When I think of dimensions in mathematics and science, I generally conceptualize dimensions in space. Although Prof. Banchoff mentions that time can also be a dimension (p. 5), it might be informative to introduce an exercise that encourages students to list a set of potential factors that could pose additional dimensions in the hypothetical example of the driver caught in traffic. For example, temperature, fatigue, frustration, and anger could be dimensions that change as the driver inches along. The more dimensions we have, the more detailed picture we have of the situation, and the more complex our knowledge becomes.

A useful exercise for higher-level students, would be to describe the limitations of two- and three-dimensional models of four-dimensional objects. This emphasizes the point that all representations we make of four-dimensional objects have limitations. By puzzling over the question of limitations, I believe I have a more sophisticated understading of a four-dimensional space. I think that noting the compromises of three and four-dimensional objects in two-dimensional space, and 4D objects in three-dimensional space will also improve students' conceptualization of 4D space in general.

Examining computer-generated pictures of the swallowtail catastrophe, I can see that color and shading our useful tools for creating the illusion of dimensions. A good exercise for middle-level and upper-level students would be to draw or, even just observe, a series of simple objects that utilize colors and shading to produce this illusion. Actually understanding how these techniques are used can enhance student's understanding of two-dimensional representations.

Are two-dimensional representations of four-dimensional objects even less concise than representations of three-dimensional ones?

In making the jump from a line to a square and from a square to a cube, I'm comfortable with the idea that one can move a line and a square perpendicularly to itself. These representations work for me in two-dimensional pictures. However, I have a lot of difficulty in the idea of trying to move a cube "perpendicularly to itself." What direction is perpendicular to every edges on a cube? The fourth-dimension I understand is the answer. But it seems misleading to say or represent this transposition on two-dimensional paper, or even in a three-dimensional model. Even with a 3D model, the transposition is in an arbitrary dimension that is not perpendicular to every side.

What are "four-dimensional figures projected into three-dimensional space."? (p. 10, caption)

Can an accurate visualization of an object in the fourth dimension be accomplished?

It seems that even by attempting to visualize a four-dimensional object, I am still making compromises. In other words, I'm still creating three dimensional objects in my head when I try to imagine four dimensions. Can even a conceived four-dimensional object be accurately visualized?

What is useful about studying the fourth dimension practically in a world that is, for most human purposes, a three-dimensional world? Are there any applications?