  In the past years classical wave-systems have constituted an excellent
platform for emulating complex quantum phenomena. This approach has been
especially fruitful in demonstrating topological phenomena in photonics and
acoustics: from chiral edge states of Chern insulators and helical edge states
of topological insulators to higher-dimensional topological states of
quasiperiodic systems and systems with synthetic dimensions. Recently, a new
class of topological states localized in more than one dimension of a
D-dimensional system, referred to as higher-order topological (HOT) states, has
been reported, offering an even more versatile platform to confine and control
classical radiation and mechanical motion. However, because experimental
research of HOT states has so far been limited to two-dimensional (2D) systems,
third and higher-order states have evaded experimental observation. Studying
higher-dimensional classical systems therefore opens an opportunity to emulate
higher-order topological insulators and explore HOT states beyond second order.
In this letter, we design and experimentally study a 3D acoustic metamaterial
supporting third-order (0D) topological corner states, along with second-order
(1D) edge states within the same topological bandgap, thus establishing a full
hierarchy of HOT states in three dimensions, co-existing robustly within the
same topological bandgap. The metamaterial is implemented over a versatile
additive manufacturing platform, which enables rapid prototyping of metaatoms
and metamolecules, which can be snapped together to form 3D metamaterials with
complex geometries. The assembled 3D topological metamaterial represents the
acoustic analogue of a pyrochlore lattice made of interconnected molecules, and
is shown to exhibit topological bulk polarization, leading to the emergence of
topological HOT states localized in all three or in two dimensions.
