Graphene +1 extra carbon bond I'm not a physicist just a curious mind, so please go easy!
I was just watching a BBC Horizon Documentary that featured a piece on the recently discovered material Graphene. One of the facts mentioned on the documentary was that a sheet of graphene, despite being a lattice of carbon atoms just one atom thick, could hold the weight of a household cat without breaking.
They showed an illustration of graphene's hexagonal structure and how a crude form of graphene could be created using Sellotape and piece of graphite.
OK so here's my question.. From what I can remember from Chemistry lessons in school, carbon has a valency of 4, so it can bind to 4 more carbon atoms. Surely (in theory at least) a sheet of carbon could exist that adopts a square grid lattice, as oppose to graphene's hexagonal structure. If so wouldn't such a material be stronger than graphene? Also would such a lattice naturally form under any circumstance or does carbon always assume a hexagonal structure when reduced to layers?
 A: Although it's not strictly what happens, you can think of the bonds around a carbon atom as repelling each other because the electrons localised into those bonds want to get as far away from each oither as possible. That's why when a carbon atom forms three bonds you get the bonds separated by 120º. When you have four bonds they arrange themselves into a tetrahedral shape with an angle of about 109º between each pair of bonds.
It is possible to force the bonds closer together, and molecules, like cyclobutane, exist that have a four carbon ring. However these tend to be more easily attacked by reagents than five or six carbon rings, so they tend to be unstable. These molecules have only two bonds separated by a 90º angle, with the other two bonds at more like the usual 109º and in a plane at right angles to the two 90º bonds. I don't know of any molecule with four bonds at 90º in the same plane. My guess is that the energy of this arrangement would be so high it would spontaneously reorganise to something more stable.
A: There is such a material where each carbon atom binds to four other atoms. It's not a square lattice (due to the character of the so-called sp3-hybridization: the energetically most stable configuration is in 3D, not 2D).
There are several standard bondings for carbon (and many other materials): the sp2-hybridization is in 2D and has three bonds (like graphite, graphene and all organic molecules), and the (stronger) sp3-hybridization has four bonds, but is in 3D. My guess is that the bond in a 2D material with a square lattice is much weaker than in graphene, whereas the diamond (sp3) structure has a much stronger bond.
You can change graphene so that it becomes diamond if you put energy into it, and you will get more out of it than you put in, because the diamond structure is energetically optimal. From a physicist's perspective, it's like igniting a balloon with hydrogen: you need to put in a little energy, but you get more out of it.
A: Apparently, one of the bonds in graphene is a double bond. 
Rarely represented in the graphics. 
