Crystal geometry is described relative to the lattice axes using integer index notation.

Directions

A crystallographic direction is written $[uvw]$. The integers are proportional to the direction's components along the lattice basis vectors and are reduced to the smallest integer ratio.

For a cubic crystal, the direction from one lattice point to another displaced by one cell in $x$, one in $y$ and none in $z$ is

$$[110].$$

A family of symmetry-equivalent directions is written $\langle uvw\rangle$.

Planes

A crystal plane is labeled by Miller indices $(hkl)$. To find them:

1. determine the plane's intercepts with the crystallographic axes in lattice-coordinate units;
2. take the reciprocals of those intercepts;
3. clear fractions to obtain the smallest integer triplet.

A plane parallel to an axis has an infinite intercept there, whose reciprocal is zero. Thus, in a cubic crystal, a plane intercepting $x$ at one lattice spacing and parallel to $y$ and $z$ is $(100)$.

Negative indices are conventionally written with an overbar, such as $(\bar110)$.

A family of symmetry-equivalent planes is written ${hkl}$.

Directions and planes are distinct geometric objects even when the same integers appear. The notation makes slip systems, lattice spacings, diffraction peaks and anisotropic properties concise enough to compare across crystal structures.