# The Shadows of a Cycle Cannot All Be Paths

@article{Viglietta2015TheSO, title={The Shadows of a Cycle Cannot All Be Paths}, author={Giovanni Viglietta and Prosenjit Bose and Jean-Lou De Carufel and Michael Gene Dobbins and Heuna Kim}, journal={ArXiv}, year={2015}, volume={abs/1507.02355} }

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… CONTINUE READING

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