No acute tetrahedron is an 8-reptile

TL;DR

This paper proves that no acute tetrahedron can be subdivided into fewer than nine similar parts, establishing a lower bound on the complexity of such dissections.

Contribution

It introduces a new geometric constraint showing that acute tetrahedra cannot be 8-reptiles or 8-gentiles, extending understanding of shape dissection limitations.

Findings

No acute spherical diangle can be dissected into fewer than nine acute spherical triangles.

No acute tetrahedron is an 8-reptile or 8-gentile.

The result sets a lower bound of nine for the number of similar parts in dissections of acute tetrahedra.

Abstract

An $r$-gentiling is a dissection of a shape into $r \geq 2$ parts which are all similar to the original shape. An $r$-reptiling is an $r$-gentiling of which all parts are mutually congruent. This article shows that no acute tetrahedron is an $r$-gentile or $r$-reptile for any $r < 9$, by showing that no acute spherical diangle can be dissected into less than nine acute spherical triangles.