Monday, February 25, 2013

More dances with hooves: a correction

Thinking about it some more, the previous post understates the constraints on the coordination of the clown paths with the horse's path, and so comes to the wrong conclusion.

The clowns actually need to remount fully in step with the horse. This is simply because the clown's chain transmutes into a continuation of the horse's chain at the join. (As a practical matter of crochet, we have more freedom in how we do the dismount, but we'll see that that cannot save the day.)

Because our ambition is to make the work truly ambidextrous and optimally strong, all remounts must be made the same way; for example, remounts from the left side of the horse must be a mirror image of a remounts from the right side of the horse. For any two remounts on the same side to be done the same way, we need them to be separated by an even number of horse steps. But on the other hand, for any two remounts from different sides to be done the same way (that is, they must be mirror images of each other,) they must be separated by an odd number of horse steps. As there is no number that is both even and odd, in walking crochet there is no way that all edges can be of equal length.

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