  Extraordinary new materials named quasicrystals and characterized by
noncrystallographic rotational symmetry and quasiperiodic translational
properties have attracted scrutiny. Study of quasicrystals may shed light on
the most basic notions related to the quantum critical state observed in
heavy-fermion metals. We show that the electronic system of some quasicrystals
is located at the fermion condensation quantum phase transition without tuning.
In that case the quasicrystals possess the quantum critical state with the
non-Fermi liquid behavior which in magnetic fields transforms into the Landau
Fermi-liquid one. Remarkably, the quantum critical state is robust despite the
strong disorder experienced by the electrons. We also demonstrate for the first
time that quasicrystals exhibit the typical scaling behavior of their
thermodynamic properties such as the magnetic susceptibility, and belong to the
famous family of heavy-fermion metals. Our calculated thermodynamic properties
are in good agreement with recent experimental observations.
