  A quasicrystal is an ordered but non-periodic structure understood as a
projection from a higher dimensional periodic structure. Some physical
properties of quasicrystals are different from those of conventional solids. An
anomalous increase in heat capacity at high temperatures has been discussed for
over two decades as a manifestation of a hidden high dimensionality of
quasicrystals. A plausible candidate for this origin has been phason, which has
excitation modes originating from additional degrees of freedom in the
higher-dimensional lattice. However, most theoretical studies on phasons have
used toy models. A theoretical study of the heat capacity of realistic
quasicrystals or their approximants has yet to be conducted because of the huge
computational complexity. To bridge this gap between experiment and theory, we
show experiments and molecular simulations on the same material, an Al--Pd--Ru
quasicrystal, and its approximants. We show that at high temperatures, aluminum
atoms diffuse with discontinuous-like jumps, and the diffusion paths of the
aluminum can be understood in terms of jumps corresponding to hyperatomic
fluctuations in six-dimensional space. It is concluded that the anomaly in the
heat capacity of quasicrystals arises from extra degrees of freedom due to
hyperatomic fluctuations that play a role in diffusive Nambu--Goldstone modes.
