The Bravais lattice is the basic building block from which all crystals can be constructed. The concept originated as a topological problem of finding the number of different ways to arrange points in space where each point would have an identical “atmosphere”. That is each point would be surrounded by an identical set of points as any other point, so that all points would be indistinguishable from each other. Mathematician Auguste Bravais discovered that there were 14 different collections of the groups of points, which are known as Bravais lattices. These lattices fall into seven different "crystal systems”, as differentiated by the relationship between the angles between sides of the “unit cell” and the distance between points in the unit cell. The unit cell is the smallest group of atoms, ions or molecules that, when repeated at regular intervals in three dimensions, will produce the lattice of a crystal system. The “lattice parameter” is the length between two points on the corners of a unit cell. Each of the various lattice parameters are designated by the letters a, b, and c. If two sides are equal, such as in a tetragonal lattice, then the lengths of the two lattice parameters are designated a and c, with b omitted. The angles are designated by the Greek letters α, β, and
γγ size 12{γ} {}, such that an angle with a specific Greek letter is not subtended by the axis with its Roman equivalent. For example, α is the included angle between the b and c axis.
Table 1 shows the various crystal systems, while Figure 1 shows the 14 Bravais lattices. It is important to distinguish the characteristics of each of the individual systems. An example of a material that takes on each of the Bravais lattices is shown in Table 2.
Table 1: Geometrical characteristics of the seven crystal systems.
System 
Axial lengths and angles 
Unit cell geometry 
cubic 
a = b = c, α = β =
γγ size 12{γ} {}= 90° 

tetragonal 
a = b ≠ c, α = β =
γγ size 12{γ} {}= 90° 

orthorhombic 
a ≠ b ≠ c, α = β =
γγ size 12{γ} {}= 90° 

rhombohedral 
a = b = c, α = β =
γγ size 12{γ} {} ≠ 90° 

hexagonal 
a = b ≠ c, α = β = 90°,
γγ size 12{γ} {} = 120° 

monoclinic 
a ≠ b ≠ c, α =
γγ size 12{γ} {} = 90°, β ≠ 90° 

triclinic 
a ≠ b ≠ c, α ≠ β ≠
γγ size 12{γ} {} 

Table 2: Examples of elements and compounds that adopt each of the crystal systems.
Crystal system 
Example 
triclinic 
K_{2}S_{2}O_{8}

monoclinic 
As_{4}S_{4}, KNO_{2}

rhombohedral 
Hg, Sb 
hexagonal 
Zn, Co, NiAs 
orthorhombic 
Ga, Fe_{3}C

tetragonal 
In, TiO_{2} 
cubic 
Au, Si, NaCl 
The cubic lattice is the most symmetrical of the systems. All the angles are equal to 90°, and all the sides are of the same length (a = b = c). Only the length of one of the sides (a) is required to describe this system completely. In addition to simple cubic, the cubic lattice also includes bodycentered cubic and facecentered cubic (Figure 1). Bodycentered cubic results from the presence of an atom (or ion) in the center of a cube, in addition to the atoms (ions) positioned at the vertices of the cube. In a similar manner, a facecentered cubic requires, in addition to the atoms (ions) positioned at the vertices of the cube, the presence of atoms (ions) in the center of each of the cubes face.
The tetragonal lattice has all of its angles equal to 90°, and has two out of the three sides of equal length (a = b). The system also includes bodycentered tetragonal (Figure 1).
In an orthorhombic lattice all of the angles are equal to 90°, while all of its sides are of unequal length. The system needs only to be described by three lattice parameters. This system also includes bodycentered orthorhombic, basecentered orthorhombic, and facecentered orthorhombic (Figure 1). A basecentered lattice has, in addition to the atoms (ions) positioned at the vertices of the orthorhombic lattice, atoms (ions) positioned on just two opposing faces.
The rhombohedral lattice is also known as trigonal, and has no angles equal to 90°, but all sides are of equal length (a = b = c), thus requiring only by one lattice parameter, and all three angles are equal (α = β = γγ size 12{γ} {}).
A hexagonal crystal structure has two angles equal to 90°, with the other angle (
γγ size 12{γ} {}) equal to 120°. For this to happen, the two sides surrounding the 120° angle must be equal (a = b), while the third side (c) is at 90° to the other sides and can be of any length.
The monoclinic lattice has no sides of equal length, but two of the angles are equal to 90°, with the other angle (usually defined as β) being something other than 90°. It is a tilted parallelogram prism with rectangular bases. This system also includes basecentered monoclinic (Figure 1).
In the triclinic lattice none of the sides of the unit cell are equal, and none of the angles within the unit cell are equal to 90°. The triclinic lattice is chosen such that all the internal angles are either acute or obtuse. This crystal system has the lowest symmetry and must be described by 3 lattice parameters (a, b, and c) and the 3 angles (α, β, and
γγ size 12{γ} {}).