Motivated by mathematical aspects of origami, Erik Demaine asked which points in the plane can be constructed by using lines whose angles are multiples of
for some fixed
. This has been answered for some specific small values of
. We answer this question for arbitrary
. The set of points is a subring of the complex plane
, lying inside the cyclotomic field of
th roots of unity; the precise description of the ring depends on whether
is prime or composite. The techniques apply in more general situations, for example, infinite sets of angles, or more general constructions of subsets of the plane.