lattice vectors are orthogonal and of equal length (usually denoted a), as are those of the reciprocal lattice. Thus, in this common case, the Miller indices (hkℓ) and [hkℓ] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (hkℓ) lattice planes is (from above) . Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes: Indices in angle brackets such as ⟨100⟩ denote a family of directions which are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions. Indices in curly brackets or braces such as {100} denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions. For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions. Case of hexagonal and rhombohedral structures With hexagonal and rhombohedral lattice systems, it is possible to use the Bravais-Miller system, which uses four indices (h k i ℓ) that obey the constraint h + k + i = 0. Here h, k and ℓ are identical to the corresponding Miller indices, and i is a redundant index. This four-index scheme for labeling planes in a hexagonal lattice makes permutation symmetries apparent. For example, the similarity between (110) ≡ (110) and (10) ≡ (110) is more obvious when the redundant index is shown. In the figure at right, the (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2π/3 rad, 120°). The [100], [010] and the [0] directions are really similar. If S is the intercept of the plane with the [0] axis, then i = 1/S. There are also ad hoc schemes (e.g. in the transmission electron microscopy literature) for indexing hexagonal lattice vectors (rather than reciprocal lattice vectors or planes) with four indices. However they don't operate by similarly adding a redundant index to the regular three-index set. For example, the reciprocal lattice vector (hkℓ) as suggested above can be written in terms of reciprocal lattice vectors as . For hexagonal crystals this may be expressed in terms of