EJMT Abstract

Constructible numbers are numbers that can obtained using only a straightedge and compass. This paper seeks to unpack the meaning of that statement. We begin with the lattice of integers, providing a discrete geometric foundation for exploring which numbers and coordinates can be produced through successive geometric constructions. We consider some granular questions like constructible distances between integral lattice points and how restricting our set of tools of the number of uses of those tools affects our ability to solve and construct. This all culminates with an extension to higher dimensions and student investigations intended to deepen understanding.