The Shadows of a Cycle Cannot All Be Paths

TL;DR

The paper proves that a cycle in three-dimensional space cannot have all three orthogonal shadows as paths, and explores various shadow properties of paths and spheres in higher dimensions.

Contribution

It establishes the impossibility of all three shadows of a cycle in R^3 being paths and investigates shadow characteristics of paths and spheres in higher dimensions.

Findings

All three shadows of a cycle in R^3 cannot be paths.

Three shadows of a path in R^3 can all be cycles, not necessarily convex.

Existence of d-spheres in R^{d+2} with shadows that deformation-retract onto a point.

Abstract

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).