  Systems with low mechanical dissipation are extensively used in precision
measurements such as gravitational wave detection, atomic force microscopy and
quantum control of mechanical oscillators via opto- and electromechanics. The
mechanical quality factor ($Q$) of these systems determines the
thermomechanical force noise and the thermal decoherence rate of mechanical
quantum states. While the dissipation rate is typically set by the bulk
acoustic properties of the material, by exploiting dissipation dilution,
mechanical $Q$ can be engineered through geometry and increased by many orders
of magnitude. Recently, soft clamping in combination with strain engineering
has enabled room temperature quality factors approaching one billion ($10^9$)
in millimeter-scale resonators. Here we demonstrate a new approach to soft
clamping which exploits vibrations in the perimeter of polygon-shaped
resonators tethered at their vertices. In contrast to previous approaches,
which rely on cascaded elements to achieve soft clamping, perimeter modes are
soft clamped due to symmetry and the boundary conditions at the polygon
vertices. Perimeter modes reach $Q$ of 3.6 billion at room temperature while
spanning only two acoustic wavelengths -- a 4-fold improvement over the
state-of-the-art mechanical $Q$ with 10-fold smaller devices. The small size of
our devices makes them well-suited for near-field integration with
microcavities for quantum optomechanical experiments. Moreover, their
compactness allows the realization of phononic lattices. We demonstrate a
one-dimensional Su-Schrieffer-Heeger chain of high-$Q$ perimeter modes coupled
via nearest-neighbour interaction and characterize the localized edge modes.
