From Flatland to the fourth dimension (part 2)

Before I begin, a quick word on inauguration day: All I can say is that I am proud to be here during this historic event. We should rejoice at the changing of the guard in US politics and at the fact that Obama’s victory represents a step in the right direction for race relations. I already talked about my predictions in “In 103 days…” and I hope that what I talked about will really come to pass.

Now, to business: yesterday I briefly mentioned the mathematical concept of the fourth dimension while discussing Edwin Abbot Abbot’s Flatland. Today, I would like to devote an entire post to this intriguing mathematical topic.

First, let me make one thing clear: I am discussing the fourth spatial dimension; today, the layman uses the term “fourth dimension” to describe time, in accordance with Einstein’s idea of “space-time.” But here I am using the term in reference to the hypothetical next spatial dimension, as I shall explain.

The best way to explain the fourth dimension is by progression and analogy. First, consider a line, which has one dimension. It has two perpendicular directions, north and south (also known as length). Now, “drag” the line in a direction perpendicular two the existing two directions, like rolling a pencil covered in paint across a table. The result is a square, with two dimensions. The second dimension has four perpendicular dimensions: north, south, east, and west (also known as length and width). Now, drag the square in a direction perpendicular to the existing four (imagine lifting a paper square with string attached to the corners straight up off a table). The result is a cube, with three dimensions. The third dimension (the one we live in) has six perpendicular directions: north, south, east, west, up, and down (also known as length, width, and depth). Now, this is where it gets tricky: move the cube in a direction perpendicular to all 6 directions. The result is a shape with 8 perpendicular directions and four dimensions. This fourth dimension is to us what the third is to second; imagine how a box differs from a piece of paper and you will get an idea of what the fourth dimension is to us. Can’t imagine it? Don’t worry: it’s technically impossible to fully visualize it. But don’t give up yet—using analogies, we can figure out many aspects of the fourth dimension.

By moving a square out of our three-dimensional “plane,” a figure called a hypercube is created. As I stated previously, this object is the 4D equivalent of a square of cube. I would like to discuss some of the properties of this shape by analogy. For example: a line has 2 vertices, a square has 4, and a cube has 8, so a hypercube must have 16. A line has 2 sides (which are its vertices), a square has 4, and a cube has 6, so a hypercube must have 8. But remember that a square’s sides are lines, and a cube’s sides are squares, so a hypercube’s sides must be three-dimensional cubes. Confused? Remember that the fourth dimension has 8 perpendicular directions. Think about how a cube can be created by stacking square pieces of paper on top of each other; likewise, a hypercube is essentially an infinite number of stacked cubes, placed on top of each other on the new perpendicular axis. Still having trouble? Look at the figure to the top right; this is a drawing of a hypercube. Can you see the 8 cubes? If not, try looking at the figure below to the left and try to see the 8 cubes there, then look back at the other one. See how they are the same picture only projected differently? (Some of the cubes are “slanted” because of perspective, like the squares in a drawing of a cube. As you can see there is actually more than one way to draw a hypercube—read this But I prefer the one to the right because it shows the extrusion into the fourth dimension.)

A 4D sphere, called a glome, can be explained in a similar way. A 3D sphere is a infinite number of circles of increasing-then-decreasing size stacked on top of each other. (Try to imagine making a sphere out of a stack of pieces of paper. How would you cut the paper? The answer is into circles of increasing-then-decreasing size.) Likewise, a glome is made out of spheres stacked in increasing-then-decreasing size. Like the hypercube, these sphere are stacked in the new perpendicular direction.

Geometry-minded people’s brains are probably going crazy with all the applications of this idea. Personally I am not particularly interested in geometry, but I would like to elaborate just a little on 4D geometry. For example, look at a triangle, the simplest polygon it is possible to create. In three dimensions, the closest equivalent is a tetrahedron, a polyhedron with four sides made of triangles. Thus, the 4D equivalent is a pentachoron, a 4D figure with 5 sides made of tetrahedrons (a polychoron is the word for a 4D shape, like polygon or polyhedron). Also, similar to how regular polyhedrons exist in 3D, regular polychora exist in 4D. However, geometry is not really my area and I am not entirely sure about all the properties of these figures. Perhaps another day I will explain more about them.

If the concept of the fourth dimension still isn’t making sense, I recommend perusing this website—it’s very helpful.

Tomorrow I will look into more aspects of the fourth dimension.