, = integers. The end points of all possible translations vectors define the lattice as a periodic sequence of points in space. Unfortunately, one and the same lattice can be defined by many different sets of vector triples as illustrated right below. construct, a succession of (infinitely small) mathematical points in space. A perfect drawing of such a lattice thus would show nothing at all. Instead of Working with lattices and crystals produces rather quickly the need to describe certain directions and planes in a simple and unambigous way. Stating that an elemental face-centered cubic crystal can be made by assigning one atom to any lattice point found on ‘that plane that runs somehow diagonally through the unit cell’ just won’t do it. How to derive the Miller indices of a certain direction or plane is easy. Here is the recipe for directions (in 2 dimensions for simplicity), the figure below illustrates it: Find the intersection points h’, k’, and l’ of the plane with the (extended) base vectors. If there is none, the value is If you wonder why this slightly awkward procedure was adopted, the answer is easy: You can use the Miller indices directly in a lot of equations needed for calculating properties of crystals. Pick a base, a collection of atoms in a fixed spatial relation (similar and often but not always identical to a molecule of the substance. The example above shows how to make a crystal of the diamond type. The base consists of two atoms. In the coordinate system of the lattice unit cell (indicated by arrows), the two atoms have the coordinates (0,0,0) and (¼,¼,¼). This looks simple. It is not. It’s the point where things get difficult and confusing. Ask yourself for any still simple crystal: how many atoms are there to a lattice plane? How many atoms are in a base? , all have an fcc lattice. Different colors of the circles my or may not denote different atoms. Can you figure out the bases? If you can, you’re ahead of my average third-term student. One last thought: Crystals in a general sense, meaning an arbitrary base arranged in a periodic way, can be found everywhere, here is an example: Source.