The Least-Area Tetrahedral Tile of Space

TL;DR

This paper identifies the tetrahedral shape with the smallest surface area that can fill space without gaps or overlaps, extending previous work by removing orientation constraints.

Contribution

It proves that Sommerville's type 4v tetrahedron remains optimal even without the restriction to orientation-preserving tilings.

Findings

Sommerville's type 4v is the least-area tetrahedral tile of space.

Removing orientation constraints does not change the optimal shape.

The result confirms the robustness of Sommerville's tetrahedron as a minimal surface tiling.

Abstract

We determine the least-area unit-volume tetrahedral tile of Euclidean space, without the constraint of Gallagher et al. that the tiling uses only orientation-preserving images of the tile. The winner remains Sommerville's type 4v.