Figure 1

Figure 1: The BKL billiard of pure four-dimensional gravity. The figure represents the billiard region projected onto the hyperbolic plane. The particle geodesic is confined to the fundamental region enclosed by the three walls 1 2 1 α1(β) = 2β = 0,α2(β) = β − β = 0 and 3 2 α3(β ) = β − β = 0, as indicated by the numbering in the figure. The two symmetry walls α2(β) = 0 and α3(β ) = 0 intersect at an angle of π∕3, while the gravity wall α1(β ) = 0 intersects, respectively, at angles 0 and π∕2 with the symmetry walls α (β) = 0 2 and α (β) = 0 3. The particle has no direction of escape so the dynamics is chaotic.