Why hexagonal breaks the three-index habit
In the last two guides you learned the whole three-index toolkit: a direction [uvw] is a set of steps along the cell edges, and a plane's Miller indices (hkl) come from taking the reciprocals of its intercepts and clearing fractions. That machinery works flawlessly for cubic, tetragonal and orthorhombic cells, where the three axes are mutually perpendicular. Hexagonal crystals are where it starts to creak — not because the recipe fails, but because it hides something the crystal is proud of.
The hexagonal cell has one special axis, c, standing straight up, and a flat basal plane at its foot containing not two but effectively three equal directions spread 120 degrees apart, related by the crystal's six-fold symmetry. If you insist on describing that flat plane with only two axes a1 and a2 (as the plain three-index scheme does), the six equivalent side faces of a hexagonal prism come out with indices like (100), (010) and (-110). Those are genuinely the same face repeated by symmetry, yet nothing in the numbers (100) and (-110) shouts 'we are the same'. The symmetry is real but invisible.
Four axes for a three-axis job
The fix is to put three axes into the basal plane instead of two, and give each its own index. We keep a1 and a2, add a third basal axis a3 pointing along the last of the three symmetry-equivalent directions, and keep c standing up. The three basal axes lie flat at 120 degrees to each other, so they are not independent: geometry forces a1 + a2 + a3 = 0, exactly the way three arrows spaced evenly around a circle must cancel. This is the Miller-Bravais four-axis frame, and it is used only for the hexagonal system.
A plane now gets four numbers, written (hkil): h, k and i are the reciprocal intercepts on a1, a2 and a3, and l is the reciprocal intercept on c. Because the three basal axes add to zero, the three basal indices are locked together by the identical rule: i = -(h + k), always and exactly. It is not a rounding, not an approximation — it follows straight from a1 + a2 + a3 = 0. So the i index carries no new information; you can cover it up and reconstruct it any time. People often write it as a dot, (hk.l), precisely to signal 'this one is forced.'
Indexing a plane the four-index way
For planes the four-index scheme is almost free, because you already know how to get (hkl) from guide 2 — you just compute the fourth number and slot it in. Take one side face of the hexagonal prism. It cuts a1 at 1, runs parallel to a2 and to c (intercepts at infinity), and, being tilted the way the basal geometry demands, cuts a3 at -1. Reciprocals of (1, infinity, -1, infinity) are (1, 0, -1, 0), so the plane is (10-10). Check: i = -(h + k) = -(1 + 0) = -1. It matches, as it must.
Looking straight down c at the basal plane (c points out of the page):
a2
\
\ a1 , a2 , a3 all lie flat, 120 deg apart
_________ \ ________ a1 and a1 + a2 + a3 = 0
/
/
/
a3
The six prism faces {10-10} -- now obvious permutations:
(1 0 -1 0) (0 1 -1 0) (-1 1 0 0)
(-1 0 1 0) (0 -1 1 0) ( 1 -1 0 0)
The basal plane on top = (0001).Now the payoff you were promised. Those six prism faces, which looked unrelated in three indices, are written (10-10), (01-10), (-1100), (-1010), (0-110) and (1-100). Every one is just a shuffle of the same four numbers {1, 0, -1, 0}. The eye reads the six-fold symmetry off the page instantly, and the whole set is captured by the family symbol {10-10}. That is exactly the readability the plain scheme threw away.
Directions are trickier — do not just insert a number
Here is the trap that catches almost everyone. For planes you could just compute i = -(h + k) and append it. For directions you cannot. A direction [uvtw] must also obey t = -(u + v), but you are NOT allowed to take an ordinary three-index [UVW], set t = -(U + V), and call it done — that gives the wrong direction. Directions and planes convert differently because a direction is an actual arrow of real steps, not a set of reciprocal intercepts.
- Start from the three-index direction [UVW] (using only a1, a2, c).
- Compute the four basal-corrected numbers: u = (2U - V)/3, v = (2V - U)/3, t = -(u + v), and w = W.
- Clear the thirds by multiplying all four by the same whole number, then reduce to the smallest integers. Write the result as [uvtw].
- To go back the other way: U = u - t, V = v - t, W = w.
Try it on the a1 axis, which is [100] in three indices. Then u = (2 - 0)/3 = 2/3, v = (0 - 1)/3 = -1/3, t = -(2/3 - 1/3) = -1/3, w = 0. Multiply through by 3 and you get [2-1-10]. Notice this is nothing like the naive [10-10] you would have got by blindly inserting t = -(u + v) into [100] — that shortcut actually points along [210], a completely different arrow. This is why families of directions like <11-20>, the three a-axes and their negatives, are always converted properly, never guessed.
What the extra index buys you
The reward is not cosmetic — it changes how easily you can reason about the real behaviour of a hexagonal metal. In the hexagonal close-packed structure the flat basal plane (0001) is the close-packed plane, the densest sheet of atoms, and the three a-directions <11-20> lying in it are the close-packed directions where atoms touch bumper to bumper. Because those are the easiest plane and directions to shear, the primary slip system of an HCP metal is basal slip, written cleanly as (0001)<11-20>. The four-index labels make that slip system, and the fact that there are far fewer of them than in a cubic metal, jump straight off the page.
That scarcity of easy slip systems is not a curiosity — it is why magnesium, zinc and titanium are so much less forgiving to bend than aluminium or copper, and why hexagonal metals show strong direction-dependent properties. The tidy indices are the bookkeeping behind real anisotropy. And the same four numbers feed forward: in the next guide the interplanar spacing d and the Miller-Bravais plane-spacing equation for hexagonal cells will use exactly this (hkil), which is in turn the plane label that Bragg's law reads to tell you where a diffraction peak lands.