Crystallography & Symmetry
patterns derived from their groups
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The fourteen Bravais lattices

The fourteen Bravais lattices
The fourteen Bravais lattices. All fourteen lattices: triclinic P, with 2 symmetries; monoclinic P, with 4 symmetries; monoclinic C, with 4 symmetries; orthorhombic P, with 8 symmetries; orthorhombic C, with 8 symmetries; orthorhombic I, with 8 symmetries; orthorhombic F, with 8 symmetries; tetragonal P, with 16 symmetries; tetragonal I, with 16 symmetries; rhombohedral P, with 12 symmetries; hexagonal P, with 24 symmetries; cubic P, with 48 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

All fourteen lattices: triclinic P, with 2 symmetries; monoclinic P, with 4 symmetries; monoclinic C, with 4 symmetries; orthorhombic P, with 8 symmetries; orthorhombic C, with 8 symmetries; orthorhombic I, with 8 symmetries; orthorhombic F, with 8 symmetries; tetragonal P, with 16 symmetries; tetragonal I, with 16 symmetries; rhombohedral P, with 12 symmetries; hexagonal P, with 24 symmetries; cubic P, with 48 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

6 essays call bravais-cell. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

The fourteen Bravais lattices. All fourteen lattices: triclinic P, with 2 symmetries; monoclinic P, with 4 symmetries; monoclinic C, with 4 symmetries; orthorhombic P, with 8 symmetries; orthorhombic C, with 8 symmetries; orthorhombic I, with 8 symmetries; orthorhombic F, with 8 symmetries; tetragonal P, with 16 symmetries; tetragonal I, with 16 symmetries; rhombohedral P, with 12 symmetries; hexagonal P, with 24 symmetries; cubic P, with 48 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything. Lattices

Twenty-five cells, and fourteen lattices

The usual picture of the fourteen Bravais lattices is a plate of fourteen boxes, which is the answer with the argument removed. The argument is one question asked of twenty-five candidates, and the question has a computable answer.

 4 lattices. 4 lattices: cubic P, with 48 symmetries; cubic C, with 16 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything. Lattices

Forty-eight becomes sixteen

Centre one face of a cube and the four threefold axes along its body diagonals are gone. That sentence is usually offered as a fact to accept; it is a computation whose answer is a number, and the number says which lattice you got instead.

 3 lattices. 3 lattices: hexagonal P, with 24 symmetries; rhombohedral P, with 12 symmetries; hexagonal R, with 12 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything. Lattices

A lattice described on somebody else's axes

R-centring a hexagonal cell does lower its symmetry, from twenty-four to twelve — and the lattice that results is the fourteenth, the rhombohedral one, which already appears on the list under its own axes. It is the only row in the enumeration where losing symmetry and being a duplicate are the same verdict.

 One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours. Symmetry at work

Two stackings, one density

Stack spheres as tightly as they will go and the third layer has a free choice. Both answers fill exactly the same fraction of space and give every sphere the same twelve neighbours — and their space groups are Fm3̅m and P6₃/mmc, which is the only thing that tells them apart.

 Five shapes, and a lattice in space has no other. The five combinatorial types a Wigner–Seitz cell can have in three dimensions — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, truncated octahedron — each drawn from a lattice that produces it. Fedorov proved in 1885 that there are no others, and that fourteen faces is the most any of them has, which is Minkowski's bound of 2(2ⁿ − 1) in three dimensions. Each solid here is cut out by the perpendicular bisectors of nearby lattice vectors and its volume checked against the primitive cell's, which is what catches a face that failed to appear. Lattices

Five parallelohedra, and no others

The cell that needs no basis and no convention has, in three dimensions, exactly five shapes. The fourteen Bravais lattices produce all five between them — and which one a lattice gives is not decided by which of the fourteen it is.

 Sublattices of index n, in space. How many sublattices a three-dimensional lattice has at each index, beside the plane's answer, with the Hermite enumeration and the coefficient of ζ(s)ζ(s−1)ζ(s−2) in separate columns. The two are computed by routines sharing no code, and a row where they disagreed would be a failure rather than a result. The last column counts the ones that survive every operation of the cubic group, and it is almost always empty. Lattices

The three that stay cubic

A lattice in space has far more sublattices than one in the plane — 651 of index sixteen against 31 — and almost none of them keeps the symmetry it came from. The ones that do exist at indices m³, twice m³ and four times m³, there is exactly one at each, and they are the primitive, face-centred and body-centred cubic lattices, arrived at by asking which sublattices keep a symmetry rather than by enumerating centrings.

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