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.

Trees of Life

What's the difference between a pile of wet dirt steaming in the sun, and a tree?

I've asked people this question before and have gotten that is-he-off-his-meds look from them. But it's simple enough: a tree and a pile of wet dirt are composed of the same stuff. The difference is that the tree is alive, has life, is a life. That means it takes the moisture and dirt and air and sunlight and transforms it through self-sustained processes into a structure, a form. In effect, the life of the tree takes the raw materials - literally earth, water, air and fire - and uses them to assemble a form to realize or manifest itself.

The form it takes is a branching fractal, and in fact a tree is the paradigm for all such geometric shapes; for example, a chart that shows the branching of lines of descent from an ancestor is known as a family "tree."

Which brings me to the next question that tends to elicit "that look" from people: Where does life come from?

I hasten to add that I'm not asking about the origin or ultimate source of all life; rather, simply this: If we look at a specific living thing, where did its life come from? And the simple answer is: from another living thing. Life comes from life.

What a family tree shows, from this point of view, is the history of transmission of life through successive generations, as it branches, re-branches and multiplies.

Trying to trace these lines of transmission, as I indicated in my last post, can get to be a pretty complex task. And it gets even more complex if you consider that your personal family tree is just one of billions of sub-branches of the overall human family tree, which in turn is just a sub-branch of the mammalian family tree, which again is a sub-branch of a yet-larger "tree of life."

Here's one attempt to represent this situation graphically (click to enlarge; right-click to open in a new browser window):


(Source: Wikipedia)


Drastically simplified as this picture is, it does get the basic idea across, with special emphasis on the tree-like nature of the relationships.

This next one is much more scientifically up-to-date and gives a better picture of how complicated these life-connections have become over time:

(Source: Cold Spring Harbor Laboratory Press)

Even this diagram, however, exhibits a tree-like branching structure that fans out from a single point of origin. Now, there's obviously a lot of room for debate about the exact nature of that point of origin - was it an act of divine creation, or a chance combination of organic chemicals, or the arrival on earth of some existing simple organism from the far reaches of space? - but we can safely ignore that question for now. The point is that all subsequent life consists of an uninterrupted transmission from that one original source.

Here's one more example of a tree-like branching fractal structure:



(Source: U.S. Geological Survey; photo of Selenga River delta on the southeast shore of Lake Baikal, Russia.)

What does this river delta have in common with a tree, apart from a branching structure? I would argue that both can be thought of as objects created by a type of outflow. That's obvious in the case of the river, but perhaps not so obvious with regard to a tree. But I think it's a true way of looking at a tree's growth: The life within the tree is creating a flow up into the sky by assembling its outstretching branching structure, just as the gravitational energy of the river causes it to flow into the sea, depositing a similar structure as it does so.

In the very same way, the tree-of-life diagrams above chart the outflow over the eons of life itself. What we see in these diagrams, in other words, is the history of the flow of a single stream with many branches, just like the one in the satellite photo.

In short, all life on earth is one life flowing from a single wellspring through myriad branches. From that perspective, each individual living thing is related to every other living thing in the same way as the separate leaves on a gigantic tree, or as the fingers of one individual human's hands: distinguishable as objects of a kind in themselves, but impossible and meaningless without their connection to a larger life.

I suggested last time that when we stand in line at the grocery store, there's a fair chance that the cashier or some of the other customers may be our cousins. But clearly, our relatedness is ultimately much closer, much more intimate: There is only one life, a single self-same life, in all of us. It enlivens each of us for a time as it flows through us from its unknown past toward its unguessable future.

Knowing that, how can I possibly regard you or any other living being with ill-will?

It Could Be Worse

Today’s economic reports, stock market decline and related news stories were all pretty ugly. But they were 1970s ugly rather than 1930s ugly, which is some consolation.

The Dow Jones Industrial Average is down, as of today’s close, 44 percent from its all-time high set a little over a year ago. The decline from the January 1973 high to the December 1974 low was 45 percent. The decline from the September 1929 high to the July 1932 low was almost exactly twice as much, 89 percent. An 89 percent decline from the October 2007 high would put the Dow at about 1558, so today’s much-discussed close slightly below 8000 may not look quite so bad.

Mathematically, stock market movements resemble earthquakes; that is, they’re fractal and scale according to a power law. In oversimplified English, which is the only way I can understand this kind of thing, what that means is that exponentially larger movements occur exponentially less often than small movements. This is why seismologists are unable to tell Californians exactly when “the Big One” will occur but insist that the more time passes without a big one, the more likely it becomes. So same thing with the stock market.

The current bear market is the second since 2000; the decline from the January 2000 high to the October 2002 low amounted to about 38 percent. From the seismological point of view, that might mean that we’re less likely to get a “big one” of near 90 percent, and instead must suffer through a series of less catastrophic but still thoroughly unpleasant medium-sized shocks.

From the standpoint of technical market analysis, 8000 on the Dow isn’t very interesting anyway. The really interesting number is the October 2002 low at about 7286, which we could stretch a bit down to the 7000 level as an idealized 50 percent decline. If that range doesn’t hold, then we’re potentially looking at a decline to 1.) the 5300-5700 area, representing the trading range of a 1996 pause, for a possible 62 percent drop; the 3600-4000 range, which is the level of a somewhat significant sideways move in 1994 and would amount to a roughly 75 percent decline; or the aforementioned near-90 percent decline to the 1500 range. And of course there's always the possibility that the world could come to an end and the Dow would fall to zero, but personally I give that pretty low odds.

Those are, of course, the worst-case scenarios. The most optimistic case would be that the market is “base-building” right around where it is now and will launch a new, multi-year uptrend from here that will replenish all the 401(k) accounts and other investments that have been stripped in the past year. I’d feel a lot more confident about that scenario if it didn’t depend so much on believing that the same people who contributed so much to the current problems - and who are still apparently more interested in self-justification and self-aggrandizement than in doing anything for their society or their world - will somehow suddenly start making all the right decisions.