Friday, February 6, 2015

Crease knotology weavers on both diagonals?

Creasing a knotology weaver along both diagonals of the squares yields a crack+hinge graph that is essentially the same as that of a standard weaving of a refined and rotated map—so what's the gain?
Because some knotology baskets cannot be immersed in 3-space without additional, non-standard creases, one may wonder: why not make this problem rarer by creasing weavers along both diagonals rather than one?

There may be practical benefits to creasing along both diagonals, but from an analytical viewpoint, it is overkill. As shown in the image above, the same folding versatility can be gained by standard knotology weaving of a map that is finer-grained and rotated. (Though their crack+hinge graphs are identical, the two baskets are not the same because in each case different edges are cracks or hinges.)

As we will see in the next post, always folding along both diagonals would make every knotology basket completely foldable. As things are, there are many interesting knotology baskets that are not completely foldable, and, imho, it is not worth excluding them from the class in return for a technique that is largely equivalent to simply refining the underlying map.

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