One lattice, many cells you could draw
The last two guides handed you two ideas we now lean on hard. First, a crystal is a lattice plus a motif: the lattice is pure geometry — an infinite array of identical points — and the motif is the group of atoms hung on each point. Second, we tile that infinite pattern with a single repeating unit cell whose shape is fixed by six lattice parameters (the edge lengths a, b, c and the angles between them). This guide asks a deceptively simple question: for a given lattice, which cell should we draw?
The surprise is that the answer is not unique. Picture a sheet of graph paper: you can outline the repeating tile as the little square between four dots, but you could just as honestly outline a slanted parallelogram that still tiles the whole sheet with no gaps. Both are legal unit cells for the same lattice. So we need a way to compare cells — and the sharpest yardstick is how many lattice points each one truly contains.
Counting points: the fractions trick
A lattice point sitting on a corner of the cell is not owned by that cell alone — it is shared with the neighbouring cells that meet there. In three dimensions, eight cells meet at every corner, so a corner point contributes only 1/8 of itself to any one cell. The full bookkeeping is: a corner point counts as 1/8, a point on a face is shared by two cells and counts as 1/2, a point on an edge counts as 1/4, and a point wholly inside the cell counts as a full 1. Adding these fractions gives the number of lattice points per cell.
Try the simplest case. A simple cubic cell has a lattice point on each of its eight corners and nowhere else. The count is 8 corners times 1/8 = 1. A cell that contains exactly one lattice point is called a primitive cell: it is the smallest cell that still tiles the whole lattice, and every lattice is guaranteed to have one. Anything with more than one point is, by definition, larger than it needs to be.
One caution: the fractions are pure bookkeeping, not physics. No atom is really 'sliced'; the trick just makes sure that when you tile the whole crystal, every point is counted exactly once across the cells that share it. Whether you count points (geometry) or atoms (points times the motif size), the same fractions apply — and that distinction between a point and an atom is the trap we return to at the end.
Centered cells: I, F and C
Sometimes a lattice carries extra points beyond the corners, sitting at special positions that are themselves identical to the corners in every way. Adding these is called centering, and there are three flavours worth naming. A body-centered cell (symbol I) adds one point at the very center of the cell. A face-centered cell (symbol F) adds one point at the center of each of the six faces. A base-centered cell (symbol C) adds a point at the center of just one opposing pair of faces.
Run the fractions and the payoff is immediate. Body-centered: 8 corners times 1/8 plus 1 whole point inside = 2 lattice points per cell. Face-centered: 8 times 1/8 plus 6 faces times 1/2 = 1 + 3 = 4 points per cell. Base-centered: 8 times 1/8 plus 2 faces times 1/2 = 1 + 1 = 2 points per cell. Each of these describes exactly the same kind of object — one lattice — but with two or four points in the box instead of one.
CELL ADDED POINTS POINTS/CELL FRACTIONAL COORDINATES ---------------- ------------------- ----------- ------------------------------------- P primitive none 1 (0,0,0) I body-centered 1 at body center 2 (0,0,0) (1/2,1/2,1/2) F face-centered 1 on each of 6 faces 4 (0,0,0) (1/2,1/2,0) (1/2,0,1/2) (0,1/2,1/2) C base-centered 1 on one face pair 2 (0,0,0) (1/2,1/2,0)
That last column is worth pausing on. Instead of giving positions in absolute length, crystallographers use fractional coordinates: a point at (1/2, 1/2, 1/2) sits halfway along each edge, whatever the cell's real size. It is the natural language of the cell because it is dimensionless and travels unchanged from a tiny metal to a giant protein. Body-centered iron, for example, is fully specified by two lines: a lattice point at (0,0,0) and one at (1/2,1/2,1/2).
Why ever choose a bigger cell?
Here is the obvious objection. If a primitive cell with one point is always available, why do textbooks draw the four-point face-centered cube for copper or the two-point body-centered cube for iron? Because a bigger cell can be honest about symmetry in a way the primitive one is not. The primitive cell of a face-centered-cubic lattice does exist, but it is a squashed rhombohedron: its three edges run along face diagonals, meet at 60-degree angles, and look nothing like a cube. Stare at it and you would never guess the lattice has full cubic symmetry.
So we make a deliberate trade. We choose the larger conventional cell — the tidy cube with 90-degree angles and four lattice points — precisely because its shape wears the crystal's symmetry on its sleeve. We pay for that clarity with a cell four times bigger than strictly necessary, and we accept it gladly. The rule of thumb: use the primitive cell when you want the absolute minimum (for theory and counting), and the conventional cell when you want the symmetry to be obvious (for teaching, indexing and everyday work).
The Wigner-Seitz cell: a fairer primitive cell
The slanted primitive cell has one ugly flaw: its lopsided shape hides symmetry. There is a beautiful fix. Pick one lattice point, draw a straight line to each of its neighbours, and cut each line with a plane exactly at its midpoint (a perpendicular bisector). The little region trapped by all these planes — every spot closer to your chosen point than to any other — is the Wigner-Seitz cell. It contains exactly one lattice point, so it is primitive, yet it is built symmetrically about that point, so it always shows the lattice's full symmetry.
A homely picture: give every lattice point a small town and draw the borders so that each town owns exactly the land nearest to it. Those borders are the Wigner-Seitz cells, and they tile all of space with no gaps and no overlaps (mathematicians call this a Voronoi partition). For a body-centered lattice the cell comes out as a handsome truncated octahedron — far more revealing than the slanted box. This construction is not just a curiosity: run the very same recipe in reciprocal space, a tool the next rung builds, and the Wigner-Seitz cell becomes the Brillouin zone that governs how electrons and waves move through the crystal.
From cells to systems and lattices
You now have every piece needed for the two big enumerations that close this rung. Sort all possible cell shapes by their axial relations — which edges are equal, which angles are 90 or 120 degrees — and they collapse into exactly seven families, the crystal systems, the subject of the next guide. Then allow each system to be primitive or centered, and you might expect seven times four combinations. But most of those are redundant: a base-centered cubic cell, for instance, can always be redrawn as a smaller primitive cell of a different system, so it is not a new lattice at all.
When you throw out every redundant combination, you are left with exactly fourteen distinct arrangements of lattice points — the Bravais lattices, the subject of the final guide. Be clear on the honesty here: fourteen is not 'fourteen so far'. It is a complete, proven count, like the fact that there are exactly five Platonic solids. The same rigour later yields exactly 32 point groups and 230 space groups. These are closed lists that mathematics has finished, not tallies awaiting the next discovery.