Shufflebrain

Our search will uncover hologramic mind not as a particular thing but as a generalization. We will begin the quest in this chapter with a theoretical look at waves. And we will continue our search through the next two chapters. It is important to know in advance that our objective will not be the geography, but the geometry of the mind. Ready? "And a one-y and a two-ey..." as my children's violin teacher used to say.

***

The central idea for our examination of waves originated in the work of an eighteenth-century Frenchman, Pierre Simon, Marquis de Laplace. But it was a countryman of Laplace who in 1822 explicitly articulated the theory of waves we will call upon directly. His name was Jean Baptiste Joseph, Baron de Fourier.

In chapter 3, I mentioned that in theory a compound irregular wave is the coalesced product of a series of simple regular waves. The latter idea is the essence of Fourier's illustrious theorem.[3] The outline of a human face, for example, can be represented by a series of highly regular waves called sine waves and cosine waves.

Such a series is called a Fourier series, and the Fourier series of one person's face differs from that of another person's face.

We're not just talking about waves, however. As, J. W. Mellor wrote in his classic textbook, Higher Mathematics, "Any physical property--density, pressure, velocity--which varies periodically with time and whose magnitude or intensity can be measured, may be represented by Fourier's series."[4] Therefore, let's take advantage of Fourier's theory, and, to assists our imagery, let's think of a compound irregular wave as a series of smaller and smaller cycles ; or better, as wheels spinning on the same axle; or perhaps even better still, as the dials on an old-fashioned gas or electric meter. Waves, after all, are cycles. And wheels and are circles. Now imagine our series with the largest wheel--or slowest dial--as the first in line, and the smallest or fastest back at the tail end of the line. The intermediate wheels progress from larger to smaller, or slower to faster. Or the faster the dial, the more cycles it will execute in, say, a second. In other words, as we progress along the series, frequency (spins or cycles per second) increases as the cycles get smaller.

If we were to transform our cycles back to wavy waves, we would see more and more wavelets as we progressed along the series. In fact, in a Fourier series, the frequencies go up--1, 2, 3, 4, 5, 6... and so on.

But wait! If the frequencies of your face and mine go up 1, 2, 3, 4, 5, 6...how can our profiles be different? What Fourier did was calculate a factor that would make the first regular wave a single cycle that extended over the period of the compound wave. Then he calculated factors, or coefficients, for each component cycle--values that make their frequencies 1, 2, 3, 4, 5, 6... or more times the frequency of the first cycle. The individual identities of our profiles, yours and mine, depend on these Fourier coefficients. The analyst uses integral calculus[5] to determine them. Fourier analysis (what else!) is the name applied to the analytical process. Once all the coefficients are available, the analyst can represent the compound wave as a Fourier series. Then the analyst can graph and plot, say, amplitude versus frequency. A graph can be represented by an equation. And an equation using Fourier coefficients to represent a compound wave's amplitude versus frequency is called a Fourier transform, which we'll discuss in the next chapter.

But wait! Isn't there something fish about coefficients? Isn't Fourier analysis like making the compound wave equal ten, for instance, and then saying 10 = 1+2+3+4? If the components don't come out just right, we'll just multiply them by the correct amount to make sure the series adds up to the sum we want. Mellor even quotes the celebrated German physician and physicist, Ludwig von Helmholtz as calling Fourier's theorem "mathematical fiction."[6] But this opinion did not stop Helmholtz and many in his day from using Fourier's theorem. Fourier's ideas gave new meaning to theoretical and applied mathematics long before the underlying conditions had been set forth and the proofs established. Why would anyone in his or her right mind use an unproved formula that had shady philosophical implications? The answer is very human. It worked!