Geometry and chemistry There's a really obvious geometric reason why the cosine of the bond angle in graphite is $-1/2$: the stuff consists of sheets shaped like honeycombs.
There's also a really obvious geometric reason why the cosine of the bond angle in methane is $-1/3$: four hydrogen atoms are arranged in a symmetric way about a center, with all six angles between them equal: tetrahedral symmetry.
Is there a geometric reason why the cosine of the bond angle in water is $-1/4$?
(I suspect chemists might dismiss this question as being about magic and voodoo and mystical and the like.  Mathematicians don't mind magic and voodoo and mysticism, but maybe they object to vagueness or chemistry.  So we'll see if this question is tolerated here.)
(Full disclosure: once upon a time I posted nearly this same question somewhere else.....)

The answer is unfortunately disappointing: No, there's no geometric reason. The only geometric prediction for the water molecule is the one J. M. already mentioned, the tetrahedral angle predicted by VSEPR theory, which is a qualitative theory that doesn't distinguish between lone pairs and bonding pairs.
That there's nothing else hidden behind this numerical coincidence is evidenced by very precise Hartree-Fock calculations for water molecules, which yield a bond angle of $106.339^\circ$, quite a bit off from $\arccos(-\frac14)$. These calculations should pick up on any geometric aspects of the problem; the difference between this value and the actual value is due to the electronic correlations that are neglected in the Hartree-Fock approximation.
