Adler’s finite-family conjecture for tetrahedral longest-edge bisection

Determine whether repeated longest-edge midpoint bisection of an arbitrary configuration of four points in general position in three-dimensional space produces only finitely many similarity classes; equivalently, establish whether every such configuration has a finite family of descendant similarity classes.

Background

Adler formulated a higher-dimensional analogue of the finite-similarity-class result known for planar longest-edge bisection. The conjecture concerns the process obtained by repeatedly bisecting a longest edge of a configuration of four points in three dimensions and asks whether the resulting configurations, considered up to similarity, form a finite family.

The paper states that the conjecture was already difficult for four points in general position in three dimensions and later constructs a degenerating bisection sequence that disproves it. The item is included because the conjecture is explicitly identified in the paper, even though the present paper resolves it negatively.