Shufflebrain

Change exists in two senses: co-variation, where changes proceed in the same direction (as when a beagle chases a jackrabbit); contra -variation, as exemplified in the two ends of a stretching rubber band. And (with tensors at least) the changes can be complex mixtures of covariance and contravariances (e. g., if the beagle gains on the jackrabbit as the quarry flags from fatigue or the rubber band offers progressively more resistance as tension increases). Tensors can attain higher rank--by simultaneously representing several variations. Packaged into one entity there can be an incredible amount of information about how things are changing. Ordinary mathematics become cumbersome beyond comprehension and eventually fail in the face of what tensors do quite naturally.

If you've seen equations for complex wave, you might be led to expect tensors to occupy several pages. But the mathematician has invented very simple expressions for tensors: subscripts denote covariance and superscript indicate contravariance.[13a] If, having applied the rules and performed the calculations, R in one place doesn't equal R in another, then the transformations aren't those of a tensor and from, Brillouin's dictum, represent local fluctuations peculiar to the coordinate system; they may have empirical but not analytical meaning. If the mathematician even bothers with such parochial factors at all, he'll call them "local constants."

Many features of holograms cannot be explained by ordinary transformations. In acoustical holograms, for instance, the sound waves in the air around the microphone don't linearly transform to all changes in the receiver or on the television screen. Tensors do. The complete construction of any hologram can be regarded as tensor transformations, the reference wave doing to the object wave what transformational rules do to make, say, Rij equal R^nm . Decoding, too, is much easier to explain with tensors than with conventional mathematics. With tensors, we can drop the double and triple talk (recall the transform of the transform), especially if we place the tensors within Riemann's continuum. The back-transform of phase codes from transform space to perceptual space becomes, simply, the shift (or parallel displacement) of the same relative values from a spatial to a temporal coordinate of the same continuum.

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We imagined mind as a version of Riemann's theoretical world, as a continuous universe of phase codes. Now we add the concept of the tensor to the picture: Tensors represent phase relationships that will transform messages, independent of any coordinates within the universe. Indeed, phase relationships, as tensors, will define the coordinates--the perception, the memory, the whatever.[13b]

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I admit that a universe constructed from Riemann's guidelines is an exceedingly abstract entity. Diagrams, because of their Euclidean features, can undermine the very abstractions they seek to depict. But, as we did with perceptual and transform space, let's let our imaginations operate in a Euclidean world and, with reason, cautiously proceeding step by step, let's try to think our way to a higher level of intuitive understanding. Ready!

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Imagine two points, A and B in our Euclidean world, 180 degrees apart on a circle. Let's begin a clockwise journey on a circlar course. When we arrive at 180 degrees, instead of continuing on around to 360 degrees,let's extend our journey, now counter-clockwise--to maintain our forward motion--into another circularly directed dimension, eventually arriving back at the 180 degree mark and continuing on our way to the origin; we form a figure 8. (Note that the two levels share one point -- at B.)

Notice that we "define" our universe by how we travel on it. On a curve, we of course, move continuously over all points. When we get to the 180 degree point we have to travel onto the second or lower dimension if it is there. If we don't, and elect not to count that second dimension, it may as well be "what ain't." But if the dimensions join, then a complete cycle on it is very different from an excursion around a single dimension. Notice on the 8, we must execute two 360-degree cycles to make it back to our origin. The point is that while curvature is our elemental rule, and while the relative values remain unchanged, an increase in dimensions fundamentally alters the nature of the system.

Suppose now that we add another dimension at the bottom of the lower circle to produce a snowman. Again our basic rules can hold up, and again relative values can transform unchanged, but, again, the course and nature of our journey is profoundly altered beyond what we'd experienced in one or two dimensions. For although we have a single curved genus of figures, each universe becomes a new species.