Shadows of Cube Vertices

TL;DR

This paper investigates how vertices of high-dimensional cubes behave under orthogonal projections onto lower-dimensional sections, revealing a threshold dimension where vertices start to project outside the section.

Contribution

It identifies a critical dimension (dimension 10) where the behavior of projected cube vertices changes, and shows that for higher dimensions, most projections exclude all vertices from the section.

Findings

Vertices are projected onto the section for dimensions up to 9.

For dimension 10 and above, most projections exclude all vertices.

The phenomenon becomes typical as the dimension tends to infinity.

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.