a crystallographic direction

Imagine you are standing inside a crystal and someone asks you to point along a particular line of atoms. You cannot just say up or left, because the crystal has its own built-in axes — the three edges of the unit cell, called a, b and c. A crystallographic direction is simply an arrow drawn in this built-in coordinate system, and we name it with three whole numbers written in square brackets, [uvw]. Think of it like giving a compass bearing, but the compass is the crystal's own cell rather than the room you are in.

To read off the indices [uvw] you draw the arrow starting at a corner of the cell (the origin), see how far it goes along each of the three cell edges measured in units of a, b and c, and then scale those three numbers to the smallest whole numbers that keep the same ratio. An arrow that goes one step along a, one step along b and none along c is [110]. The arrow to the far body-corner of the cell, going one step along every edge, is [111]. If the arrow runs backwards along an axis the index is negative, written with a bar over it (printed here as a minus, so backwards along a is [-100]). Because the direction is really just a ratio, [110], [220] and [330] are all the same line — we always reduce to lowest terms.

The square brackets are not decoration; the bracket style tells a reader exactly what you mean. Square brackets [uvw] mean one specific direction, angle brackets <uvw> mean a whole family of symmetry-equivalent directions, round brackets (hkl) mean a plane, and curly braces {hkl} mean a family of planes. A trap for beginners: in a cubic crystal the direction [hkl] happens to point straight out of (be perpendicular to) the plane (hkl), which makes them feel interchangeable, but this coincidence fails in every non-cubic system, where the cell edges are not at right angles.