The Shadows of a Cycle Cannot All Be Paths
This addresses a geometric problem in topology and projection theory, with implications for computational geometry and visualization, but it is incremental as it builds on known shadow concepts.
The paper proves that a simple closed curve in 3D Euclidean space cannot have all three orthogonal projections be simple open curves, while also showing that a simple open curve can project to closed curves and that higher-dimensional spheres can project without holes.
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 $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, 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).