  We propose a simple family of valence-bond crystals as potential ground
states of the $S=1/2$ and $S=1$ Heisenberg antiferromagnet on the pyrochlore
lattice. Exponentially numerous in the linear size of the system, these can be
visualized as hard-hexagon coverings, with each hexagon representing a
resonating valence-bond ring. This ensemble spontaneously breaks rotation,
inversion and translation symmetries. A simple, yet accurate, variational
wavefunction allows a precise determination of the energy, confirmed by DMRG
and numerical linked cluster expansion, and extended by an analysis of excited
states. The identification of the origin of the stability indicates
applicability to a broad class of frustrated lattices, which we demonstrate for
the checkerboard and ruby lattices. Our work suggests a perspective on such
quantum magnets, in which unfrustrated motifs are effectively uncoupled by the
frustration of their interactions.
