## Monday, July 7, 2008

### What Would a Linelander Really See?

In mathematics, no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space E n. The point E 0 is 0-dimensional. The line E 1 is 1-dimensional. The plane E 2 is 2-dimensional. In general, E n is n-dimensional.
- from the wikipedia article on dimensions

Because there are so many different ways to define the word "dimensions", some people with their own narrow definition of the word have claimed that my use of the word is not correct. The definition above, from wikipedia, works very well for the way of imagining the dimensions that we've been playing with here: each additional dimension adds an additional "degree of freedom" that was unavailable from the previous one. If the ten dimensions that physicists are talking about are truly spatial dimensions (or as some prefer to say "space-like dimensions"), then a seven dimensional space must have seven co-ordinates that would define the position of a unique point within that seven-dimensional system, and so on: the space E 7 is 7-dimensional, and that is the kind of dimensions we are talking about in Imagining the Tenth Dimension.

When the website for this project first went live two years ago, a number of people had questions about the animation, since parts of it didn't just parrot what the mainstream had been saying about how our reality is constructed. Two years later, I'm pleased that a number of new theories have come forward from respected physicists which can be easily connected to the supposedly "out there" ideas that were in this original video and its accompanying book, but there are still many more ideas attached to this project that will eventually be proved or disproved in the months and years to come.

One of the most important basic assumptions about this project is that if physicists are really saying that these ten dimensions are spatial, then we should be able to use what we know about the first three or four dimensions to deduce some things about the additional degrees of freedom that these extra spatial dimensions would add. Saying "you can't imagine the fifth dimension and above because they're totally unlike the dimensions of spacetime" is, I think, taking the easy way out*: are we saying these additional dimensions are spatial or not? Of course it becomes harder to harder to imagine each additional dimension beyond the spacetime we're familiar with, but that doesn't mean it's impossible!

"Don't say Higher, say Extra"
Here's a brief side note: you may have noticed that the term "extra dimensions" is often used rather than "higher dimensions" when talking about the dimensions beyond spacetime: I think this is a useful distinction, because saying "higher" somehow sets the idea up in our minds that we should be gazing skyward as we think about these additional dimensions. A more valid way of thinking about each new dimensions is that it is somehow "outside" the current dimension, and that there is no way for us to get to the states that are within that additional dimension if we stay within the current one. For instance, with three-dimensional space we need to add the fourth dimension or there is no way for us to get from 3D space in one state to 3D space in another state: the directions of time and "anti-time" are the two new directions that we are able to reach by adding the fourth dimension into the equation. Saying that each new dimension is at "right angles" to the one below is another way of thinking about that same idea.

The "Linelander"
Last blog, we talked about the viewpoint of a Flatlander, an imaginary two-dimensional creature. Now, let's try this as a thought experiment: imagine yourself on a one-dimensional line. If that were your world, anything other than "forward on the line" and backward on the line" would be unfathomable. Imagining some kind of a wormhole that "folded" your line to allow instantaneous jumping from one position to another might possibly be something you could wrap your head around, and that would give you a way to imagine the second dimension, but dimensions beyond that would be so completely outside your experience that you would probably be inclined to say that the extra dimensions beyond the second dimension were theoretical constructs only, with no way for one-dimensional people to be able to imagine such an outlandish geometry.

For a one-dimensional Linelander, a second dimension at right angles to the dimension he lived in would possibly be imaginable, but something that was at an additional right angle that was somehow different from the first right angle already considered (to create a three-dimensional space) might well seem beyond the ability for our poor Linelander to imagine: the difficultly is particularly compounded for the Linelander, since he can easily imagine only one way of extending out to an additional dimension, so visualizing each additional dimension would require him to imagine a repetitive "right angles" operation over and over again. The advantage we have over a Linelander is that we are intimately familiar with a space of three dimensions, so for us, to imagine three unique right angles that together could help to form a cube is much easier - and this is why using "line/branch/fold" works so well for us as a visualization tool, because it gives us an ordered way of imagining how three different dimensions are at right angles to each other.

"Bend Me, Fold Me, Any Way You Want..."
So here we are, imagining ourselves as a Linelander on a one-dimensional line, trying to imagine how our 1D world might be able to be bend or fold to allow instantaneous teleportation to other points on our line. Einstein suggested that we should think of gravity as a "bending" of spacetime. Whether we're talking about folding, branching, twisting, bending or any other spatial manipulation word you care to think of, all of these are ways to start thinking about the next dimension up, the dimension that is moving at right angles to the one currently being examined. To be clear, then, the point/line/branch/fold concepts that this project starts from are really just interchangeable spatial manipulation terms that repeatedly find different ways to describe the same idea of moving to the next dimension, which is why this concept can work no matter where you start. A second dimensional plane can be thought of as a thick line joining two other lines, or it can be thought of as a branch off of a line, or it can be thought of as being created by the folding of a line: these are all ways of thinking how the one dimension we are looking at is at right angles to the other.

Like our Linelander, right angles are relatively easy for us to imagine for the dimensions we live in -- which in our case as 3D "Spacelanders" would be the first three or four dimensions. But for us, saying that "the sixth dimension is at right angles to the fifth dimension" doesn't create a particularly useful mental image. This is why the line, branch, fold metaphor is so powerful - it lets us visualize something that is easier to hold in our minds than if we were to just imagine line/line/line etc, branch/branch/branch etc., or fold/fold/fold.

Time is at Right Angles
So. Each of these terms is a different way of thinking about that same "this is how a dimension is at right angles to the one below" concept. This is also why I say "time" is in the next dimension up no matter dimension you're examining, because "time" is another way of moving at right angles to the dimension below, and this is why "time" for a 2D flatlander would be a direction in the third dimension.

Recently I posted a blog entry about wormholes, extending a discussion that's in the tenth dimension faq: the question in the faq is "can you really fold a dimension?". And my answer is yes, you can fold a dimension and science calls that a wormhole. But with the logic of what we're examining here, different wormholes would have different effects depending upon the dimension being folded, and ultimately this gives us ways of thinking how the solid reality we are seeing at this very instant could be nothing more than shadows of higher dimensional shapes and patterns, connecting our reality together in ways that are beyond our ability to directly witness from down here in spacetime.

Enjoy the journey,

Rob

Related entries:
Time is a Direction
The Flipbook Universe
You Can't Get There From Here
How to Make a Universe
What Would a Flatlander Really See?
Hypercubes and Plato's Cave

* P. S. - for anyone who is unfamiliar with my project, please read blog entries like The Fifth Dimension Isn't Magic to see how the idea that the extra dimensions are "compactified" is easily dealt with in the visualization we're using in Imagining the Tenth Dimension.

Next: The Annotated Tenth Dimension Video