  We perceive the world through images formed by scattering. The ability to
interpret scattering data mathematically has opened to our scrutiny the
constituents of matter, the building blocks of life, and the remotest corners
of the universe. Here, we deduce for the first time the fundamental symmetries
underlying image formation. Intriguingly, these are similar to those of the
anisotropic "Taub universe"' of general relativity, with eigenfunctions closely
related to spinning tops in quantum mechanics. This opens the possibility to
apply the powerful arsenal of tools developed in two major branches of physics
to new problems. We augment these tools with graph-theoretic means to recover
the three-dimensional structure of objects from random snapshots of unknown
orientation at four orders of magnitude higher complexity than previously
demonstrated. Our theoretical framework offers a potential link to recent
observations on face perception in higher primates. In a later paper, we
demonstrate the recovery of structure and dynamics from ultralow-signal random
sightings of systems with no orientational or timing information.
