Closed-form entrelac

OK, I think calling it “3-D Entrelac” is over. From now on, I’m calling it “closed-form entrelac.” After all, every piece of entrelac knitting exists as a 3-D object.

I have been ruminating over the last few weeks about what forms are possible in entrelac. Here are the limitations I have placed on the closed-form entrelac knitting I like to do:

1. Single continuous thread of yarn. Start making one entrelac square and keep adding squares, each connected to a previous square, until you have finished the piece. Never break the yarn and pick up the knitting again somewhere else on the piece.
2. Minimize the number of grafted joins. The only time I want to use a grafted join between two entrelac squares is on the last seam, where you are trying to connect the top of an entrelac square to the side of a previous one.

Substitution tiling

One way to generate shapes in entrelac pieces is to start with a regular piece of entrelac knitting, then substitute a structure made up of more than one square for one of the squares. Repeat ad libitum.

Turns out there’s all kinds of ways to do this. I haven’t knitted most of them out, but in my head I've come up with several.

The first one I thought of was to "pop out" a cube from a square. For one of the squares in an entrelac piece, substitute a cube with one of its faces missing. So you have a flat piece of entrelac with a cube popping out of it. You can always turn the 5/6 cube inside out, so you have an innie rather than an outie.

If you keep going with this substitution, you can make any shape composed of cubes that are stuck together, and it will obey the constraints above.

More later.