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.

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.