Realizing (and stellating) deltahedra

Not every triangulation of the sphere can be realized as a deltahedron, that is, a solid having faces that are all equilateral triangles. Furthermore, if such a deltahedron does exist, its trirectangular stellation (which is what knotology weaving produces) may have faces that intersect. The former problem has been investigated in Tsuruta, Naoya, et al. "Enumeration of deltahedral graphs with up to 10 vertices." Proceeding of the 16th International Conference on Geometry and Graphics (ICGG2014). The images shown here have been marked up from that paper.

Basically, from our road code weaving perspective, three things can go wrong with a triangulation on the sphere: it might be multicursal (marked up with an X in the figure above), it might be non-deltahedral (an N in Tsuruta et al.'s code for the candidate deltahedron), or it might be non-stellate-able using trirectangular pyramids (marked up with a strike-through).

The first condition can be easily tested in the abstract graph: the graph is unicursal (in all of its embeddings in the plane) if its spanning tree count is odd.

The second condition is difficult to prove, but Tsuruta et al. have calculated computer models of presumed deltahedra on a small number of vertices that are equilateral to within 1 part in 100,000.

The third condition is sometimes easy if there is a dihedral angle in the deltahedron that is not obtuse enough to accomodate trirectangular pyramids on each of the adjoining faces. A trirectangular pyramid has dihedral angles of 54.74 degrees at its base, so a dihedral angle in the deltahedron less obtuse than 2 x 54.74 = 109.48 degrees cannot accomodate stellation. Such an angle is noticeably more obtuse than 90 degrees, so any dihedral angle of 90-degrees or less is disqualifying on sight.

It is worth noting that Tsuruta et al. use the same Plantri-based naming convention as here, but they count indexes from one rather than zero. Thus the pentagonal dipyramid is 7₄ in their nomenclature, but 7-3 here.

Also Tsuruta et al. exclude deltahedra candidates having dihedral angles of exactly 180 degrees; from a basketry perspective there is not really a need to object to these passages of flat weaving. For example in the figure below, Tsuruta et al. exclude 5 of the 14 candidate deltahedra on 8 vertices (N in their code stands for non-deltahedral) because of they 180-degree dihedral angles. Coincidentally we would exclude all the same (and more) for not being unicursal (marked up with X).