*Matt Strassler 11/14/11*

If you’ve read the article on Worlds of 1 Spatial Dimension, you know that what makes something one-dimensional is the fact that

- A location within this space is specified by one piece of information

Also, it is continuous (or close to continuous for all practical purposes). I discussed several examples of dimensions: the Income Line, which is infinite, and represented by an infinite line; the Rainbow-Color Line, which is finite with end-walls, and represented by a line segment; and the Aeolian Line of wind directions, which is finite but periodic, and represented by a line segment whose left end is the same as its right end, or, equivalently, by a circle. (I also mentioned in passing a fourth example: a world that is infinite in one direction and has an end-wall in the other.) In a follow-up article, I emphasized that there are many types of dimensions, but physical dimensions of space have unique and special (and rather obvious) properties that distinguish them from other dimensions.

Now what about two dimensions? Not surprisingly, there are many more types of two-dimensional spaces than one-dimensional spaces. A few examples are shown in Figure 1. We can have a world that is infinite in both directions: a plane. (Top left of Figure 1.) We can have a world that is infinite in one dimension, but in the other forms either a line segment or a circle. These worlds are naturally called a strip and a tube. (Bottom left of Figure 1.) Or we can have a world which is not infinite in either dimension. (Right side of Figure 1.) And look at all the possibilities! In that one figure, from top to bottom, you can see a square; a cylinder (the round part of a can, without its lids or its interior); a disk (like a coin); a torus (which looks like a car tire), a sphere (not the interior, just the outer shell, like a tennis ball), or a double-tire. These are not the only options, either. Looking ahead, it is clear that by the time we get to three dimensions and beyond, we’re clearly not going to be able to make lists like this…

As with one-dimensional spaces,

- A location inside a two-dimensional space is specified by two pieces of information.

For instance, an example of a sphere (to a pretty good approximation) is the earth’s surface: any location can be specified by a latitude and a longitude. An ant walking on a garden hose moves along a two-dimensional tube, and at any particular moment is located a certain distance from the spigot and at a certain angle off the vertical. A multi-lane highway is essentially a two-dimensional strip, with a very long direction and a short direction: the two pieces of information you need to determine where you are include (1) what distance have you traveled since the start of the road, and (2) how far are you from the right edge of the road.

Let’s recall the Income Line. Quoting from Worlds of 1 Spatial Dimension: “How much money you made last year is a certain number in your home currency. It could have been positive or negative; it could have been small or large; it can be represented as a point on a line … which we’ll call the `income line’. Every point on the line represents a possible income.” Well, if you’re married and both you and your spouse have incomes, the two income streams into (or out of) your household can be represented on the Two-Income Plane. If you describe (1) your income, and (2) your spouse’s income, then those two numbers specify a point on that plane.

*Here’s a tricky example of a torus, one that shows how we can still talk about interesting two-dimensional shapes whose dimensions are not physical space. We saw in Worlds of 1 Spatial Dimension (see Figure 3 of that post) that the possible directions from which the wind can blow form a one-dimensional world in the form of a circle (or equivalently, a line-segment with the left end the same as the right end.) The possible directions in which a sailboat may travel also form a similar circle. But anyone who has gone sailing knows that you don’t have to sail in the same direction as the wind is blowing; if you angle your sail, you can head east even if the wind is blowing from the north. So if I ask for two pieces of information — (1) from what direction is the wind blowing, and (2) in what direction is my boat sailing —- both pieces of information are points on a circle. Two pieces of information, each on a circle, form a point on a torus.*

Before we go on, there’s a natural and common confusion that I need to nip in the bud. I’ve already alluded to it in my description of the various spaces above. *You musn’t confuse dimensions or shapes themselves with a particular way of representing those dimensions or shapes!* What is intrinsic to a circle is that if you walk along it in either direction you will come back to where you started. A circle has no inside or outside. That said, the representation of a circle as a closed loop inside a two-dimensional plane

*looks*as though has an inside and an outside.

**You can learn more about this, and how the same confusion occurs in the context of the expanding universe and the Big Bang, here.**

*But that’s a property of representing the circle inside a plane, not a property of the circle itself.**More on extra dimensions — showing how a two-dimensional shape can look at first like a one-dimensional shape, and explaining how one may discover it is really two-dimensional — is coming soon.*

“Today, however, we do have the opportunity not only to observe phenomena in four and higher dimensions, but we can also interact with them. The medium for such interaction is computer graphics. Computer graphic devices produce images on two-dimensional screens. Each point on the screen has two real numbers as coordinates, and the computer stores the locations of points and lists of pairs of points which are to be connected by line segments or more complicated curves. In this way a diagram of great complexity can be developed on the screen and saved for later viewing or further manipulation .” THOMAS BANCHOFF – http://www.geom.uiuc.edu/~banchoff/ISR/ISR.html

Again I am not trying to move ahead to where I think you might be going but to fill in the gaps of where my inexperience shines as you help to orientate.

Best,

The phrase “the representation of a circle as a closed loop inside a two-dimensional plane looks as though has an inside and an outside. But that’s a property of representing the circle inside a plane, not a property of the circle itself” is most important, thank you. Please make similar notes when talking about 3-dimensional and higher analogies.

Is it a problem that to describe a point in a finite, periodic, one-dimensional world (e.g. a circle), that we are using the notion of an angle – which is a measure from a point outside the one-dimensional world? An income-line had real numbers associated with each point. Perhaps if the real angle numbers are simply assigned to the points of the circle without reference to a center point, then there is no problem. The starting point is arbitrary. But assigning a center point is assigning a point outside the dimension and that seems – problematic.

No, because the angle can be measured also as an arclength around the circle divided by the circle’s total circumference, both of which can be measured just by walking on the circle and not with reference to a point at the center, times 2 pi.

An illustration of how two points on a circle are but one point on the torus would be helpful. I can “see” one point, but the second won’t come into focus. It’s a bit like Heisenberg’s uncertainty principle.

I didn’t say that two points on a *single* circle are one point on a torus, though I can see how what I wrote can be hard to understand.

There are two circles here. One circle tells us where the wind is blowing. The second circle tells us where the sailboat is going. Taken together, the two circles form a torus, and whatever the wind direction and the sailboat’s direction are give us *one point* on that torus.

Does that help?

Ok, so maybe only a very small bit…