* Coordinates in square brackets such as  denote a direction vector (in real space).
* Coordinates in angle brackets or chevrons such as &lt;100&gt; denote a family of directions which are related by symmetry operations. In the cubic crystal system for example,  or the negative of any of those directions.
* Miller indices in parentheses such as (100) denote a plane of the crystal structure, and regular repetitions of that plane with a particular spacing. In the cubic system, the normal to the (hkl) plane is the direction [hkl], but in lower-symmetry cases, the normal to (hkl) is not parallel to [hkl].
* Indices in curly brackets or braces such as  denote a family of planes and their normals. In cubic materials the symmetry makes them equivalent, just as the way angle brackets denote a family of directions. In non-cubic materials, <hkl> is not necessarily perpendicular to {hkl}.