crystallographic directions
To point somewhere inside the crystal box, you give a little recipe of steps along the cell edges: so many steps along x, so many along y, so many along z. A crystallographic direction is exactly this — a standard way to name a straight direction through the crystal.
A direction is written as [uvw] in square brackets. To find the indices, take a vector along the direction, express its components as multiples of the edge lengths a, b, c, clear any fractions to the smallest whole integers, and mark a negative component with a bar over the number. For a cube, an edge is the [100] direction, a face diagonal is [110], and the body diagonal from corner to opposite corner is [111]. A whole family of symmetry-equivalent directions is written with angle brackets, like the six cube edges together as <100>.
This labelling is not busywork: metals slip, and therefore deform, most easily along their close-packed directions, so naming directions lets engineers predict and describe how a crystal yields. Directions also underlie anisotropy — the stiffness of iron measured along [100] is very different from that along [111].
In a cubic cell, a cube edge is the [100] direction, a face diagonal is [110], and the body diagonal running corner-to-opposite-corner is [111]. The set of all six cube-edge directions is written together as <100>.
Naming directions in a cube: [100] edge, [110] face diagonal, [111] body diagonal.
Square brackets [uvw] mean one specific direction; angle brackets <uvw> mean the whole family of equivalent directions. Directions use integers directly, not the reciprocals used for planes.