  The optical conductivity of quasicrystals is characterized by two features
not seen in ordinary metallic systems. There is an absence of the Drude peak
and the interband conductivity rises linearly from a very low value up to
normal metallic levels over a wide range of frequencies. The absence of a Drude
peak has been attributed to a pseudogap at the Fermi surface but a detailed
explanation of the linear behavior has not been found. Here we show that the
linear conductivity, which seems to be universal in all Al based icosahedral
quasicrystal families, as well as their periodic approximants, follows from a
simple model that assumes that the entire Fermi surface is gapped except at a
finite set of Dirac points. There is no evidence of a semiconducting gap in any
of the materials suggesting that the Dirac spectrum is massless, protected by
topology leading to a Weyl semimetal. This model gives rise to a linear
conductivity with only one parameter, the Fermi velocity. This picture suggests
that decagonal quasicrystals should, like graphene, have a frequency
independent conductivity, without a Drude peak. This is in accord with the
experimental data as well.
