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 $\mathbb R^3$ 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).

💡 Research Summary

The paper investigates the topological constraints imposed by orthogonal projections—called “shadows”—of curves in Euclidean space. A shadow of a set S ⊂ ℝⁿ is defined as the orthogonal projection onto one of the coordinate hyperplanes, denoted π_i(S). The authors focus on two fundamental questions: (1) Can a simple closed curve (a cycle) in ℝ³ have all three of its orthogonal shadows be simple open curves (paths)? (2) Can a simple open curve (a path) have all three shadows be simple closed curves (cycles)?

To answer the first question, the paper introduces the notion of an x_i‑strand: a minimal sub‑path of a curve whose endpoints attain the extreme x_i‑coordinates of the whole curve. Several observations and Lemma 1 establish that if a projection is a path, the corresponding strands in other projections are preserved. Lemma 2 shows that if a cycle γ is not contained in any coordinate plane and both π₂(γ) and π₃(γ) are paths, then π₃(γ) must contain at least two distinct x₁‑strands. Lemma 3 proves that in the plane, an x₁‑strand and an x₂‑strand of a path must intersect, and Lemma 4 demonstrates that a planar path cannot simultaneously have two distinct x₁‑strands and two distinct x₂‑strands. Combining these results yields Theorem 5: a cycle whose three shadows are all paths cannot exist, because the required multiple strands would contradict Lemma 4.

The second question receives a constructive answer. The authors exhibit explicit examples of paths whose three shadows are cycles. An axis‑aligned polygonal chain (Figure 6) provides a simple illustration, while a more refined construction with only six vertices (Figure 7) is shown to be optimal. A proof by exhaustion shows that any polygonal path with fewer than six vertices cannot have all three shadows as cycles: with ≤3 vertices a shadow cannot close; with 4 vertices the shadows would force the start and end points to coincide; with 5 vertices the geometry forces overlapping shadows that contradict the openness of the path. Moreover, they prove (sketch) that it is impossible for all three shadows to be convex cycles, using a cylinder‑intersection argument that reduces the problem to a finite enumeration of embedded graphs, none of which admit the required convexity.

The paper then generalizes the construction to higher dimensions. Starting from Rickard’s classic 1‑sphere curve S₁, they define inductively S_{d+1} = { (1−|λ|)·x , λ } for x ∈ S_d and λ ∈