  We study closed liquid membranes that segregate into three phases due to
differences in the chemical and physical properties of its components. The
shape and in-plane membrane arrangement of the phases are coupled through
phase-specific bending energies and line tensions. We use simulated annealing
Monte Carlo simulations to find low-energy structures, allowing both phase
arrangement and membrane shape to relax. The three-phase system is the simplest
one in which there are multiple interface pairs, allowing us to analyze
interfacial preferences and pairwise distinct line tensions. We observe the
system's preference for interface pairs that maximize differences in
spontaneous curvature. From a pattern selection perspective, this acts as an
effective attraction between phases of most disparate spontaneous curvature. We
show that this effective attraction is robust enough to persist even when the
interface between these phases is the most penalized by line tension. This
effect is driven by geometry and not by any explicit component-component
interaction.
