In what sense is the geometry of the dual lattice "reciprocal" to the original lattice?

Question

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.

Answer 1

How should I be interpreting this?

I don't think the quoted statements can be used to get a concrete understanding of these matters. Rather, they attempt to give a general idea.

how someone might arrive at the notion to begin with

On the one hand, lattices are $\mathbb{Z}$-modules, real vector spaces are $\mathbb{R}$-modules, and the definition of the dual lattice echoes the definition of the space of linear functionals on a vector space, which is a concept known to be useful.

On the other hand, there is a solid state physics view. Roughly speaking, whenever a function is invariant with respect to a discrete set of translations (which is inevitably a lattice), its Fourier tranfsorm only has components corresponding to the sites of the dual lattice. Physicists use the term "reciprocal lattice", and the corresponding Wikipedia article is referenced in the preamble of the Wikipedia article on the dual lattice.

how the geometry or the dual helps explain the geometry of the original lattice.

Well, there is a list of relations between a lattice and its dual in that Wikipedia article. Since you are not satisfied with that collection of statements, I think one could answer this question only if it is specified more concretely what is it meant by explaining the geometry of a lattice, if not those relations.

Answer 2

The writer uses the label reciprocal in analogy with its arithmetic meaning. The fraction $2/3$ has as its reciprocal $3/2$, and many students see the operation of taking the reciprocal as flipping a fraction upside down.

Likewise, since lattices are posets, consider the dual of any poset $P=(X, \leq)$, which is the poset $P^{\rm d}=(X, \leq^{\rm d})$, where $\leq^\rm d$ is defined by $$\mbox{for all }x,y\in X, x\leq^{\rm d}y \mbox{ precisely when } y\leq x.$$ So the Hasse diagram of $P^\rm d$ can be obtained by flipping $P$’s Hasse diagram top for bottom. In that sense, their Hasse diagrams are reciprocals, and by extension, they themselves might be called reciprocals.

And, of course, anything one can learn from studying $P$—or its Hasse diagram—amounts to a corresponding fact about $P^\rm d$. In practice, this is perhaps most useful in the case of finite posets, where they and their Hasse diagrams are, at least in principle, guaranteed tractable in their entirety, and their automorphism groups might be smaller.