What is meant by "a plane wave that has the periodicity of a Bravais lattice"?

Question

I'm reading through Ashcroft & Mermin's chapter on reciprocal lattices and am confused about this sentence:

Consider a set of points R constituting a Bravais lattice, and a
plane wave, e^{i k*r}. For general k, such a plane wave will
not, of course, have the periodicity of the Bravais lattice,

I understand what a Bravais lattice is in that it's infinitely repeating and looks the same no matter which unit's perspective you take. I'm just puzzled by what this part means.

hwang · Answer

I guess it simply means $e^{ivec{k}cdot(vec{r} + vec{R})} = e^{ivec{k}cdotvec{r}}$, for any Bravais lattice vector $vec{R}$.

Claudio Saspinski · Answer

A wave as:$psi = Ae^{i mathbf{k.r}} = A$ always where: $mathbf{k.r} = 2npi$. Or $k_1x + k_2y + k_3z = 2npi$. It is the equation of a family of parallel planes, all of them normal to $mathbf k$.
The distance between that planes is not related to the lattice interpanar distance.

What is meant by "a plane wave that has the periodicity of a Bravais lattice"?

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