Every 6-net with at most 50 vertices has an undented Euclidean realization from unit equilateral triangles, and prime 6-nets have unique convex ideal hyperbolic realizations.
Filling a triangulation of the 2-sphere
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abstract
Define the tet-volume of a triangulation of the 2-sphere to be the minimum number of tetrahedra in a 3-complex of which it is the boundary, and let $d(v)$ be the maximum tet-volume for $v$-vertex triangulations. In 1986 Sleator, Tarjan, and Thurston (STT) proved that $d(v) = 2v-10$ holds for large $v$, and conjectured that it holds for all $v \geq 13$. Their proof used hyperbolic polyhedra of large volume. They suggested using more general notions of volume instead. In work that was all but lost, Mathieu and Thurston used this approach to outline a combinatorial proof of the STT asymptotic result. Here we use a much simplified version of their approach to prove the full conjecture. This implies STT's weaker conjecture, proven by Pournin in 2014, characterizing the maximum rotation distance between trees.
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Neoplatonic solids
Every 6-net with at most 50 vertices has an undented Euclidean realization from unit equilateral triangles, and prime 6-nets have unique convex ideal hyperbolic realizations.