Ambidextrous crochet: the parable of the two-legged horse

There is something sad in the 21st century about doing one right-handed stitch after the other. This post looks at what happens in crochet when the ambidextrous walking stitch is substituted for the chain stitch.

Imagine a two-legged horse that walks around in a closed circuit on a small, dusty planet. Its path, however serpentine it might be, never crosses over itself. The second time around the circuit, and all later times around it, the horse steps exactly in its own hoof prints.

Imagine also, there are clowns on the planet that like to ride horseback. This particular time around the horse's circuit, at a point in time and space we call START, all the clowns are astride the horse's back.

When afoot, clowns never cross over their own tracks, the horse's tracks, or anyone else's tracks. Clearly then, it matters to each clown which side of the horse he dismounts from, since that limits where he can walk. One at a time, at their own whim, clowns dismount from whichever side of the horse they choose, and wisely waddle over to some future point on the horse's route (which is to say, they never go to a point on the circuit that the horse has already passed since START.) There they wait to remount the horse. (A little thought will convince you that a clown that dismounts on the horse's right must also remount on its right—assuming the little planet is a topological sphere—and likewise for the left; this is a consequence of the Jordan Curve Theorem.) 

When the horse reaches START again, we find all the clowns have remounted the horse.

The footprints left in the little planet's dust represent one way an article that is a topological sphere can be crocheted in walking stitches--and this is about how the stitches will look. The horse's tracks are the plain walking stitches, the clowns' tracks are crochet stitches with walking stitches inside (rather than the chain stitches of the traditional technique.)  Of course, since a crocheter can only work in one place at a time, the clown tracks will actually be made by waiting till the point where the clown remounts the horse, making a straight run to where he dismounts, turning and chaining back astride the straight run.

We don't need a sphere to draw the diagram on. If we imagine the small planet is covered with a rubber skin, we can pick someplace where nobody happened to step, cut a small round hole there, and stretch the perimeter of the hole until the rubber skin pulls flat (this gives us a Schlegel diagram.) Now, spotting the hoofprints of the horse, we can stretch the rubber skin still more until the horse's circuit becomes a circle. (Notice that none of the paths crossed on the planet, and none will cross after all this stretching.) So our diagram of footprints and hoofprints can always be simplified to circular horse path  connected by inner and/or outer chords of non-crossing clown paths.

One constraint needs to be added to the story. In the interest of the strength of the fabric, walking stitches should engage each other as fully as possible wherever they meet: preferably they will engage at two footsteps, not just one. In terms of the fable, a clowns first two steps after dismounting, and last two steps before remounting, should superimpose on horse steps. That creates a phasing constraint between horse steps and clown steps that is easier to draw than put in words (see the diagram above.)

A natural question is whether this constraint interferes with other constraints we may wish to impose, such as equal path lengths. If path lengths are even, it does not.