Shufflebrain

In terms of our search, a periodic event in perceptual space is a transformation, as a least curvature, to any other coordinates within the mental continuum. The same thing would be true of a series of periodic events. Since phase variation must be part of those events, memory (phase codes) becomes transformable to any coordinates in the mental continuum. A specific phase spectrum--a particular memory--becomes a definite path of least curvature in transformations from sensations to perceptions to stored memories to covert or overt behaviors, thoughts, feelings or whatever else exists in the mental continuum. If we call on our e 's for imagery, the least pearly path will not change merely because the coordinates change. Behavior, then, is an informational transform of, for instance, perception.

Transformation within the Riemann-style mental continuum is the means by which hologramic mind stores itself and manifests its existence in different ways. But we need some device for carrying out the transformation in question. For the latter purpose, I must introduce the reader to other abstract entities, quite implicit in Riemann's work but not explicitly worked out until some time after his death. These entities are known as tensors.

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The mathematician Leon Brillouin credits the crystal physicist W. Voigt with the discovery of tensors in the world of mass-energy.[12] It's no news that a crystal's anatomy will deform in response to stress or strain. But what stunned Voigt was that certain relative values within the distorted crystal remained invariant before and after the deformation. Tensors represent those invariant relative values. Like Riemann's invariant curvature relationships, tensors survive transformation anywhere, any place, any time. Just in time for Einstein, mathematicians worked out and proved the theorems for tensors. In the process they found tensors to be the most splendid abstract entities yet discovered for investigating ideal as well as physical changes. Tensors provided mathematicians with a whole new concept of the coordinate. And tensors furnished Einstein with a language in which to phrase relativity, as well as the means to deal with invariance in an ever-varying universe. As to their power and generality, Brillouin tells us: "An equation can have meaning only if the two members [the terms on either side of the equals sign] are of the same tensor character."[13] Alleged equations without tensor characteristics turn out to be empirical formulas and lack the necessity Benjamin Peirce talked about.

Tensors depict change, changing changes, changes in changing changes and even variations of higher orders. Conceptually, tensor resemble relative phase. Tensors relationships transform in the same way that relative phase does. This transformational feature affords us an impressionistic look at their meaning.

Do you recall the dot transform experiments from the last chapter:

Notice the diagonal arrangement of both the dots and the rings. If we rotate the rings so as to bring them into a horizontal orientation (tilt your head), the dots also take on a horizontal arrangement.

The rings and dots are transforms of each other. The fundamental direction of change remains constant in the transform as well as its back-transform. And during rotation, the basic orientation of the rings and dots--relative to each other--remains invariant. Absolute values differ enormously. But as we can see for ourselves, the relative values transform in the same way. This is the cardinal characteristic of the tensor: it preserves an abstract ratio independent of the coordinate system. If we stop to think about it, we realize that if tensors carry their own meaning wherever they go, they should be able to define the coordinate itself. And so they do.

Ordinary mathematical operations begin with a definition of the coordinate system. Let's ask, as Riemann did, what's the basis for such definitions? With what omniscience do we survey the totality of the real and ideal--from outside, no less!--and decide a priori just what a universe must be like? The user of tensors begins with a humble attitude. The user of tensors begins, as did Riemann, ignorant of the universe--and aware of his or her ignorance. The user of tensors is obliged to calculate the coordinate system only after arriving there and is not at all free to proclaim the coordinate system in advance. Tensors can work in the Cartesian systems of ordinary graphs; they can work in Euclid's world. But it is almost as though they were created to mediate travel in Riemann's abstract universes.