  The properties of topological systems are inherently tied to their
dimensionality. Higher-dimensional physical systems exhibit topological
properties not shared by their lower dimensional counterparts and, in general,
offer richer physics. One example is a d-dimensional quantized multipole
topological insulator, which supports multipoles of order up to 2^d and a
hierarchy of gapped boundary modes with topological 0-D corner modes at the
top. While multipole topological insulators have been successfully realized in
electromagnetic and mechanical 2D systems with quadrupole polarization, and a
3D octupole topological insulator was recently demonstrated in acoustics, going
beyond the three physical dimensions of space is an intriguing and challenging
task. In this work, we apply dimensional reduction to map a 4D higher-order
topological insulator (HOTI) onto an equivalent aperiodic 1D array sharing the
same spectrum, and emulate in this system the properties of a hexadecapole
topological insulator. We observe the 1D counterpart of zero-energy states
localized at 4D HOTI corners - the hallmark of multipole topological phase.
Interestingly, the dimensional reduction guarantees that one of the 4D corner
states remains localized to the edge of the 1D array, while all other localize
in the bulk and retain their zero-energy eigenvalues. This discovery opens new
directions in multi-dimensional topological physics arising in
lower-dimensional aperiodic systems, and it unveils highly unusual resonances
protected by topological properties inherited from higher dimensions.
