Knotology surfaces: keeping rangles and nuses apart

Knotology "cuboid" by Dassa. Image quoted from https://www.flickr.com/photos/dasssa/sets/72157622174934552/detail/?page=19

Staircase Pipe, an example of knotology weaving. Image quoted from mechiz, http://momentodecatarsis.blogspot.com/2007_11_01_archive.html

Knotology Egg by Dassa. Image quoted from https://www.pinterest.com/beckymckeehan/origami-instructions/

Inspired by Heinx Strobl's knotology weaving, define a knotology surface as a closed geometric surface facetted entirely by congruent, isosceles, right triangles.

If we assemble a knotology surface by joining its component right triangles edge-to-edge, hypotenuses must mate up with hypotenuses, and legs must mate up with legs, simply because hypotenuses and legs have different lengths.

Knotology, however, adds one more constraint: right angles (rangles for short) and base angles, or hypotenuse angles (nuses for short,) must always be kept apart! That is, every vertex in the surface will be either a pure rangle vertex or a pure nuse vertex.

This seemingly discriminatory constraint is closely related to the needs of weaving. Ignoring the hypotenuse folds (after all, they are merely creases in the weavers,) the residual square-faced graph drawn on the surface must be bipartite in order for the surface to be weave-able. Bipartite means that the vertices of the abstract graph fall into two classes such that no edge connects two vertices of the same class. Once we ignore the hypotenuses, all the edges in the graph are triangle legs, and each triangle leg connects a nuse vertex to a rangle vertex, so our insistence on keeping rangles and nuses apart guarantees weave-ability.

In the weaving, the physical distinction between these two classes of vertices is that the small opening or engagement window at the vertex is either right-handed or left-handed.

The creases are needed to make the surface adaptable enough to have an immersion in 3-space without crumpling (unstructured folding.) The creases do not really need to have a fixed relationship to the weave openings—unless we want all the weavers to be identical, which indeed we do. The only option is whether rangle vertices correspond to right-handed or left-handed engagement windows, a question which really doesn't matter to anyone but the weaver. Strobl's design is the one inevitable design for this type of digital weaving.

Knotology weaving is truly remarkable. With straight, all-identical weavers, a huge class of surfaces can be woven including all of the enveloping surfaces of the vox-solids (voxel-based 3-D objects in computer graphics) and stellated duals of all the fullerenes and graphene allotropes in theoretical chemistry, plus hybrids of these and many more….