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 ${\mathbb{R}}^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$canallbecycles(althoughnotallconvex)and,forevery$d\geq 1$, there exists a $d$-sphere embedded in ${\mathbb{R}}^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformation-retract onto a point).