I began to show the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 3^2.
“… I suppose 3^3 must mean something in Geometry; what does it mean?”
“Nothing at all,” replied I, “not at least in Geometry; for Geometry has only Two Dimensions.”
- Edwin Abbott, Flatland
A lot of mathematical art concerns itself with objects that exist in four or more dimensions. Even physicists have been telling us that we live in a ten dimensional universe. How are we, as three-dimensional beings, able to understand this?
Starting Out Simple
Let’s begin by reviewing the lower dimensions:
0 - Point: A point has zero dimensions, and is usually represented graphically by a dot. Some people get confused about how it is possible for something to have zero dimensions. Try holding up two fingers in front of you, and imagine the point in space exactly halfway between your two fingers. This point exists; it has a definite location. However, measuring its length, width, or height would be impossible. It is zero-dimensional.
1 - Line: Now, suppose we move the point and look at the trail it leaves behind. The path of this point is a line. It is one-dimensional - it has length, but no height or depth.
2 - Square: If a line moves parallel to itself (or as mathematicians would say, “is extruded along an orthogonal axis”), we end up with a two-dimensional square.
3 - Cube: A square that is extruded along an orthogonal axis (in this case, the axis would be coming out of the computer screen) becomes a three-dimensional cube.
But we have to pause here, because the last figure in the diagram above is misleading. Borrowing from an idea of Rene Magritte:
The image on the left is indeed not a pipe. Pipes are solid objects that are used for smoking. What’s shown above is a painting of a pipe (or, rather, a computer image of a painting of a pipe). Likewise, what is shown above is not a cube. Cubes are solid objects. Unfortunately, it is impossible to display a solid object on a flat computer screen, so we are forced to use a two-dimensional representation of a cube.
4 - Hypercube: So what happens if we extrude a cube along an orthogonal axis? Well, naturally, we end up with a four-dimensional hypercube - also called a tesseract. We can attempt to make a two-dimensional drawing of a tesseract like so:
One question, however, still remains for us three-dimensional beings: where is that orthogonal axis?
But isn’t time the fourth dimension?
Well, that depends. Physicists, especially those interested in relativity, usually consider time to be the fourth dimension (or rather, one of the four dimensions). Mathematicians, however, can construct as many spatial dimensions as they want simply by adding on more variables. Mathematicians say that calling the fourth dimension “time” is an application of four-dimensional geometry.
Depicting 4D Objects in 3D Space
Because humans cannot see four dimensions, objects have to be reduced to three dimensions or fewer in order to be viewed. There are two basic ways of doing this.
The first is to slice the object and look at its cross section. For example, taking the cross-section of a sphere results in a circle. Just as a cross-section of a 3D object is a 2D figure, a cross section of a 4D object is a 3D figure. This would be called “intersecting” the object with a plane. Of course, what the cross section looks like will depend on where and at what angle the object is sliced. To get a feel for the nature of a four-dimensional object, it is helpful to examine a series of slices along an axis perpendicular to the slicing plane.
The second method is to look at the shadow of the object. If you shine a light on a three-dimensional object, you will see a two-dimensional shadow projected on the wall. Likewise, the shadow of a 4D object is a 3D object. What the shadow looks like will depend on the location of the light source relative to the object, as well as the orientation of the projection plane.
Experiencing Four Dimensions
Will human beings ever be able to directly experience four spatial dimensions? Psychologist Frances Wang has been doing research at the Beckman Institute to see how humans interpret four-dimensional space. Early results are showing that after a period of trial and error, subjects can gain some intuitive knowledge of how 4D space works.
Further Reading
Flatland: A Romance of Many Dimensions - Flatland, a book by Edwin Abbott Abbott, tells the story of A Square, his life in two-dimensional Flatland, his visit to Lineland, and his voyage to Spaceland. You can order the book at Amazon (they even made a DVD out of it), or you can read it for free at Project Gutenberg.
Fourfield - Fourfield:Computers, Art & the Fourth Dimension, by Tony Robbin gives some explanation of four dimensions and how it has been applied to art. It even comes with a pair of red and blue anaglyph 3D glasses and a printout of hypercube tiles. He has another book, Shadows of Reality, but I haven’t read that one yet.
