A bit more on the math-stuff of stability that I was talking about, with a variation on the metaphor:
It's ... consider a quartic, say x^4 - x^2 + a*x, with initially a=0. Basically a parabola-like thing, except with two dips in the bottom, one on each side of the y axis; the bottom of each dip is an equilibrium point if you have a metaphorical marble rolling along the curve. Suppose the marble is in the right-hand dip, and you increase a. As you increase a, the right-hand dip goes up, and the left-hand dip goes down. As it goes up, the left-hand dip becomes more and more of a "preferred" equilibrium, but the marble stays in the right-hand equilibrium instead. Eventually, the right-hand dip will stop being a dip, and the marble will suddenly roll into the left-hand dip.
Or, alternately, you can adjust things so that the right-hand dip is still a dip, but just barely one, and then jostle the marble slightly -- and it will go over the bump and end up in the left-hand equilibrium.
This could be a representation of cold water -- one dip is "solid", the other "liquid". And what happens is that, at the freezing point, the two dips are equally low; they're both equally stable. But, as you chill the water, the "liquid" side becomes less and less of a dip, and it very quickly becomes a very tiny dip, although it takes a lot of chilling for it to disappear entirely. So, the water stays liquid, but give it a bit of jostling, and that causes the "marble" for a spot near the jostling to fall over into the "solid" side, and that jostles the ones near it, and so forth across the glass of water.
What I think is happening in your "normal" modes is something sort of like this: instead of a marble, you've got a quantum superposition of states. And so, when the two dips are equal, the quantum-marble is half in one and half in the other -- sort of like Schroedinger's cat, except this one doesn't collapse its wavefunction if you observe it. If you raise one dip, what happens is that the amount of quantum-marble in it decreases and the quantum-marble in the other increases in a smooth fashion; there's never a situation where this thing is in a state other than its preferred "best" equilibrium. And the marble, as a whole, is in the two dips -- never in the middle. So, the presence of quantum-marble in multiple dips corresponds to having multiple facets fronting, and the fraction of quantum-marble in each dip corresponds to the amount that the corresponding facet is front.
Re: A thought and a question (not connected to each other)
It's ... consider a quartic, say x^4 - x^2 + a*x, with initially a=0. Basically a parabola-like thing, except with two dips in the bottom, one on each side of the y axis; the bottom of each dip is an equilibrium point if you have a metaphorical marble rolling along the curve. Suppose the marble is in the right-hand dip, and you increase a. As you increase a, the right-hand dip goes up, and the left-hand dip goes down. As it goes up, the left-hand dip becomes more and more of a "preferred" equilibrium, but the marble stays in the right-hand equilibrium instead. Eventually, the right-hand dip will stop being a dip, and the marble will suddenly roll into the left-hand dip.
Or, alternately, you can adjust things so that the right-hand dip is still a dip, but just barely one, and then jostle the marble slightly -- and it will go over the bump and end up in the left-hand equilibrium.
This could be a representation of cold water -- one dip is "solid", the other "liquid". And what happens is that, at the freezing point, the two dips are equally low; they're both equally stable. But, as you chill the water, the "liquid" side becomes less and less of a dip, and it very quickly becomes a very tiny dip, although it takes a lot of chilling for it to disappear entirely. So, the water stays liquid, but give it a bit of jostling, and that causes the "marble" for a spot near the jostling to fall over into the "solid" side, and that jostles the ones near it, and so forth across the glass of water.
What I think is happening in your "normal" modes is something sort of like this: instead of a marble, you've got a quantum superposition of states. And so, when the two dips are equal, the quantum-marble is half in one and half in the other -- sort of like Schroedinger's cat, except this one doesn't collapse its wavefunction if you observe it. If you raise one dip, what happens is that the amount of quantum-marble in it decreases and the quantum-marble in the other increases in a smooth fashion; there's never a situation where this thing is in a state other than its preferred "best" equilibrium. And the marble, as a whole, is in the two dips -- never in the middle. So, the presence of quantum-marble in multiple dips corresponds to having multiple facets fronting, and the fraction of quantum-marble in each dip corresponds to the amount that the corresponding facet is front.