The Shadows of a Cycle Cannot All Be Paths

A shadow of a subset S of Euclidean space is an orthogonal projection of S into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in R to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in R can all be cycles (although not all convex) and, for every d ≥ 1, there exists a d-sphere embedded in R whose d + 2 shadows have no holes (i.e., they deformation-retract onto a point).