Shufflebrain

Consider a pretzel and a doughnut (or a bagel, if you're avoiding sweets). To keep the discussion simple, imagine them on a plane. Our doughnut has two apparent surfaces, the outside and the lining of the hole (Riemann called it a doubly connected surface). Our pretzel has four surfaces ("quadruply" connected) : one for each of the three holes, plus the exterior. Let's assume (with Riemann) that all things with the same number of surfaces belong to the same topological species, and let's not fret about whether the doughnut is round, oval or mashed down on one side. Also, let's consider surface to be a manifestation of dimension.

How can we convert a pretzel to a doughnut and vice versa? With the pretzel we could make the conversion with two cuts between two apparent surfaces. To go the other way, we (or the baker) can employ the term join.--two joins to regenerate a pretzel from a doughnut.

We've said that at either end of our universe there's a dimension (surface) we can't reach, positive and negative infinity. Thus, as we move up the scale (by joining), there's always one dimension more in the continuum than we can count. If we actually observe two surfaces on the doughnut, we know our overall system (or ideal universe) has at the very minimum three positive dimensions: the two we can count and the one we can't.[10a] What happens when we move in the other direction--when we apply (or add) one cut to our doughnut? We create a pancake with a single (singly connected) surface. But since we have the one countable surface, we know that still another surface must exist in the negative direction. And in the latter dimension is active zero.

Wait! We can't do that! you may insist. But look at it like this. We admitted that we can't count to infinity, right? If we don't put those uncountable dimensions on either end of our universe, then we end up in a genuine bind: either we can't count at all or we can reach unreachability. We can see for ourselves that we can cut or join--add and subtract (count)--to make pretzels and doughnuts out of each other. But to assert that we can reach unreachability may be okay for a preacher, but it's preposterous for a scientist.

Suppose we add a cut to a pancake--eat it up, for instance. Where are we? We now have another goofy choice: between active zero (zero surface) and "what ain't." We can't define "what ain't"; or, if we do define "what ain't" by definition, it won't be "what ain't" anymore. If we don't define it, it disappears from the argument, leaving us with good grammar and the zero-surfaced figure. And we can define the zero-surfaced dimension from our counting system: the dimension sandwiched between the one-surfaced pancake and the minus-one dimension.[11]

If we can conceive of a zero-dimensional surface, we can certainly appreciate a zero-curvature without making it "what ain't." And the zero-curvature, as part of the continuum of curvatures, is the curvature of the Euclidean world of experience.

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Before we put hologramic mind into a Riemannian context, I would like to emphasize three important principles.

First, Riemann's success followed from in his basic approach. He didn't begin with an already-assembled coordinate system; he didn't erect the superstructure before he attempted to describe what the coordinates were like. In a system formulated from Riemann's approach, the elements define the coordinates, not the other way around as is usually the case, even in our own times.

The second principle is related to the first but is a direct outgrowth of Riemann's discovery that measurable relationships in the vicinity of one point are the same around all points. What does this mean in terms of different coordinate systems? If we can find those relationships in different coordinates, as we did in our imaginary experiments with pearly e 's, then the corresponding regions can be regarded as transformations of each other.

The third principle, an extension of the second, underlies our experiment with pearly e 's. An entire coordinate system becomes a transformation of any other coordinate system via their respective paths of least curvature. Thus the entire abstract universe is a single continuum of curvatures. Curvature is an abstraction. But we can always mimic curvature in our thoughts with our imaginary pearly pearly e 's.

Now let's make a preliminary first fitting of hologram mind to the world of Riemann.

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