Considering the potential importance of pentagonal nanochannels
(PNCs) in determining the structure of the amorphous ice, we introduce
topological criteria to identify the PNCs with an optimal fivefold
symmetric environment. The basic building block in our criteria, termed
as a bicyclo octamer, is an axisymmetric cluster formed by combination
of three 6-membered boat-shaped rings. This bicyclo octamer unit serves
as a seamless interface of the PNCs with cubic-hexagonal stacking
patterns of the amorphous ice. This interface results in a fivefold
symmetric mesostructure of relatively high stability: a central PNC
with extended five branches of two-dimensional (2-D) hexagonal (wurtzite)
crystalline monolayers stacked with cubic (diamond) layers. We also
unearth a hierarchy of symmetric structures in amorphous ice: the
PNCs, together with dodecahedron cages, form a network consisting
of (nearly) equilateral triangular patterns. At the next hierarchical
level of symmetry, such triangular patterns combine to form triangular
pyramids with dodecahedron cages as the vertices, PNCs as the edges,
and confined 2-D hexagonal crystalline monolayers as the triangular
faces of the pyramids. The central core of the pyramids consists of
cubic (diamond) regions with a strong local tetrahedral order. The
overall structure of the amorphous ice states is found to be profoundly
affected by PNCs. Specifically, in states with a relatively large
number of PNCs, the cubic-hexagonal stacking is primarily in the form
of hexagonal crystalline monolayers stacked from both sides with cubic
crystalline layers.