The hexagonal ice formation on the surfaces of several substrates which have different crystallographic configuration of oxygen atoms from that on the preferentially crystallized surface of ice was studied by taking rate and direction of cooling into consideration. It was found that the cooling direction is the dominating factor for the preferential orientation of crystal axis of ice formed on a substrate. That is, the hexagonal ice grows preferentially along c-axis for thin layer ice and along c-axis or a-axis for fine columnar ice parallel to the cooling direction, regardless of the degree of epitaxy of oxygen atoms on surfaces between substrate and ice, or of the degree of wettability between substrate and water. But it has been also found that the freezing temperature lowering and the temperature range for single crystal formation of ice in the whole capillary space of glass tubing on substrates were significantly influenced by the epitaxy, i.e. the degree of misfit of interatomic spacing of oxygen on crystallographic surfaces of substrate and ice (00l).