  To understand the formation of the Bergman cluster in the F-type icosahedral
quasicrystal, crystallographic relations between the quasicrystal and the
intermetallic-compound H and Zn$_{23}$Y$_6$ phases in the Zn-Mg-Y alloy system
were investigated mainly by transmission electron microscopy. It was found
that, although sample rotations of about 1{\deg} were required to obtain simple
crystallographic relations, the orientation relationship was established among
the cubic-$Fm\overline{3}m$ Zn$_{23}$Y$_6$ structure, the hexagonal-$P6_3/mmc$
H structure, and the F-type icosahedral quasicrystal (IQ); that is,
$[\overline{1}13]_{\rm c}$ // the fivefold axis in the IQ // N$(2\overline{1}
\overline{1}0)_{\rm H}$, and $[110]_{\rm c}$ // the twofold axis in the IQ //
N$(05\overline{5}3)_{\rm H}$, where N(hkml)$_{\rm H}$ means the normal
direction of the (hkml)$_{\rm H}$ plane in the H structure. The correspondences
between atomic positions in the Bergman cluster and in the Zn$_{23}$Y$_6$
structure and between those in the cluster and in the H structure were
investigated on the basis of the established relationship. As a result, an
assembly of six short-penetrated-decagonal columns was identified as an
appropriate nucleus in the formation of the Bergman cluster from these two
structures.
