While thinking about that problem, contemplate the experimentally determined phase diagram of water as shown above and appreciate the many possible phases that exist as temperature and pressure are varied. Note that there is actually an ice-nine, the phase region in the upper left labeled by the Roman numeral IX, but it exists only for pressures much greater than atmospheric pressure and for temperatures less than room temperature. There are 15 known crystalline phases of water, all known as ice, and these phases are of especial interest to astronomers and astrogeologists since many of these phases presumably exist in our solar system, e.g. in Jupiter's icy moon Europa or Saturn's moon Enceladus.
This phase diagram raises many interesting questions: how do the phases---which all consist of water molecules---differ from one another? For a given substance like water, is there a limit to how many phases can exist? What are the rules, if any, that govern how one phase changes into another phase as the temperature or pressure is varied? For example, in the figure you can see so-called triple points where three phases (such as solid, liquid, and vapor phases, or two solid phases and a liquid phase) can coexist. Can a quadruple or higher-order point exist?
We will answer some of these questions during the
semester. Indeed, if we think of the different phases that
form from "structureless" vapor as a kind of pattern
formation (say through the formation of different
crystalline structures), phase transitions become an
important conceptual and technical metaphor for
understanding how complex structure emerges from simpler
structure as some parameter is varied. Experiments and
theoretical work concerning phase transitions in fact have
strongly influenced how people think about questions such as
nonequilibrium pattern formation (say formation of cloud
patterns or snowflakes), the transition from inanimate to
animate molecules (origin of life), the formation of matter,
stars, and galaxies from the superhot plasma of the Big
Bang, consciousness in brains (is this an abrupt transition
as brains become bigger?), and the properties of social,
economic, and computer networks.