I was just recently looking into some of the details of the different crystal structures that ice can freeze in. I was aware that many of the unusual properties of ice $I_h$ (the structure of “everyday ice”) are a result of the proton disorder necessary for the hydrogen-bonding arrangement it adopts.
What I also found is there are other arrangements of ice which are proton ordered and have nearly identical density to ice $I_h$. Now, I know density has no predictive value on what crystal structure will be taken (rather crystal structure can be used to predict density), but it seems unusual to me that there is such massive divide between ice $I_h$ and the rest of the ice srystal structures.
Is the entropic gain of proton disorder an explanation why ice freezes the way it does? Also, is there a large energy minimization associated with the hexagonal lattice that ice $I_h$ takes on? Doesn't it seem like a proton-ordered arrangement would be preferable over a proton-disordered arrangement? Or is it all much more complicated than that? It just seems like there should be a crystal structure of ice which is comparable to ice $I_h$ in energy, stability, etc. and yet there isn't.
I suppose a true answer to this question, because I admit it's rather broad, would deal with why the hexagonal lattice is so much more preferable for water than some other lattice arrangement. By dealing strictly with water, rather than crystals in general, I believe an answer to this question is more within reach.