[Watermelon]

Carbon atoms in graphite apparently share one electron with three other carbon atoms, resulting in layers of hexagons.
If I'm thinking right, Carbon needs four electrons to fill up its shell... yet it only shares 3. Therefore there is a spare electron.

But why? Why doesn't it rearrange itself so each carbon atom bonds with FOUR other carbon atoms?

Or is my general idea of chemistry all wrong? I thought atoms always 'try' to get full outer shells..

[AWK]

compare
http://en.wikipedia.org/wiki/Prismane

[fledarmus]

The structures in graphene are actually benzene rings, not cyclohexane rings. Benzene is usually drawn as two resonance structures, each having three double bonds and three triple bonds, for bookkeeping purposes - this allows you to see that each carbon can be seen as having four bonds. In reality, however, the electrons appear to be delocalized into a ring formed by the overlap of one p orbital from each of the six carbons in the benzene ring. If you take an entire sheet of benzene rings, like the graphene structure, the electrons can be delocalized across the entire sheet, which is why graphite, like metals, can carry an electrical current.

If it helps you see better, you can draw it in either of the ways shown below. Rather like the blind men viewing an elephant - each view emphasizes something different, and none are totally accurate, but they may help you build up a picture.