A _Bravais [lattice](Lattices)_ is an infinite array of points where the following necessary properties are met: ^2f23a3 1. The points in the lattice or oriented in such that the orientation and arrangement of the lattice appears exactly the same regardless of which one of these points the lattice is looked at from. ^585816 2. A $d$ dimensional Bravais lattice contains points with positions given by position [position vectors](Geometry%20(index).md#Vectors) $\mathbf{R}=\sum_{i=1}^d n_i\mathbf{a}_i.$ The vectors $\mathbf{a}_i$ are referred to as _primitive vectors_ and these vectors are said to _span_ or _generate_ the [lattice.](Lattices) and $n_i$ are integers $\{-\infty ... -2, -1, 0, 1, 2, ...\infty\}$. ^a50b5e Properties [1.](Bravais%20lattices#^585816) and [2.](Bravais%20lattices#^a50b5e) may be considered equivalent definitions for the Bravais Lattice which means one may show that 1. implies 2. and vice versa ([proof](Bravais%20lattices.md#Proof%20of%20the%20equivalence%20of%20definitions%201%20and%202%20for%20Bravais%20lattices)). %%Are primitive vectors only something that exists in non-Bravais lattices? if so this should be taken from a subsection of a more general lattice entry.%% %%Here there's more to explain about the non-uniqueness of primitive lattice vectors and so on.%% ^0abbf3 # Reciprocal lattices ([... see more](Reciprocal%20lattices.md)) # Three dimensional Bravais lattices A three dimensional [Bravais lattice](Bravais%20lattices.md) has points whose position vectors are [$\mathbf{R}=n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3$](Bravais%20lattices#^a50b5e) Shown below is an illustration of this with an example of a [cubic lattice.](Cubic%20lattice.md)  ^cd8772 And the reciprocal lattice is described by $\mathbf{k}=x_1\mathbf{b}_1+x_2\mathbf{b}_2+x_3\mathbf{b}_3$ ^0fdaee --- # Proofs and examples ## Proof of the equivalence of definitions 1 and 2 for Bravais lattices %%This proof is left as an exercise on page 83 of Ashcroft and Mermin. It should be obvious that 2 implies 1 and this is discussed in the text itself but the exercise deals with proving that 1 also implies 2 and that's less obvious.%% --- # Recommended Reading Since Bravais lattices model ideal [crystal lattices](Crystal%20lattices.md) they are a fundamental topic in [solid state physics](Solid%20State%20Physics%20(index).md) and thus their elementary properties are introduced in many solid state physics textbooks such as * [Ashcroft N. W., Mermin, N. D., _Solid State Physics_, Harcourt College Publishers, 1976.](Ashcroft%20N.%20W.,%20Mermin,%20N.%20D.,%20Solid%20State%20Physics,%20Harcourt%20College%20Publishers,%201976..md) pgs. 64-70. #MathematicalFoundations/Geometry/LatticeGeometry