Hypercube Unfoldings that Tile R^3 and R^2

TL;DR

This paper demonstrates that the hypercube can be unfolded into space-tiling and plane-tiling nets, introducing the concept of a 'dimension-descending tiler,' and explores related unfoldings with open questions.

Contribution

It proves the existence of specific unfoldings of the hypercube that tile space and the plane, advancing understanding of polyhedral unfoldings and tilings.

Findings

Hypercube has a face-unfolding that tiles space

Hypercube has an edge-unfolding that tiles the plane

The hypercube cross unfolding tiles space, but its plane-unfolding remains open

Abstract

We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a "dimension-descending tiler." We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an edge-unfolding that tiles the plane.