Crystallography: The 14 Bravais Lattices Explained
Every crystalline solid — from a grain of table salt to a diamond to a superconducting ceramic — repeats its atomic arrangement in three dimensions according to one of exactly 14 possible lattice geometries. Not 13, not 15: fourteen. This is not an experimental observation but a mathematical theorem, first proved by French physicist Auguste Bravais in 1848, and it remains one of the cleanest examples of pure symmetry mathematics dictating what nature can and cannot build.
1. What Is a Lattice?
A Bravais lattice is an infinite array of points generated by all integer combinations of three primitive translation vectors a⃗, b⃗, c⃗:
Crucially, every point of a Bravais lattice has an identical environment — the same neighbours at the same distances in the same directions — as every other point. This is what distinguishes a true lattice from an arbitrary periodic pattern: a honeycomb (hexagonal) arrangement of points, for instance, is not a Bravais lattice on its own, because alternating points have differently oriented neighbours; it is instead described as a hexagonal Bravais lattice with a two-point basis.
The unit cell is the parallelepiped spanned by a⃗, b⃗, c⃗, defined by six numbers: three edge lengths (a, b, c) and three angles between them (α, β, γ). Different constraints on these six numbers — imposed by requiring different rotational symmetries — define the seven crystal systems.
2. The 7 Crystal Systems
Group theory shows that only certain rotational symmetries (1-fold, 2-fold, 3-fold, 4-fold, and 6-fold — never 5-fold or 7-fold or higher, a fact called the crystallographic restriction theorem) are compatible with a lattice's strict translational periodicity. Combined with reflections and inversions, this yields exactly seven distinct unit-cell shapes:
| System | Constraints | Example |
|---|---|---|
| Cubic | a=b=c, α=β=γ=90° | Table salt (NaCl) |
| Tetragonal | a=b≠c, α=β=γ=90° | White tin |
| Orthorhombic | a≠b≠c, α=β=γ=90° | Sulphur |
| Hexagonal | a=b≠c, α=β=90°, γ=120° | Graphite, ice |
| Trigonal (Rhombohedral) | a=b=c, α=β=γ≠90° | Calcite, quartz |
| Monoclinic | a≠b≠c, α=γ=90°, β≠90° | Gypsum |
| Triclinic | a≠b≠c, α≠β≠γ≠90° | Turquoise, K₂Cr₂O₇ |
3. Centring Types: Why Not More Than 14
For each crystal system, lattice points can in principle be placed only at the corners of the unit cell (Primitive, P), or additional lattice points can be added at the centre of the cell (Body-centred, I), at the centre of all six faces (Face-centred, F), or at the centre of just one pair of opposite faces (Base-centred, C).
Naively this suggests 7 systems × 4 centrings = 28 possible lattices. But most of these combinations are not distinct — a "centred" lattice in one crystal system frequently turns out to be identical (after choosing a different, smaller unit cell) to a primitive lattice of a different, higher-symmetry system, or simply mathematically redundant given the required point-group symmetry. Working through every combination rigorously, only 14 produce genuinely distinct lattices:
Total: 3+2+4+1+1+2+1 = 14
4. The Complete Table of 14 Lattices
These 14 lattices are formally called the Bravais lattices, and every crystalline material's underlying translational symmetry is one — and exactly one — of them. Note that this classifies only the translational symmetry; combined with the point-group symmetry of the basis (the atoms placed at each lattice point) and additional symmetry operations like glide planes and screw axes, the full classification yields the 230 space groups, which describe every possible crystal structure in three dimensions.
Crystal Structures Simulation
Rotate and explore all 14 Bravais lattices interactively in 3D, with unit cells and coordination spheres highlighted.
5. Why Some Combinations Reduce to Others
A classic example: a face-centred tetragonal lattice (which might seem geometrically distinct) is mathematically identical to a body-centred tetragonal lattice, just described using a smaller, rotated unit cell — so it is not counted separately. Similarly, a base-centred cubic lattice turns out to be identical to a primitive tetragonal lattice once the correct (smaller) unit cell is chosen, since the C-centring breaks the four-fold symmetry required for a cubic classification. This kind of reduction — always choosing the smallest unit cell consistent with the required symmetry, called the primitive cell when it contains exactly one lattice point — is what eliminates the naive 28 down to the genuine 14.
6. Real Materials and Their Lattices
Metals
Most metallic elements adopt one of three simple, densely-packed lattices: face-centred cubic (FCC — aluminium, copper, gold, nickel), body-centred cubic (BCC — iron at room temperature, chromium, tungsten), or hexagonal close-packed (HCP — magnesium, zinc, titanium), the last of which is technically a hexagonal Bravais lattice with a two-atom basis. See our companion article on BCC, FCC, and HCP for the packing-factor mathematics.
Semiconductors
Silicon and germanium crystallise in the diamond cubic structure — an FCC Bravais lattice with a two-atom basis (the same lattice as diamond itself), which is why silicon wafers can be cut along specific crystallographic planes to control cleavage and electronic properties.
Minerals
Quartz (SiO₂) crystallises in the trigonal system; calcite (CaCO₃) is also trigonal (rhombohedral), which is directly responsible for its characteristic double-refraction (birefringence — see our article on crystal optics). Gypsum is a classic monoclinic mineral, its lower symmetry visible in its characteristically oblique crystal habit.
Frequently Asked Questions
Why are there exactly 14 Bravais lattices and not some other number?
The number 14 emerges from combining the 7 possible unit-cell shapes (constrained by the crystallographic restriction theorem, which permits only 1-, 2-, 3-, 4-, and 6-fold rotational symmetry) with the 4 possible centring types (primitive, body-centred, face-centred, base-centred). Most of the 28 naive combinations turn out to be geometrically identical to another combination once the smallest consistent unit cell is chosen, leaving exactly 14 genuinely distinct lattice types — a result proved rigorously by Auguste Bravais in 1848 (building on earlier, slightly flawed work by Moritz Frankenheim in 1835, who originally proposed 15).
Is a Bravais lattice the same thing as a crystal structure?
No. A Bravais lattice describes only the underlying translational symmetry — an abstract grid of points. A real crystal structure combines a Bravais lattice with a "basis": one or more atoms (or ions, or molecules) attached to every lattice point in a fixed arrangement. Diamond and rock salt (NaCl) both have an FCC Bravais lattice, but diamond's basis is two carbon atoms while rock salt's basis is one Na⁺ and one Cl⁻ ion — producing very different physical properties from the same underlying lattice.
What is the difference between the hexagonal and trigonal systems?
Both share a=b≠c, but the hexagonal system has γ=120° (with α=β=90°) and 6-fold rotational symmetry, while the trigonal system in its rhombohedral setting has a=b=c with α=β=γ≠90° and only 3-fold symmetry. Confusingly, some trigonal crystals (like quartz) are conventionally described using hexagonal axes for convenience, even though their true underlying Bravais lattice is rhombohedral (R) — this is a common source of confusion in introductory crystallography.
Who was Auguste Bravais?
Auguste Bravais (1811-1863) was a French physicist and naval officer who, in an 1848 memoir to the French Academy of Sciences, rigorously derived that exactly 14 distinct lattice types exist in three dimensions, correcting an earlier 1835 classification by the German mineralogist Moritz Ludwig Frankenheim, who had proposed 15 lattices (one of which turned out to be a duplicate). Bravais's broader scientific work spanned meteorology, astronomy, and botany (notably including a mathematical study of phyllotaxis — the spiral arrangement of leaves).
Why can't crystals have 5-fold symmetry?
The crystallographic restriction theorem proves that translational periodicity (the defining feature of a lattice) is incompatible with exact 5-fold, 7-fold, or any rotational symmetry higher than 6-fold, except for 1, 2, 3, 4, and 6. Intuitively, just as regular pentagons cannot tile a flat plane without gaps, no arrangement of points with 5-fold local symmetry can be extended to fill all of space with perfect translational repetition. Quasicrystals, discovered by Dan Shechtman in 1982 (Nobel Prize 2011), display forbidden symmetries like 5-fold and 10-fold precisely by sacrificing strict periodicity in favour of a more subtle, aperiodic long-range order — see our article on Penrose tiling and quasicrystals.
How does X-ray diffraction reveal the Bravais lattice of a material?
X-ray diffraction measures the reciprocal lattice of a crystal — Bragg's law nλ = 2d·sinθ relates observed diffraction angles θ to the spacing d between lattice planes. The pattern of allowed and systematically absent diffraction peaks (a consequence of destructive interference from centred lattices) directly reveals both the crystal system and centring type, allowing crystallographers to determine the full Bravais lattice, and with further analysis the complete space group, from a diffraction pattern alone. See our companion article on X-ray crystallography for the full derivation.
Can the same chemical substance have different Bravais lattices?
Yes — this is called polymorphism (or allotropy for elements). Carbon forms diamond (FCC lattice, tetrahedral bonding) and graphite (hexagonal lattice, layered bonding) from identical atoms arranged differently. Iron transforms between BCC (α-iron, below 912°C), FCC (γ-iron, 912-1394°C), and back to BCC (δ-iron) as temperature rises, a transformation central to steel heat-treatment metallurgy.
What is the difference between a unit cell and a primitive cell?
A primitive cell contains exactly one lattice point (accounting for points shared between adjacent cells) and has the smallest possible volume — but it does not always display the full rotational symmetry of the lattice as clearly. The conventional unit cell (used in the 14-lattice classification) may contain more than one lattice point (2 for body-centred, 4 for face-centred) but is chosen specifically because its shape makes the point-group symmetry — cubic, hexagonal, etc. — immediately visible, which is far more useful for communicating and reasoning about crystal structure.