Continuation of analytical tetrahedron packings beyond the studied diameter range

Determine how far the analytical dense-packing constructions for regular tetrahedra in circular cylinders, including the Tetra Helix and Dimer Chain families, can be continued as the cylinder diameter-to-tetrahedron edge-length ratio D/a increases beyond 1.21045.

Background

The paper analytically constructs two families of dense regular-tetrahedron packings in circular cylinders: chiral Tetra Helices and achiral Dimer Chains. The study establishes these structures only over the range 1 ≤ D/a ≤ 1.21045, with the Open Dimer Chain becoming physically invalid above the upper threshold because successive dimers overlap and the Compact Dimer Chain occurring at the limiting diameter.

The authors explicitly leave unresolved whether the analytical constructions can be extended to larger cylinder diameters. Simulations beyond D/a = 1.21045 suggest that other structures, including pentagonal stacks, may achieve ideal packing, but these observations are not analytically proven.