Shadows of Cube Vertices

Abstract

Suppose that a finite-dimensional cube is orthogonally projected onto a central section of itself by a subspace of one dimension less. Up to dimension $9$, at least one vertex is projected onto the section, but for dimension $10$ or larger, there are orthogonal projections for which all the vertices are projected outside the section. In fact, this is the case for "most" orthogonal projections, as the dimension tends to infinity.

This is an Original Manuscript of an article published by Taylor & Francis in The American Mathematical Monthly on July 9th, 2019