Sunday, April 7, 2013

The Form

After many false starts, I've finally realized that the best way in to the convoluted story I want to tell is by simply drawing your attention to one of the most unexceptional connections imaginable between nature and human design. The connection could have been made in a variety of ways, so I'll imagine one to stand for all possibilities.

One morning, well over two millennia ago, a man strolled out onto the beach as he had done thousands of times before. He leaned down and picked up an object that had washed ashore.  But this time, rather than just looking at this object which was so familiar to him, he "saw" it. This was the object:

And today we have proof of what he saw thousands of years ago, because he left a record of his in-sight:

The volute or scroll or spiral form is ubiquitous in nature and design, and has been thoroughly studied and remarked upon. We know its equations. We know the logarithmic spiral whose arms get further apart with every turn as in the examples above––and we know the relationship of the general logarithmic spiral to the golden mean and Fibonacci spirals.  Simplest of all is the arithmetic or Archimedean spiral:

All of this is in two dimensions. If we ask what a spiral looks like in three dimensions, we are led to the helix, another well-studied form. But the seed question I wish to pose here is: What does a spiral "look like" in one dimension?

Imagine a collection of eight "things"––we'll label them A through H. 

It doesn't matter what the things are––letters, numbers, words, colors, pitches, durations, mangoes, rabbits, playing cards, shot glasses, walnut shells, dried peas––, but what does matter is how they are arranged and how we are going to REarrange them. (Think, if you will, of the old shell game.) We order the original things along a line (1 dimension) and then we temporarily "go into" 2 dimensions and superimpose a spiral: 

Then, beginning with H (we could also begin with D, or we could rotate the spiral and begin with or A or E, with similar but not identical results) we follow the spiral around and collect the letters in a new ordering: 

We then superimpose the same spiral and, beginning in the same place  on the spiral, which is now D, repeat the process (which we can now call an "algorithm").

This results in the new order:

If we continue to do this, boring as it may seem to the reader at this point, we notice that we arrive back at the original ordering after only four "moves":

If we reduce the number of "things" to 4, then we find it only takes three "moves" to return to the original order:

We also observe that in this case the "C" is stationary.  This quirk as well as the shortcut return to original order in both of these examples is in contrast to a maximal run of the spiral algorithm such as this one with six things that takes six moves to return the original order:

There is one more interesting feature of the spiral algorithm––its inverse: how it works backward––but I will save that for the two target applications of the spiral algorithm that I am now headed for, one very old one in poetry and one several relatively recent one in music.

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