Chiral map operations and tensegrities

Circle diagrams for chiral map operations including propellor, capra, gyro, and snub. The chiral edges, shown in orange, do not have their symmetrical pairs—they count 2x instead of 4x for the black edges.

Tensegrity fabrics related to the map operations snub, capra, truncate, and medial. See this earlier posting.

Circle diagrams of common achiral map operations

Circle diagrams for common achiral map operations: identity, dual, subdivide, parallel, radial, medial, etc.

Map operations that are achiral (as opposed to chiral operations such as snub) can operate on non-orientable as well as orientable maps. Since the edges of maps are also undirected, an achiral map operation must have two perpendicular lines of mirror symmetry allowing the truchet tile to be played in either rotation, flipped over or not. For an achiral map operation, we only need to see one quadrant of the quadrilateral tile—the rest follows from symmetry.

A quadrant of a quadrilateral can be deformed into a circle having three 120° arcs: an arc representing the primal half-edge on the left, an arc representing the dual half-edge on the right (both these arcs are also mirror lines) and an arc representing a hypotenuse edge (quadrilateral edge) at the bottom. The hypotenuse arc at the bottom is not a mirror line.

Map operations are drawn on these diagrams as graphs with vertices along the circumference of the circle (and possibly also in the interior.) Black vertices are the real vertices, white vertices are simply where lines continue across the boundaries of the representation.

A map operation, O(), is associated with three other map operations: O*() = Du(O()) and O'()=O(Du()), and also O'*=Du(O(Du()). Two map operations O() and Q() are dual if Q = O'* = Du(O(Du()). For example, Ki and Tr are dual operations.

Two routes to complete foldability

Two routes to a completely foldable knotology basket. A knotology basket is completely foldable if it is directed by a bipartite map. There are basically two ways to turn an arbitrary map, m, into a bipartite map: subdivide edges, Su(m), or take the radial, Ra(m). The resulting baskets have the same crack+hinge graph, but rather different weaving.

Di Francesco has shown that a triangulation of congruent triangles can be completely phantom folded into a single triangle iff it is tricolorable (the vertices can be colored in three colors such that no edge joins vertices of the same color.) The radial map operation that turns an arbitrary map, M, into directions for weaving a knotology basket, Ra(M), creates a new class of vertices and returns edges that exclusively connect new vertices to old vertices. If the underlying map is itself bipartite, then Ra(M) will be tricolorable, and it will remain so even after the edges of M are added back as hinge lines turning the quad-faced map into a triangulation.

The image above illustrates two ways to obtain a bipartite map from an arbitrary map. One way is to use Ra(). This results in the same √2 refinement and rotation discussed in the previous post. Another is to subdivide edges with new vertices, which is the subdivide map operation Su(). The radial method results in extended weaving, the subdivide method results in chain mail.

A ring of knotology chain mail is a closed belt of eight, diagonally folded squares.

Crease knotology weavers on both diagonals?

Creasing a knotology weaver along both diagonals of the squares yields a crack+hinge graph that is essentially the same as that of a standard weaving of a refined and rotated map—so what's the gain?

Because some knotology baskets cannot be immersed in 3-space without additional, non-standard creases, one may wonder: why not make this problem rarer by creasing weavers along both diagonals rather than one?

There may be practical benefits to creasing along both diagonals, but from an analytical viewpoint, it is overkill. As shown in the image above, the same folding versatility can be gained by standard knotology weaving of a map that is finer-grained and rotated. (Though their crack+hinge graphs are identical, the two baskets are not the same because in each case different edges are cracks or hinges.)

As we will see in the next post, always folding along both diagonals would make every knotology basket completely foldable. As things are, there are many interesting knotology baskets that are not completely foldable, and, imho, it is not worth excluding them from the class in return for a technique that is largely equivalent to simply refining the underlying map.

Second prototype magnetic tiles for Rangles and Nuses

Magnetic construction tiles for "Rangles and Nuses"

Right-triangular tiles with disk magnets arranged as shown only go together when the rules of "Rangles and Nuses" are obeyed.

Disk magnets can be affixed to the corners of isosceles right-triangular tiles in a way that enforces the rule that right-angles (rangles) and base angles (hypotenuse angles or nuses) be kept apart. The tile shown above (or alternatively its enantiomorph) can be used. These tiles look the same when flipped over.

Here are some basic Rangles and Nuses plays with prototype right-triangular magnetic tiles:

A larger array of tiles shows how similar the two classes of vertices are:

An arrangement of magnetic tiles for "Rangles and Nuses" showing the similarity of the two classes of vertices. 

Modeling knotology surfaces with inside-out Bassetti hinges

The square pillow (the knotology basket directed by the digon graph) realized with inside-out Bassetti rubber band hinges. The bold black edge coincides with the diagonal hinge or crease.

The preceding post showed how knotology surfaces can be assembled from congruent isosceles right triangles. Because every hypotenuse edge in a knotology surface must pair up with another hypotenuse edge, knotology surfaces can also be seen as being composed of squares that are hinged, creased, or folded along a designated diagonal. This is probably the preferred way to model these surfaces because the number of pieces needed is reduced by half.

Square tiles with appropriately color-marked corners (as illustrated in the images here) will be called bi-squares, because they force construction of a square-faced, bipartite crack-graph—the graph made evident by separating the square tiles slightly—when the bi-squares are assembled with their corner colors matching.

A cardstock bi-square that has been cut and notched for attachment with Bassetti rubber band hinges. The overall size is 3" x 3". The creased internal square is 2-3/16" x 2-3/16". The notches were cut straight across with scissors after the flaps were folded flat. This size works well with the widely available Rainbow Loom elastic bands.

The same card stock bi-square as above with the ears folded flat.

In 1959, architect Frederick F. Bassetti invented a good way to hinge polygonal cards together with elastic bands to form polyhedral shapes. The main drawback to making a model with Bassetti hinging is that the outside of the finished model wears protruding flaps and rubber bands. I find that it is possible, though a bit more time consuming, to work Bassetti hinges inside-out so that the flaps and elastic bands are hidden inside the model.

The bi-squares illustrated here were made from 3" x 3" pieces of 65-lb (176 g/m2) card stock, that was creased along the edges of a 2-3/16" square diamond and also along one of its diagonals. After the "ears" were folded flat, the corners were truncated with straight scissor cuts leaving clean-cornered rectangular notches that hold Rainbow Loom elastic bands fairly securely. Working carefully, one elastic band engagement at a time, and with the help of a small gauge (2-1/4 mm) crochet hook, I find it is possible to assemble the Bassetti hinges inside out. In the end, there is just a peek of elastic band showing at the corners. It also helps to pre-curl the ears a bit so that they can pass around behind the elastic band.

View from the inside of a bi-square with two pre-curled ears. 

Stage in the closure of a particularly difficult inside-out Bassetti hinge.

Using a narrow gauge crochet hook to properly set the elastic band in an inside-out Bassetti hinge.

Close-up view of the inside-out Bassetti hinges in the smallest knotology surface: the triangular pillow (the surface directed by the line segment graph.)

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….