabricating an effective hyperbolic space using a 2D network of superconducting circuits. Each circuit is a coplanar waveguide resonator, a platform for so-called cavity quantum...
