I've always struggled to get good intuition for how the geometry of the dual lattice relates to the geometry of the original lattice, even in the Euclidean case!
Looking for more sources on this, I went to Wikipedia, where I found the following vague statements:
In certain respects, the geometry of the dual lattice of a lattice $L$ is the reciprocal of the geometry of $L$, a perspective which underlies many of its uses.
One of the key uses of duality in lattice theory is the relationship of the geometry of the primal lattice with the geometry of its dual[.]
How should I be interpreting this? What does have the "reciprocal" geometry mean, and why would the geometry of the dual be useful for understanding the geometry of the original lattice?
I'm familiar with lots of examples of duals being useful (most notably the different in algebraic number theory), so I'm looking more for an intuitive explanation of how someone might arrive at the notion to begin with, and how the geometry or the dual helps explain the geometry of the original lattice.