Do discrete geometric objects exist in the same space as non-discrete geometric object?

Question

Do discrete geometric objects exist in the same space as non-discrete geometric object? I am wondering if the spaces are defined differently in discrete geometry. After reading a little about it, I am still confused as to what purpose discrete geometry serves exactly. It seems to be about the study of graphs, but I don't see how it relates to geometry.

Answer 1

I'd say discrete geometry mostly takes place in $d$-dimensional Euclidean space for some $d$, often denoted $\mathbb{E}^d$, which is the same thing as $d$-dimensional real space $\mathbb{R}^d$ with the usual Euclidean metric. Plenty of work is done in other settings, though (hyperbolic space, toroids, etc.)

It includes the study of polytopes, like the regular polyhedra (the Platonic solids), Archimedean solids, and so on. This is the focus of Euclid's Elements, making it probably the oldest and longest-studied branch of mathematics there is.

By contrast, geometry which isn't discrete, which I think of as "swoopy" geometry, has only been developed much more recently.

Whenever you're assembling discrete components, like making polyhedra by attaching polygons, or tiling a surface, or designing a space-filling honeycomb or weight-supporting truss, you're doing discrete geometry. The study of crystals, quasi-crystals, and other periodic or aperiodic systems crop up here.

As you mention, embedding graphs in space is another example, like planar graphs, or with distance constraints in three dimensions, which gives us protein folding. (And global positioning, although the geometric aspects aren't the most challenging part of the GPS system.)

Computer graphics are another "purpose" served by discrete geometry. Objects are modeled by a very large number of polyhedra, to which various Euclidean transformations are applied as the viewpoint moves.