  We show that it is possible to have a topological phase in two-dimensional
quasicrystals without any magnetic field applied, but instead introducing an
artificial gauge field via dynamic modulation. This topological quasicrystal
exhibits scatter-free unidirectional edge states that are extended along the
system's perimeter, contrary to the states of an ordinary quasicrystal system,
which are characterized by power-law decay. We find that the spectrum of this
Floquet topological quasicrystal exhibits a rich fractal (self-similar)
structure of topological "minigaps," manifesting an entirely new phenomenon:
fractal topological systems. These topological minigaps form only when the
system size is sufficiently large because their gapless edge states penetrate
deep into the bulk. Hence, the topological structure emerges as a function of
the system size, contrary to periodic systems where the topological phase can
be completely characterized by the unit cell. We demonstrate the existence of
this topological phase both by using a topological index (Bott index) and by
studying the unidirectional transport of the gapless edge states and its
robustness in the presence of defects. Our specific model is a Penrose lattice
of helical optical waveguides - a photonic Floquet quasicrystal; however, we
expect this new topological quasicrystal phase to be universal.
