"Pizza" (or zero-based-index Plantri 7-1) is the smallest unicursal deltahedron that can derived as a connected sum of other unicursal deltahedra.
The smallest composite unicursal deltahedron (composite in the sense that it is a connected sum of two smaller unicursal deltahedra) is the shape we have been calling pizza, a V-shaped, symmetrical composition of two triangular bipyramids. Pankov and Tyc prove that the sum is unicursal, and in fact, in the the photo the shape has been made by wrap forming.
Two triangular bipyramids can be summed to make "pizza".
Though the triangular dihedron is considered a degenerate deltahedron, we make exception for it here because it is identity element in connected sums of deltahedra (and it is in fact unicursal.) Connected summation with the triangular dihedron does not change a thing, since removing one of its triangles leaves an exact replacement for the triangle being removed from the other deltahedron.
Apart from the triangular dihedron, the only unicursal deltahedron smaller than the trio of shapes with 7 vertices, shown in the figure below, is the triangular bipyramid (TBP). Since connected summation with a TBP adds a net of 5 - 3 = 2 vertices, we must look among the 9 vertex unicursal deltahedra for the next composites.
The omnicapping (kis) of any unicursal deltahedron is also a unicursal deltahedron, so we know that this shape, the omnicapping of the triangular bipyramid (J12) is unicursal; but it can also be seen as the connected sum of two tricorns.
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, one trianglular face to another, and then delete the two glued faces. 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 in detail. It turns out that the rules for combination are fairly simple, and the technique promises to be 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 strip of triangles.) Whenever two arcs cross, four sectors are created. Given an arbitrary orientation of each arc, exactly one of those four sectors will be a convergent or entry sector, as in the diagram above. In wrap-forming the orientation of the arcs is given by the direction of numerical increase, and the weaver usually works with the convergent sector downward, as shown, so that the numbers on both strip segments are easily read.
An edge of the deltahedron is made by overlapping creases of the two strip segments. The paired-up creases can have one of two orientations in relation to the convergent sector: either the convergent sector lies all on one side of the edge (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 vote for a different orientation; in the straddle case, both arcs agree on the orientation of the edge.
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 ccw, to the whole triangle,) or three straddle edges all agreeing on a cw or ccw orientation of the whole triangle. We will call the resulting orientation of a triangle its straddle orientation.
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 in whether a triangle’s straddle orientation is cw or ccw, per se, rather we will be interested in whether the triangle’s straddle orientation is in concord (or discord) with an entirely different orientation of the triangle.
In a deltahedron, the centerline of a strip of triangles (or indeed any line parallel to the edges of the strip) traces a geodesic of the surface. In a unicursal deltahedron, such a line forms a closed geodesic that visits every triangle three times before closing on itself, each time running parallel to a different edge of the visited triangle. Since each edge of a triangle can be identified with its opposite vertex, the three visits the geodesic makes to a given triangle constitute a cyclic ordering of its vertices, and therefore can assign an orientation to the triangle.
In wrap-forming, the three visits to a triangle result in a first, second, and third layer of material, so we will call this orientation of a triangle its layer orientation. We will use the term concord when straddle and layer orientations agree, discord when they do not.
Pankov and Tyc found there are 4 types of triangle face in a unicursal deltahedron (Pankov and Tyc's labels in parentheses):
o = unison and concord (M1)
* = unison and discord (M2)
+ = solo and discord (M3)
- = solo and concord (M4)
Using "compatible" to mean "this pairing of triangle types yields a unicursal connected sum," Pankov and Tyc found:
o is compatible with *, and sometimes with +
* is compatible with o, *, +, -
+ is compatible with *, +, -, and sometimes with o
- is compatible with *, +, -
There is one more issue: which of the 3 possible relative angular rotations at the place where the two deltahedra meet yields a unicursal connected sum? In Pankov and Tyc the phrase for this is "a homeomorphism g : ∂F → ∂F′ transferring vertices to vertices."
If there is a triangle of star or circle type, {*, o}, involved, angular rotation is immaterial. If both triangles are drawn from the set {+, - } then the triangles must be aligned polar-vertex to polar-vertex, where the meaning of "polar-vertex" becomes clear in the two exemplars, respectively, of M3 (+) and M4 (-): the triangular bipyramid and the pentangular bipyramid.
Notice that type o (M1) triangles are the only possible barricade since any pair of triangles drawn from the set {*, +, -} is compatible. Tyc has shown that no unicursal deltahedron has exclusively type o triangles for its faces; thus, given two unicursal deltahedra, there is always (up to geometric constraints) at least one way to sum them and get a larger unicursal deltahedron. [Theorem 9, corrollary 3 in Tyc's Doctoral Thesis, Zigzag structure in triangulations of surfaces.]
For example, with a Galway strip, an edge coded 8.11 (because the integer pair begins with its even number) is a non-straddle edge.
In contrast, an edge coded 11.8 (because the integer pair begins with its odd number) is a straddle edge.
Notice that when an edge is straddled the arc that enters on the right makes a left turn. When using the style of triangle strips we used at the Galway Bridges Family Day in August,
strips only turn left at even numbers, so strips of this style make straddle edges when the integer pair in the roadcode is odd.even. Therefore, 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, which is generally the easiest number to see on the completed face. (Some of the roadcodes here got abruptly truncated, but there is probably enough information to sketch out the shape.)
At the terminus of a tetrahelix (a Boerdijk-Coxeter helix of finite length) is a tetrahedron with three vertices at its base which can be individually distinguished by valence: 4-, 5-, and 6-valent.
In a tetrahelix, there are six straight strips of equilateral triangles that trace three types of helical path around the central axis. The three types are conventionally identified by numbers that correspond with their cardinality: there are three helical strips of Type 3 (these wrap slowly around the axis, each covering a third of the triangles,) two helical strips of Type 2 (which wind around the axis more rapidly, each covering half of the triangles,) and a single helical strip of Type 1 (which wraps rapidly around the axis, not missing any triangles.) As useful visual clues, Type 2 strips run parallel to the edges that are just slightly convex, while the single Type 1 strip runs parallel to the edges that are concave.
The rope-like appearance created by the Type 3 strips can serve to classify the tetrahelix, just like a rope, as Z- or S-twist. Without loss of generality we will talk about a Z-twist tetrahelix.
In a Z-twist tetrahelix, the cyclical order of the 4-, 5-, and 6-valent vertices at either end is counterclockwise. The figure above shows diagramatically the edges at the base of the terminal tetrahedron with the vertex valences indicated by bold numerals. The nonbold numerals indicate the type of strip that crosses these edges. The gray arcs show how the strips interconnect.
In the terminal tetrahedron, helical strips connect to each other in the following way (the second time a given connection is listed, it is indented)
CROSSING 4/5:
Type 3 -> Type 2 crossing 5/6.
Type 2 -> Type 1 crossing 6/4
CROSSING 5/6
Type 3 -> Type 3 crossing 6/4
Type 2 -> Type 3 crossing 4/5
CROSSING 6/4
Type 3 -> type 3 crossing 5/6
Type 1 -> Type 2 crossing 4/5
Now consider a tetrahelix of a length (six tetrahedra for example) that makes it unicursal. Since the lone Type 1 strip can only connect to Type 2 strips, a tetrahelix with a unicursal strip must have a 3-leg sequence, 2-1-2. That accounts for half of the total strips. Furthermore, since there is a 3-to-3 connection at each end, but only three Type 3 strips in the tetrahelix, a unicursal strip must have a 3-leg sequence, 3-3-3. All six strips are now accounted for. What remains is to connect these two sequences by the 2-3 connection available at each end, as in the figure above.
Anticipating practical issues, it may not be advantageous to wrap-form a tetrahelix that leaves a Type 3 strip on top as tension in this type of strip does not anchor it very well to the surface. The more tightly wound Type 2 or Type 1 might be a better top layer. In that case, assuming we want the terminal splice to be at one end, the candidate orders are 3-3-3-2-1-2, 1-2-3-3-3-2 or 2-3-3-3-2-1.
The 1-2-3-3-3-2 pattern may be the best for fabrication. The initial Type 1 helical strip can completely cover a cylindical form (where the altitude of the triangles is 1, the diameter of the form needs to be about 0.95.) After the second helical strip (Type 2) is glued down to the first, the cylindrical form can be pulled out. Freed from the form, the concave creases can be forced inward by bending to form an uncompleted tetrahelix, the remaining four passages can be wrap-formed in the usual way.
Tetrahelical struts composed of 29 tetrahedra (BC29) are three times more acurate in terms of angular divergence between the two terminal edges than BC18. If bent around to form a loop, one additional tetrahedron is needed to fill the space above the two mated terminal edges. So BC29 is fundamentally the same structure as the 30 tetrahedron exact rings that can exist in the curved 3D space S3.
BC29 is not unicursal, but it can be made unicursal by adding a 1-tetrahedron side branch at one end without altering its essential strut geometry or interfering with its attachment at nodes.
The wrap-forming technique for unicursal deltahedra motivates interest in strut-and-node construction sets that are composed of unicursal deltahedra. The 18-tetrahedron Boerdijk-Coxeter helix (BC18) is both unicursal and has a relatively small (15 degrees) misalignment between its two terminal edges. The stellated triangular bipyramid (STB) is a unicursal deltahedron with three symmetrically-arranged cavities that can mate with terminal edges of three BC's. It can serve, with some unspecified fastening technology, as a trivalent node joining three BC18's. In the photo above: one tetrahelix strut and two stellated triangular bipyramid nodes.
Since tetrahedral chains always have a choice of 3 triangles where the next tetrahedron will be attached, they can be described by words on the 3-letter alphabet, {-. 0, +}. On a 3-letter alphabet, up to reversals and cyclic rotations, there are only twenty-one 4-letter words. To be a helix, a tetrahedral chain must have a dominant twist. Arbitrarily choosing to prefer a '+' twist, the 4-letter words with the number of '+' greater than 0 and greater than the number of '-' are only eight: '++++', '-+++', '000+', '00++', '0+0+', '0+++', '-0++', '-+0+':
A tetrahedral chain is a particular sort of deltahedron that can be formed step-by-step, incrementally adding a single regular tetrahedron face-to-face at each step, and always adding the new tetrahedron at one of the three most recently added triangular faces. Notice that there is only one shape for three tetrahedra attached together face-to-face: the tritetrahedron or boat polyhedron. That shape can indeed be can be made by chaining tetrahedra, but the chaining process will differ according to which of three triangular faces at the stern of the boat is considered to have been the origin. In the space curve diagrams here, each tetrahedron is represented by a 70.53-degree circular arc---the dihedral angle of a tetrahedron---at the beginning of which a discrete twist of {-120, 0, +120} degrees occurs, depending on which of the three candidate faces is chosen for the attachment. The interest here is in periodic patterns in {-, 0, +} so the exact start and finish are of little consequence.
According to Adam Tyc a tetrahedral chain is at most 3-cursal, and a bit more than half are expected to be unicursal.
