[to replace the first paragraph minus its last two sentences]

The Archimedean spiral provides a convenient way to generate families of permutations.

Traveling along the curve from the origin, labeled with a one, one intersects the X-axis at locations, on either side of the origin, that are successively farther from the origin.  If these locations are labeled with the successive Natural Numbers, and then read off from left to right, one obtains a string of length k (k>=2), which identifies a permutation.

Depending on the multiple of Pi that one chooses for the angle at the origin (whether 2 is placed to the left or right of the origin), two sequences - and therefore, two permutations - result: ...3 1 2... or ...2 1 3... .