This post relies heavily on Mark Pankov and Adam Tyc's paper "Connected sums of Z-knotted triangulations" and later publications by the same authors. This is my attempt to rephrase their results in terms more suited to weaving.
The exhaustive search technique of finding unicursal deltahedra by generating all triangulations having a given number of vertices, and then testing each triangulation for unicursality (z-knottedness in Pankov and Tyc's terms) is clearly size-limited: the complexity increases by almost an order of magnitude for each vertex (or pair of triangular faces) that we add.
Perhaps this is a better idea: we make a new unicursal deltahedron by gluing two unicursal deltahedra triangle to triangle. Mathematically, this operation of fusing two topological spheres into a single topological sphere is called a connected sum. Under what circumstances does a connected sum of two unicursal deltahedra result in a unicursal deltahedron? Pankov and Tyc have looked at this question. It turns out that the rules for combination are fairly simple, and the technique promises to be fairly productive in finding new and larger unicursal deltahedra.
Every edge in a unicursal deltahedron is created by a crossing of two arcs (i.e., a crossing of two different portions of one long triangle strip.) Whenever two arcs cross, four sectors are created. Given an arbitrary orientation of each arc, exactly one of those four sectors will be converging, as in the diagram at the top. In wrap-forming the orientation of the arcs is given by the direction of numerical increase, and the weaver usually works with the converging sector downward, as shown, so that the numbers on both strips are easily read.
An edge of the deltahedron is made by overlapping the creases of the two strips. The paired-up creases can have one of two orientations in relation to the converging sector: either the converging sector lies all to one side of the creases (left in the diagram above) or it straddles the edge (right in the diagram above.) Notice that in the former case the strips cannot lend an orientation to the edge they are crossing because each arc would favor a different orientation, whereas in the straddle case the vote would be unanimous.
Pankov and Tyc have shown that every triangle in a unicursal deltahedron has either a lone straddle edge (which then suffices to assign an orientation, cw or cccw, to the whole triangle,) or three straddle edges all agreeing on a cw or ccw orientation for the whole triangle.
Let’s call a triangle with a lone straddle edge a solo triangle, and a triangle with three straddle edges (always all in agreement,) a unison triangle.
We will not be interested, per se, in whether a triangle’s straddle orientation is cw or ccw, rather we will be interested in whether the triangle’s staddle orientation is in concord (or discord) with a different way to orient the triangle.
In a deltahedron, the centerline of each strip (or indeed any line parallel to the edges of the strip) is a geodesic of the surface. In a unicursal deltahedron, such a line forms a closed geodesic that visits every triangle three times, each time running parallel to a different edge of the triangle. Since each edge of a triangle can be identified with its opposite vertex, the three visits of the geodesic to a given triangle constitute a cyclic ordering of its vertices, and therefore assign an orientation to the triangle. The three vists of the strip are also constitute a first, second, and third layer of material, so we call this orientation the triangle's layer orientation. We use the term concord when straddle and layer orientation agree, discord when they do not.
There are 4 types of triangle face in a unicursal deltahedron: o = unison and concord * = unison and discord - = solo and concord + = solo and discord
Notice that straddle edges are crossed when the arc that enters on the right makes a left turn. When using the style of triangle strips I showed at the Galway Bridges Family Day in August, strips turn left at even numbers, so strips of this style make straddle edges when the integer pair in the roadcode is: odd.even. In discussing triangle types, we can delete from the roadcode all the even.odd integer pairs. We will call the result of that deletion the deltahedron's stradcode.
Here are stradcodes for the smaller unicursal deltahedra. The letter codes indicate the type of the triangle that lies on the respective side of the edge. Unison triangles, which would otherwise merit three letters, are only annotated on the highest integer on their face. (The roadcodes here got abruptly truncated, but there is probably enough information to sketch the shape.)
Triangular dihedron 3.0 1.4 5.2 3.0 1.4 *5.2* Triangular bipyramid, 9 edges, 6 triangles 2.7 9.0 10.5 3.12 8.13 14.1 15.6 16.11 4.17 +9.0+ +3.12+ +15.6+ Pentangular bipyramid, 15 edges, 10 triangles 0.7 10.3 6.13 16.9 17.2 12.19 5.20 22.15 23.8 24.1 18.25 11.26 4.27 28.21 29.14 -17.2- -5.20- -23.8- -11.26- -29.14- Pizza, 15 edges, 10 triangles 9.0 6.11 14.3 1.16 17.8 10.19 5.20 22.13 23.2 15.24 4.25 26.21 27.12 28.7 18.29 9.0- 1.16 *17.8- -5.20- -23.2 -15.24* -27.12- Tricorn, 15 edges, 10 triangles 11.0 5.12 14.7 9.16 18.3 1.20 21.10 15.22 8.23 24.17 25.2 19.26 4.27 28.13 29.6 11.0 -5.12 -9.16 1.20 *21.10 -15.22* -25.2 -19.26* -29.6* Plantri 8-0, 18 edges, 12 triangles 2.11 13.0 14.9 5.18 20.7 15.22 10.23 1.24 25.12 26.3 28.17 6.29 19.30 31.4 32.27 33.16 21.34 8.35 13.0+ 5.18+ -15.22 1.24- *25.12- 19.30+ o31.4+ -33.16 -21.34* Plantri 8-1, 18 edges, 12 triangles 10.5 13.0 2.15 8.17 19.6 20.11 14.23 1.24 25.12 26.21 3.28 16.29 7.30 31.18 32.9 33.4 27.34 22.35 13.0- 19.6+ 1.24- *25.12+ +3.28 7.30- *31.18- -33.4 -27.34* Plantri 8-3 18 edges, 12 triangles 4.13 15.6 12.17 3.18 20.1 22.11 23.16 5.24 14.25 26.7 9.28 0.29 19.30 31.2 32.21 33.10 27.34 8.35 +15.6 3.18- +23.16 +5.24o -9.28 19.30+ o31.2+ -33.10 -27.34* Plantri 8-8 18 edges, 12 triangles 2.11 13.4 9.16 0.17 19.6 22.15 10.23 1.24 18.25 26.7 28.21 29.14 3.30 12.31 32.5 20.33 27.34 8.35 +13.4 -9.16+ +19.6- +1.24- -29.14 +3.30o -27.34+
