## Wednesday, August 3, 2011

### Imagining the Second Dimension

Let's go back to basics.

The most important thing to remember as we're talking about these ten dimensions is that they're spatial dimensions. Some people get confused by that word, "spatial" because they think it's intended to only apply to the third dimension: the length, width and depth of the "space" we see around us. In fact, some physicists do prefer to use the term "space-like" when talking about the extra dimensions. No question, when you take a three-dimensional space and add an additional "right angle" to it, you are entering a realm which is difficult for us to picture, and it certainly behaves in ways that are beyond the limitations of 3D space: that's what Imagining the Tenth Dimension is all about.

Likewise, some people have difficulty with discussing the first and second dimension. "How can something with no depth even exist?", they ask. It is a bit of a mind-bender! We start with a point that has no size and no dimension. We make a second point some place else, and the line that passes through those points is a representation of the first dimension. If you can imagine a third point that isn't on the line you've just created, you have a way of thinking about the second dimension: a line passing through this new point and the old line defines a plane. But if the lines you're thinking about are like pencil lines, which already have a length, width and depth, then you're really not visualizing the right thing. Do you know what I mean by that?

If I say "imagine you're on a boat in the middle of the ocean", and you say "I don't own a boat and I hate the water", what does that prove? The concept of boats and oceans doesn't change whether you're willing to imagine them or not. Likewise, if I say "imagine something that has length and width and no depth", and you say "I refuse to imagine that because something that has no depth can't exist", where have we gotten? Nowhere. But does refusing to discuss an idea mean the idea doesn't exist? Of course not! We can have a perfectly good discussion about dragons, which don't appear to exist in our world, because dragons are an idea which we are capable of describing and thinking about.

Same goes for the second dimension. It's not part of our 3D world, it's something separate, but it's still something we can think about and talk about. In "What Would a Flatlander Really See?", we looked at the imaginary 2D creatures invented by Edwin Abbott for a book he published way back in 1884: "Flatland: A Romance of Many Dimensions".  Could Flatlanders really exist? Perhaps no more than dragons. But we can have a perfectly good discussion about the idea of Flatlanders, and what it would be like to live in a world that has length and width, but no depth. What would it be like to look around you within that world, where all you can make out is lines all within the same plane? That is a mind-bending exercise, good food for thought regardless of whether it would really be possible for some kind or awareness to exist within such a ridiculously limited frame of reference or not.

Over the next nine entries, we're going to look at each of the dimensions from the second all the way up to the tenth, and see what kind of a mental castle we can build for ourselves, one brick at a time. Along the way, I want you to keep reminding yourself about something called the "point-line-plane postulate", which uses the same kind of logic as the "line/branch/fold" of the Imagining the Tenth Dimension project. This postulate is the accepted method used to imagine any number of spatial dimensions, using the same repeating pattern:

0 - a Point: Whatever spatial dimension you're currently thinking about, imagine a geometric point within that dimension. Remember, when we say this point has "no size", what we really mean it that the point's size is indeterminate. "Indeterminate" means that any and all sizes you care to imagine, from the infinitely large to the infinitesimally small, are true for that point. Let's say that this is a point within "dimension x".

1 - a Line: From the current dimension you're examining, find another point not within that dimension. An easy way to do that is to imagine the first point at its largest possible size within the constraints of its own dimension, and then ask where a different point would be that isn't encompassed by that first point in its infinitely large state. Once you've found a second point, draw a line through both points, and now call what you're looking at "dimension x+1".

2 - a Plane: Again, think of both of those points encompassing their largest possible version within the constraints of the current dimension, and find a point that isn't part of what those two points are encompassing. Now you're thinking about "dimension x+2".

This logic can start from any spatial dimension, and it can be repeated infinitely: that is, once you've imagined "dimension x+2" you can rename it "dimension x" and repeat the pattern as many times as you want. Here's an important thing to remember: if we're not assigning any meaning to the dimensions we're visualizing, there's no reason to stop at ten. However, with this project, by the time we've arrived at the tenth dimension, we do find a way to say that we have arrived at the most all-encompassing version of the information that becomes reality, or the underlying symmetry state from which our universe or any other patterns emerge through the breaking of that symmetry.

And I do hope you'll enjoy the journey as we work our way through this logical presentation, one step after another.

Rob Bryanton

Next: Imagining the Third Dimension