  Generic models of cosmological inflation and the recently proposed scenarios
of a recycling universe and the string theory landscape predict spacetimes
whose global geometry is a stochastic, self-similar fractal. To visualize the
complicated causal structure of such a universe, one usually draws a conformal
(Carter-Penrose) diagram. I develop a new method for drawing conformal
diagrams, applicable to arbitrary 1+1-dimensional spacetimes. This method is
based on a qualitative analysis of intersecting lightrays and thus avoids the
need for explicit transformations of the spacetime metric. To demonstrate the
power and simplicity of this method, I present derivations of diagrams for
spacetimes of varying complication. I then apply the lightray method to three
different models of an eternally inflating universe (scalar-field inflation,
recycling universe, and string theory landscape) involving the nucleation of
nested asymptotically flat, de Sitter and/or anti-de Sitter bubbles. I show
that the resulting diagrams contain a characteristic fractal arrangement of
lines.
