How does eletricity flow through a metal square? Take a metal square about 5cm x 5 cm.
Put wires on alternate corners.
Cover the square with insulation.
How would the electricity flow from corner to corner?
By the law of least action, should we expect it to flow in a straight line from corner to corner? Or in comparision with lightning would it fork out in various directions?
In fact is there any type of material that one could make the electricity flow in a thin line from corner to corner? Or does it flow in some smooth flow pattern like a liquid? Or turbulantly?
(The reason I was thinking about this was in relation to storing data on a square of some material and using electricity to flip the bits along a line).
 A: 
By the law of least action, should we expect it to flow in a straight
  line from corner to corner? Or in comparision with lightning would it
  fork out in various directions?

It will spread out with the maximum current being in the diagonal path corner to corner, since it is the shortest path, and minimum in the center region moving away from the diagonal towards the other two corners since the paths become longer. This is due to the resistance of any path being proportional to the length of the path, inversely proportional to cross sectional area of the path, and proportional to resistivity (a property of the material). It assumes the resistivity of the material and the thickness is uniform throughout the plate.
The plate can be thought of as consisting of an infinite number of parallel resistors. The diagram below is a crude attempt to convey the idea of the current density in the plate (imagine it as a square rather than a parallelogram, and the lines themselves curved, not straight except at the diagonal). The density of the lines is intended to represent the current density, showing the current density greatest at the corners and along the diagonal, and less towards the center moving away from the diagonal. 

In fact is there any type of material that one could make the
  electricity flow in a thin line from corner to corner? Or does it flow
  in some smooth flow pattern like a liquid? Or turbulantly?

For a given material and its electrical resistivity, the only way to concentrate more and more current along the diagonal, is the make the material thinner and thinner away from the diagonal. Alternatively, if the material was a composite of some sort that the material moving away from the diagonal had a much higher electrical resistivity than along the diagonal. Then there is the combination of the two, a plate that is thinner and having higher resistivity moving away from the diagonal. 
Hope this helps.

