In 1997, Ashtekar, Baez, Corichi and Krasnov quantized the classical phase space of the exterior of a black hole in vacuum General Relativity. They showed that the geometry of spacetime outside a black hole is described by spin networks, some of whose edges puncture the event horizon, contributing area to it, and that the quantum geometry of the horizon can be described by a U(1) Chern–Simons theory. The appearance of the group U(1) is explained by the fact that two-dimensional geometry is described in terms of the rotation group SO(2), which is isomorphic to U(1). The relationship between area and rotations is explained by Girard's theorem relating the area of a spherical triangle to its angular excess.