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