A hexagonal crystal structure (and trigonal ones usually as well) uses a base with equivalent vectors a and b at an angle gamma = 120 between these vectors. The third vector c is normal to the a-b-plane and of any length compared to a and b. The Miller indices hkl of a lattice plane are calculated by taking the intercepts opq of the lattice plane with the three base vectors. The intercepts are first converted to 1/o, 1/p, 1/q where a o,p,q=infinity results in the reciprocal value zero. For surface facets the values 1/o, 1/p, 1/q are multiplied by a common multiplier to give the smallest set of integers free from common dividers, these are the hkl.

In the hexagonal base the vector -(a+b) is symmetrically equivalent to vectors a and b ( it is obtained by a 120° rotation). Thus this third (redundant) vector can equally well be used as base vector. The Miller index with respect to this third axis is referred to as "i" and the triplet of Miller indices becomes a quadruplet:

h k l => h k i l

with i = -(h + k)

The advantage of this notation is that the 120° rotation transforms the Miller indices in a cyclic fashion : h k i l => i h k l => k i h l

which makes the symmetry relationship more obvious. (yes only hki do the cyclic permutation!)

Thus, yes 1 0 1 and 1 0 -1 1 (== 1 0 bar1 1) are the identical lattice planes in a hexagonal or trigonal structure.

Thank you so much Steven Van Petegem and R. B. Neder