The weave pattern at the ends of a tetrahelix

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.

29 Tetrahedron Boerdijk-Coxeter Struts

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.

Deltahedral Struts and 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.

The period 4 tetrahelices

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+':

Periodic Tetrahedral Chains

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.

Deltahedron DNA at Bridges Galway 2026 Family Day

On Saturday August 8, 2026, I will be leading an activity at the University of Galway, for the Bridges Math Art Conference Family Day. We will be making deltahedra of various shapes from the same strips of shiny cardboard as above by interpreting numerical codes generated by computer. If you succeed in making a deltahedron, you get to take it home.

Saturday, August 8, 1:00–5:00 pm in various locations in Áras na Mac Léinn, on the campus of the University of Galway.

MORE DELTAHEDRON DNA HERE

An .svg file to make your own triangle strips on a Cricut Maker is here.

A pdf with some hints on making your own is here.

Try making a reverse slow-motion of your deltahedron:

Joining up link components to form knotted deltahedra

The tetrahedral graph is a Tait graph of a 3-component link.

Many interesting deltahedra, foremost among them the triangle-faced Platonic solids, are not knotted, so it is worthwhile looking at joining up link components to yield knots. Following Q. Xing, E. Akleman, J. Chen, and J. L. Gross in “Single-cycle plain-woven objects,” 2010 Shape Modeling International Conference, IEEE, pp. 90-99, 2010, we find that a link with n components can be converted to a knot by making n-1 small edits. For example, the tetrahedron, octahedron, and icosahedron with 3, 4, and 6 components, become knotted after 2, 3, and 5 edits respectively.

If an edge of a Tait graph carries two different link components (e.g., every edge in the tetrahedral graph illustrated above), then simply subdividing that edge with a 2-valent vertex will join the two link components into one.

A knotted triangle-faced graph based on the tetrahedral graph.

The edited Tait graph will no longer be a triangle-faced, but this is easily fixed. We can add a pair of parallel edges between the two neighbors of the new vertex (because this is a graphical Reidemeister move that does not alter cursality), and then we change the embedding so that the two parallel edges straddle the subdivided edge (Shank's theorem implies that changing embedding in the plane does not alter unicursality.) Now we have recovered a triangle-faced Tait graph with one less link component. Though the graph is no longer 3-connected, simple, or non-degenerate as a deltahedron, we can nonetheless weave it.

In the tetrahedron, performing this edit on any two edges that share a vertex joins the three link components into one. Road code: 3.6 8.1 9.12 2.13 14.7 15.0 17.10 11.18 19.16 21.4 5.22 23.20, where the four edges in boldface get folded to a 180° dihedral angle.

The 2-eared tetrahedron, a knotted degenerate deltahedron.

The minimal set of edges that must be edited always constitutes a spanning tree in the dual of the Petrial, or DP. The Petrial of the tetrahedron is the hemicube, and the dual of the hemicube is the hemi-octahedron. Since the spanning trees of a graph are independent of its embedding, it suffices to draw the DP in the plane in any convenient way, for the hemi-octahedron that is most simply a triangle with doubled edges.

The hemi-octahedron, the DP of the tetrahedron, drawn as an abstract graph.