  Nature harbors crystals of dimensionality ($d$) only up to three. Here we
introduce the notion of \emph{projected topological branes} (PTBs):
Lower-dimensional branes embedded in higher-dimensional parent topological
crystals, constructed via a geometric cut-and-project procedure on the Hilbert
space of the parent lattice Hamiltonian. When such a brane is inclined at a
rational or an irrational slope, either a new lattice periodicity or a
quasicrystal emerges. The latter gives birth to topoquasicrystals within the
landscape of PTBs. As such PTBs are shown to inherit the hallmarks, such as the
bulk-boundary, bulk-dislocation correspondences and topological invariant, of
the parent topological crystals. We exemplify these outcomes by focusing on
two-dimensional parent Chern insulators, leaving its signatures on projected
one-dimensional (1D) topological branes in terms of localized endpoint,
dislocation modes and the local Chern number. Finally, by stacking 1D projected
Chern insulators, we showcase the imprints of three-dimensional Weyl semimetals
in $d=2$, namely the Fermi arc surface states and bulk chiral zeroth Landau
level, responsible for the chiral anomaly. Altogether, the proposed PTBs open a
realistic avenue to harness higher-dimensional ($d>3$) topological phases in
laboratory.
