In materials science right now we are learning about miller indices and crystallographic directions, including planes.
What is the aspect ratio (height/width) of an AFM (atomic force microscope) silicon tip in the shape of a square pyramid where each face of the pyramid is a (111) silicon plane? The silicon tip is etched out of a  silicon wafer. The questions says that for a  wafer the  direction points normal to the surface.
The book doesn't give any actual equations, only the miller indices definitions, which match what is given on http://en.wikipedia.org/wiki/Miller_index Basically each digit within the brackets represents one of three directions the plane takes, but not the actual value.
Silicon has a diamond crystal structure that is actually two FCC crystal structures offset along the vector (a/4, a/4, a/4) where a represents the lattice constant.
The Attempt at a Solution
I don't have a good grasp of the crystallographic direction material, unfortunately.
I think the tip of the pyramid relative to the center of the base can be expressed by the miller indices  in terms of vector notation. After that it would be a matter of relating the  vector to the (111) planes with the Pythagorean theorem.
Would I interpret the (111) plane as having a height of 1 in the z direction and the base having length 1 for both the x and y direction? Then the height would be 1/(sqrt2) (read it as 1 over root 2). But then it doesn't seem to connect with the  normal vector, which confuses me.
P.S. I have also derived that in silicon's crystal structure the atomic radius and lattice constant can be related by the equation r = (a(3)^0.5)/8 (read it as "a root 3 over 8"), but I don't get how this is relevant to the question, however the information was provided so I think it relates somehow.