Image from Dr. S. A. Nelson's petrology notes online at Tulane Earth & Environmental Sciences Mineralogy 211
From when I took this class at Tulane, I remember a big emphasis on this diagram (which can also be found in the lecture notes at the above link):
Of course, this is the 2 front faces (Ne-Di-Fo and Fo-Di-Qz) unfolded, projected from Plagioclase (Pl). I decided to investigate all the binaries that make up the ternaries of these two faces. Here's what I spent my Sunday afternoon making:
(A) Blue circles represent 1 atm experimental natural basalt liquids. The two front faces of the basalt tetrahedron (Di-Ne-Fo and Di-Fo-Qz) are plotted projected from Plagioclase (Pl). The Pl vertex has been projected onto the Di-Fo join (the low pressure “thermal divide”).
(B) and (C): Blowing up the back face of the tetrahedron (Ne-Di-Qz) elucidates some more mysteries involving the "thermal divide" (although technically the join Di-Ab is not the true thermal divide, but Fo-Ab is. Still don't really understand exactly why, but that's for another Sunday afternoon...). I highlighted the back face in green, and in (B) show the 2 binary eutectic systems (Ne-Ab and Qz-Ab) that comprise this face. Ne-Qz is just a binary system with an intermediate compound, Ab.
But the presence of Ab is the whole reason why the thermal divide exists in the first place. On the Ne-Qz, Ab melts at a local maximum, forming a shape that looks like a skateboarding "halfpipe" (or half a cylinder). If we parachute down onto the Ab surface (a thought exercise we were all variably subjected to in Intro Petrology), we can only go one way - either to E1 (and ultimately evolve to a phonolite) or to E2 (and ultimately evolve to a rhyolite). So that little halfpipe of Ab is responsible for the fact that no Si-undersaturated basalt can give rise to a critically Si-undersaturated basalt. It's actually more complicated than this when you consider the system Fo-Ne-Qz, which according to Morse (1981) is the ternary where you can find the "true" thermal divide.
Of course, we all know this already, but it's still a fun (and frustrating but ultimately rewarding) exercise to look at every face of the tetrahedron. Next time: the inner tetrahedron Di-Ab-An-Fo?