Two discs and a missing triangle: the maximally extended Kerr black hole revisited

The region spanning the ring singularity in the Kerr black hole solution is described here as two (flat) discs rather than one. From this viewpoint it follows that the usual depiction of the worldsheet θ =π /2 θ = π / 2 , φ ϕ constant (in Boyer–Lindquist coordinates) is missing (multiple copies of) the region $$r<0$$ r < 0 , representable as a triangle in the usual conformal diagram style.