  Multipolar degrees of freedom and their hidden orders have been widely
discussed in the context of heavy fermions, frustrated magnets and exotic Kondo
effects. Although there has been extensive search for multipolar degrees of
freedom in magnetic systems, there are few examples that allow pure multipolar
degrees of freedom in the absence of magnetic dipoles. In this work, for the
first time, we show that the magnetic behavior in an icosahedral quasicrystal
is generally described by multipolar degrees of freedom, and in a specific case
by the pure magnetic octupoles in the absence of dipoles, resulting from the
interplay of spin orbit coupling and crystal field splitting. Importantly, we
point out the non-crystallographic symmetries lead to multipolar degrees of
freedom, only allowed in quasicrystals but forbidden in conventional crystals.
For both Kramers and non-Kramers doublets, the characteristics of multipoles
are classified and the effective spin Hamiltonian on symmetry grounds are
derived. Based on the self-similar triangular structure of the icosahedron, we
argue the long-range frustration in terms of the Ising model. We further
classify the possible quantum phases including quantum fluctuations, in terms
of the instantaneous entanglement generation of the ground state. Our study
offers the magnetic icosahedral quasicrystal as a new platform to search for
the novel multipolar degrees of freedom and their exotic phenomena.
