  One-dimensional quasiperiodic systems, such as the Harper model and the
Fibonacci quasicrystal, have long been the focus of extensive theoretical and
experimental research. Recently, the Harper model was found to be topologically
nontrivial. Here, we derive a general model that embodies a continuous
deformation between these seemingly unrelated models. We show that this
deformation does not close any bulk gaps, and thus prove that these models are
in fact topologically equivalent. Remarkably, they are equivalent regardless of
whether the quasiperiodicity appears as an on-site or hopping modulation. This
proves that these different models share the same boundary phenomena and
explains past measurements. We generalize this equivalence to any
Fibonacci-like quasicrystal, i.e., a cut and project in any irrational angle.
