From Flatland to the fourth dimension (part 5)

I find it extremely amusing that this series, which is now tied with the “law and order” series for the most posts, is about math. I am by no means a math-oriented person. However, as I have said before I find this topic fascinating, and for that reason I feel it deserves this many posts. But I’m digressing. The point is: today’s post will cover the following in topics in four dimensions: water and islands. If you have not read posts 2, 3, and 4 in this series, please do so before reading this one.

First, water. In all dimensions water fills up its container, but the nature of the container is obviously dependent on the number of dimensions. In two dimensions, in order to create a water-tight container the vessel’s surface is a line and the “seal” is two points. In three dimensions, a water-tight container’s surface is a plane (like the surface of a bowl) and the “seal” is a line (imagine saran-wrap or tinfoil stretched over a bowl). In four dimensions, the container’s surface must be 3D and the seal a plane, since the water has so many directions to escape. This is difficult if not impossible to visualize, but try to picture it. Along with this is the concept of surface area: in 2D surfaces are lines, in 3D they are planes, and the 4D they are three-dimensional. This is almost certainly impossible for us to visualize, but think about it like this: imagine a warped sphere, full of curves and indents…this what a body of water in zero-g in 3D would look like. But in four dimensions this is only the surface of the water…the rest of the water is sloshing about in the other perpendicular directions. And, if the water surface is not four-dimensionally flat, we wouldn’t even be able to see all of the surface (this is really hard to visualize—try to use the 3D-2D analogy).

Along with the concept of water go the concepts of river and bridges. In two dimensions, a river moves from higher elevation to lower elevation, but it covers the entire ground surface and cannot be crossed with a bridge (unless the bridge spans the entire length of the river). In three dimensions, rivers still move from higher elevations to lower ones, but they and they zigzag depending on the terrain. A 3D river can be crossed with a bridge. In four dimensions, a river is very similar, but there are two key differences: a river in 4D would corkscrew, and there is no need for a bridge—one could simple walk around a river.

Related to rivers is the idea of lakes. In 2D, there is only one kind of lake. In 3D, though, there are two kinds: linear and round. The surface of a linear lake, as the name implies, is thinner, like a river, but a round lake is more circular. In 4D, there are three kinds of lakes, linear, flat, and globular. The surface of a linear lake is like a cylinder; the surface of a flat lake is flat, and a globular lake has a spherical surface.

Next, and somewhat related to water, is the concept of 4D islands. First, though, let’s look at an island in two and three dimensions. In 2D, the surface of an island is a line; to “search” all of it, one must simply move from one end to another. In three dimension, land surface is a plane. To search a 3D island, one must move in a 2D grid to cover all the land area. Like the surface of water in four dimensions, the surface of land is three-dimensional. To search this surface, one must move in a 3D grid along 3 axes.

The series on the fourth dimension will continue, unless for some reason I grow bored with it all of a sudden.