  The Berry phase associated with energy bands in crystals can lead to
quantized quantities, such as the quantization of electric dipole polarization
in an insulator, known as a one-dimensional (1D) topological insulator (TI)
phase. Recent theories have generalized such quantization from dipole to higher
multipole moments, giving rise to the discovery of multipole TIs, which exhibit
a cascade hierarchy of multipole topology at boundaries of boundaries: A
quantized octupole moment in the three-dimensional (3D) bulk can induce
quantized quadrupole moments on its two-dimensional (2D) surfaces, which then
produce quantized dipole moments along 1D hinges. The model of 2D quadrupole TI
has been realized in various classical structures, exhibiting zero-dimensional
(0D) in-gap corner states. Here we report on the realization of a quantized
octupole TI on the platform of a 3D acoustic metamaterial. By direct acoustic
measurement, we observe 0D corner states, 1D hinge states, 2D surface states,
and 3D bulk states, as a consequence of the topological hierarchy from octupole
moment to quadrupole and dipole moment. The critical conditions of forming a
nontrivial octupole moment are further demonstrated by comparing with another
two samples possessing a trivial octupole moment. Our work thus establishes the
multipole topology and its full cascade hierarchy in 3D geometries.
