On Edge

The twisted-universe model I wrote about last time clearly must involve curved space. The idea that space can be curved is pretty familiar by now, mostly because of Einstein’s idea that gravity results from a bending or curving of space by a mass of matter. That seems to be the consensus these days about how gravity works; the most popular alternative, that gravity is somehow transmitted from one mass to another, suffers somewhat from the failure so far to detect any of the “gravity waves” this theory would require.

I always had a hard time figuring out how gravity could have an effect on space, which essentially has no properties of its own to be affected. (Under relativity and quantum mechanics, empty space seems to be occupied by a “quantum vacuum” that does have some properties, but that’s not the same thing as space.) However, when I started thinking about the properties of my twisted space, I realized that any space does have one property: dimension. Which means that the only kind of change you can make to a space is a change of dimension.

As I explained last time, giving a spherical universe a half-twist through the fourth dimension raises the dimension of the universe as a whole to 4. The dimension at any point within the universe would appear to a normal physical observer to be 3, but in fact it would be slightly more than 3; we’ll say it’s D=3+(n<1). And if you add up that n<1 for all the points (at some arbitrarily selected but uniform scale, such as a light year or a parsec) in an orbit of the universe, the sum will be 1.

Now, even though this cosmos is finite, it’s still very large. So the n<1 – which I’m going to declare a fundamental universal constant, and call Ü, mostly because I like umlauts – is going to be very small, almost ininitesmal. The 3+Ü that exists at any point would then be the natural dimension of space in this bent universe of mine.

This whole idea of a non-integer dimension is exactly what Benoit Mandelbrot means by the term “fractal” that he coined to describe objects with a “fractional dimension.” And he has demonstrated that a very wide variety of objects are fractals, which means that many (perhaps most) of the objects we think of as, say, three-dimensional are in fact three-plus dimensional. The more complex the object, the higher the fractional excess; so a big, many-branched tree would be “more fractal,” if you will, than, say, a bowling ball.

But it seems likely that the fractional dimension of even a fairly smooth 3D+ object, like a planet, would have to be greater than the near-infinitesmal value of Ü. And as a result, the planet would “stretch” the dimension of the space around it, causing the kind of contour that Einstein’s conjecture associates with gravity. This would be true even with very small objects, which would account for the kind of “clumping” that scientists believe took place in the early universe, leading eventually to the formation of galaxies and so on.

There’s something else that the structure of the twisted-universe model might help account for that has puzzled me for a long time. I think some visual aids may help here.

We often hear airplane pilots talking about flying “straight and level.” But they’re actually doing anything but. What they’re really doing is flying at a constant altitude above the Earth’s surface. Because the Earth’s surface is curved, however, the plane’s actual path is also curved, as shown above. What would happen if an airplane really flew “straight and level” looks like this:

Now, the thing I’ve wondered about for years is this: We all know that the speed of light is a sort of universal speed limit, that nothing can go faster without violating all sorts of natural laws. But I’ve always wondered why it’s precisely the speed it is, 299,792.5 kilometers per second, or about 186,000 miles per second.

We’re used to the idea, again thanks to Einstein, of light traveling a curved path around massive, high-gravity objects. But since I’m supposing here that all light must travel a curved path in a curved, twisted universe, the “normal” path of light would look something like this:

Obviously, the curvature of this “universe” is highly exaggerated, but it illustrates how the rays of light in a sense “flow” along the contour of the space. What I’m going to suggest is that the speed at which light (and of course other forms of radiation) travels is actually determined by that contour or curvature, because if it travelled at a higher speed, this would happen:

What this would actually mean is hard to say. It might mean that the energy disappears into the fourth dimension, or it could even mean that it exits the universe, whatever that might entail.

Mention of the fourth dimension brings up one final point I want to make before closing this largely pointless expostulation. You’ll remember the Möbius strip from last time:

In looking at this illustration, I want you to imagine that the strip is actually transparent, because what we’re talking about here is empty space, not paper. So there’s really nothing separating point A from point B, or C from D, except space; or rather, except the twisted structure of this space. But the separation is nevertheless complete and inviolable: The only way to get from point A to point B is to go around the strip; you can’t go through it.

Why not? Well, if you travel around the strip from A to B, you’re in effect adding up Ü units, or in a sense travelling uphill dimensionally. By the time you reach point C, you’re in a dimension that’s 0.5 higher in relation to A, and when you reach B, the space you’re in is a full 1 dimension away from A. It’s still 2+ÜD from a local point of view (in the illustration; in the twisted universe, it would be 3+ÜD), but A is 3D from the perspective of B (4D in the real universe), and vice versa. So naturally, there’s no way to perceive one space from the other, much less to go there directly.

This is precisely what constitutes the boundary or “edge” of the universe, this dimensional barrier. And what that means is that every point in the universe is on the edge of the universe.

Somehow, that reminds me of the famous Hermetic saying quoted by Giordano Bruno and Pascal, among others: “God is an intelligible sphere whose center is everywhere and whose circumference is nowhere.” In the twisted universe, the circumference is everywhere, but I’m not sure whether there’s a center anywhere.