The idea of 'Dimension' is to design coordinate systems for various shapes, ranging from cubes to barbells. As a point of reference, we can use the sphere, which happens to be the shape of our very own Earth.

Beginning with the cube, it is fairly simple to design a coordinate system reminiscent of our own. All you need is two perpendicular lines along the cube, each one going through the exact centers of four faces. To visualize it, picture a present wrapped up in a ribbon. Or, alternatively, refer to the unfolded cube below. The lines have been drawn in such a way that when the cube is folded, they will be in the right place.

These two lines will be the 0s of latitude and longitude, equivalent to the equator and the prime meridian on Earth. We now have the power to specify the location of any point of the surface of the cube with relation our two lines. Just say how many units 'north' it is from the equator, and how many units 'east' it is from the prime meridian, and that's all we need. In terms of specifying points, the only difference between the cube and Earth is that on the cube, we need to turn a corner every now and again. This distinction is purely superficial, and makes no difference to the actual difficulty of the problem beyond our not being used to it (as a matter of fact, the sphere is actually much more unintuitive, since the units get smaller as you near the poles).

So, after the relatively simple problem of the square, we move on to the donut (or 'torus', if you want to be fancy about it). With the donut, the two lines are going to need to be placed a little more creatively.

One will run along the inside of the donut, and the other will run perpendicular to the first. See the image below.

In this illustration, we see the donut divided into sections. Any two perpendicular lines you see above you could be chosen to be the 'equator' and the 'prime meridian'; what matters is that you can describe any other point on the donut in terms of its relation to those two lines.

Now, that's about as far as I've gotten so far, but there is certainly more to think about the topic. For example, what if we tried, instead of a donut, a croissant:

Well, as you can tell from the lines already drawn in, it's obviously possible. Don't let the fact that it comes to a point deceive you: there is little theoretical difference, I think, between the croissant and Earth (with the two points corresponding to the two poles of Earth). How about this:

We are now entering a realm which I am really not prepared to discuss, so I'll leave it open to you guys. How would one even go about trying this? Is it possible to do with the methods we've been able to use so far? What do all those lines scribbled on it mean? (I actually have no idea, I just grabbed it off google images).

You seem to have gotten away with using 2 coordinates for these 3 dimensional objects, but are there any 3D surfaces that need 3 coordinates? I think 3D shape has to have some kind of symmetry to only need 2 coordinates, but I'm not sure which kind it is.

ReplyDeleteCan the same thing be said for 2 dimensions, meaning that some 2D shapes need 2 coordinates to map but some only need one? I can't think of any currently.