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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.....)

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    "I suspect chemists might dismiss this question as being about magic and voodoo and mystical and the like." - I, for one, don't. On the other hand, you're probably aware of the two lone electron pairs in water contributing to the deviation from the ideal tetrahedral angle, so I'm not entirely sure that the reasons are entirely geometric.2011-11-14
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    Maybe people are less accustomed to thinking about the $-1/3$ situation than about the honeycomb, so here it is: Picture four unit vectors pointing out from the origin in $\mathbb{R}^3$, arranged so that all of the six angles between two of them are equal. What are the components of the four vectors in the direction of one of them? The component of one of them is $1$. The components of the other three are equal to each other, and the components of all four must of add up to $0$.2011-11-14
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    Michael: a probably more geometrically intuitive way to see the point you're presenting is to consider the embedding of the tetrahedron into a cube...2011-11-15

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