  In the hydrodynamic model description of heavy ion collisions, the
final-state anisotropic flow $v_n$ are linearly related to the strength of the
multi-pole shape of the distribution of nucleons in the transverse plane
$\varepsilon_n$, $v_n\propto \varepsilon_n$. The $\varepsilon_n$, for
$n=1,2,3,4$, are sensitive to the shape of the colliding ions, characterized by
the quadrupole $\beta_2$, octupole $\beta_3$ and hexadecapole $\beta_4$
deformations. This sensitivity is investigated analytically and also in a Monte
Carlo Glauber model. One observes a robust linear relation,
$\langle\varepsilon_n^2\rangle = a_n'+b_n'\beta_n^2$, for events in a fixed
centrality. The $\langle\varepsilon_1^2\rangle$ has a contribution from
$\beta_3$ and $\beta_4$, and $\langle\varepsilon_3^2\rangle$ from $\beta_4$. In
the ultra-central collisions, there are little cross contributions between
$\beta_2$ and $\varepsilon_3$ and between $\beta_3$ and $\varepsilon_2$, but
clear cross contributions are present in non-central collisions. Additionally,
$\langle\varepsilon_n^2\rangle$ are insensitive to non-axial shape parameters
such as the triaxiality. This is good news because the measurements of $v_2$,
$v_3$ and $v_4$ can be used to constrain simultaneously the $\beta_2$,
$\beta_3$, and $\beta_4$ values. This is best done by comparing two colliding
ions with similar mass numbers and therefore nearly identical $a_n'$, to obtain
simple equation that relates the $\beta_n$ of the two species. This opens up
the possibility to map the shape of the atomic nuclei at a timescale
($<10^{-24}$s) much shorter than probed by low-energy nuclear structure physics
($<10^{-21}$s), which ultimately may provide information complementary to those
obtained in the nuclear structure experiments.
