Monday, March 18, 2024

Coding corrugated baskets

Corrugated baskets present two different coding problems: how one might mark assembly information directly on a custom-punched tape, and how one might write down (on a piece of paper) assembly information for a tape pre-marked with a standard consecutive numbering. The latter is the topic here.

In the standard consecutive numbering, orienting the tape from the smaller to the larger numbers, each segment of the tape (i.e., a segment between crossings) should be marked near its terminal end as shown above, so that the number naturally becomes a sub-address for the nearest crossing.

All the crossings in a corrugated basket look the same. Rotated so that the converging quadrant is at the bottom, as shown above, they all look like the diagram. The crossing has a binomial address (represented here generically by 0,1): an even sub-address on the left and an odd sub-address on the right. (Those positions might just as well be reversed, but, to make things easier for the basket maker, we will always choose a basepoint for the counting that makes the above diagram correct.)

The oriented geodesic path is always a boundary between white and a tinted region. At each crossing, white switches sides; “black” also switches sides but its tint changes as well. For example, in the diagram, the 0 strand, before this crossing, has white on the right, and the dark tint on the left. After this crossing, white is on the left and the light tint is on the right. Also, the sub-address will have incremented by one and so changed parity.

If we are going to be walking along the path a lot in the oriented direction, it might be good to have the right leg shorter than the left because there is always going to be either a dale on the left or a hill on the right.

0 and 1 in the diagram merely symbolize even and odd sub-addresses, their relative magnitude is unknown: either could be the bigger sub-address and thus the later strand. If, say, the even sub-address is bigger, then we are colliding with the left side of the earlier strand and are potentially closing a dale region, conversely, if the odd address is bigger, then we are colliding with the right side of the earlier strand, and are potentially closing a hill region.

If we always write the full address in "even-odd" order, it is easy to remember on which side of the smaller sub-address the bigger sub-address is approaching.

Thursday, March 7, 2024

The ouroboros splice

When finishing a unicursal basket a splice is needed to connect the ribbon's head to its tail. The method advocated here is simple, and is sturdy enough for cardstock baskets. The ribbon is cut just long enough at both ends to doubly-cover the first crossing and the last crossing as shown above. The cutting is not done at 90°, but at the angle determined by the two outer notch positions, as seen in the photo below. The first crossing is unlocked, allowing the ribbon’s tail to be threaded under both the last and first crossing. Those crossings are then locked with a doubled thickness of ribbon underneath. Made this way, the splice is scarcely noticeable in the finished basket.

Locked crossing profiles for cardstock

I have been using the above notch pattern for 2cm-wide 65-lb cardstock weaving elements cut on a Cricut Maker 3 (courtesy of the DC Public Libray's FabLab.) The 7.5° alternately plus or minus rotation of the crossed diagonals makes locked crossings that are 75°/105°. These can be used to corrugate crossings that are nominally 90°/90° (tabby weave) as deficit/excess, as well as crossings that are nominally 60°/120° (kagome weave) as excess/deficit.

Friday, August 11, 2023

Notch detail for locked crossings in aluminum

The notch profile above works for aluminum sheet metal having a springy temper, such as is widely sold to the building trades as flashing (9 mil thickness,) trim stock (18 mil thickness,) and gutter stock (27 mil thickness.)

For example, with 9 mil (0.009") aluminum flashing a width of 1.5", tangent circles of 0.125" diameter, and outer rounds of 0.5" diameter are suitable for weaving with locked crossings.

If the weavers are to cross at an angle other than 90-degrees, the diagonals of the crossing rhombus remain perpendicular to each other, but rotate together to align with the crossing's planes of symmetry. The notches translate to maintain tangency.

Wednesday, August 2, 2023

Hill-and-valley weaving of voxel object surfaces

Hill-and-valley weaving must follow the medial of a bipartite map. The skeletal surface graphs of voxelized objects (example above) are bipartite. Pasting the truchet tile (below) onto each square face (with corner colors matching) gives a face 3-coloring of the medial graph (the new edges trace the boundaries between colors).

For example, for a single voxel, i.e., a cube, decorating its six faces with the truchet tile, shows that its 3-colored medial is a cuboctahedron (below) with its square faces colored 'saddle' and its triangular faces colored alternately 'hill' and 'valley'.

A spherical cuboctahedron (see below; art by Watchduck) is four great circles in an arrangement of maximum symmetry, so the smallest angles between these planes is equal to the dihedral angles of the tetrahedron, or approximately 70.5288 degrees.

To weave the cuboctahedron weave 'flat' (i.e., without hills and valleys) weavers must cross each other such that the internal angles of the triangular faces are about 70.5288 degrees, a bit wider than the 60 degrees these angles would measure on the plane. To make the triangular faces into hills and valleys, and, correspondingly, the square faces into saddles, we need even wider internal angles in the triangular faces. Below is a hill-and-valley weaving of the cuboctahedron with 100 degree internal angles in the triangular faces. Because of the hills on alternate triangles of the cuboctahedron, the basket appears strongly tetrahedral.

Friday, July 28, 2023

More on the math of hill-and-valley weaving

Above, the quadrilateral domain of a map edge.

Above, the hill-and-valley coloring of the quadrilateral domain of a map edge via Mrs. Stott's expansion, Me(Me(m)). Black = Up; White = Down; Pink = Saddle.

Above, the hill-and-valley coloring of the quadrilateral domain of a map edge via Ring, Me(Su(m)). Black = Up; White = Down; Pink = Saddle.

From the previous post it is clear that we must start from a bipartite map (a bipartite graph embedded in a surface) in order to design a basket that can wear a hill-and-valley corrugation of its surface. There are many such maps. For example, any quadrangulation of the sphere is a bipartite map. Also, since the 3D cartesian grid of points with integer coordinates can be bicolored (e.g., simply color each vertex according to the parity of x+y+z), the surfaces of polycubes and voxelized objects (image below) are bipartite maps regardless of their (necessarily orientable) topology.

If we must start from a non-bipartite map, the map operation Radial (a.k.a., Quadrangulation), which doubles the number of edges in the map, always yields a bipartite map--the partition of vertices being respectively, the original vertices and the new vertices that are added in the center of each original face. Taking the Medial of this map (at the cost of another doubling of edges) yields a weavable pattern. So the entire chained operation on a non-bipartite map m is Me(Ra(m)). Since Medial gives the same result whether applied to a map or its dual; and Ra() and Me() of any given map are in fact duals, Me(Ra(m)) = Me(Me(m)) = Me^2(m). This compound operation, "the medial of the medial", or "medial squared", is also known as Mrs. Stott's expansion.

Another way to make a given map bipartite is to subdivide the edges, in other words, insert degree-2 vertices in the middle of each original edge. So the compound map operation needed in this case is Me(Su(M)), a map operation which I have elsewhere called Ring ("Extra ways to see: An artist's guide to map operations." Hyperseeing: Proceedings of ISAMA 2011, pages 111–121, Summer 2011. See chart below.) because it generates weave patterns associated with chainmail.

Thursday, July 27, 2023

The math of hill-and-valley weaving

A weaving pattern for any graph drawn on a surface can be derived by taking the medial of the graph, a construction that fills the rhombic domain of each edge in the graph with black and white regions as below, (original edge in red, original vertices in green):

That construction suffices for ordinary weaving, but not for hill-and-valley texturing because the black regions, which correspond to the locations of the original vertices, must be partitioned into alternating hills and valleys. So we need to start with a bipartite graph (for example, see below: each edge connects a green vertex to a blue vertex), then the medial will inherit the needed bipartition of the black regions into hills and valleys from the bipartition of the original vertices.