Exercises for Chapter 1, Beyond The Third Dimension

Jeremy Kahn

Uses of Dimension

Banchoff (p. 7) describes several ways of dealing with the concept of dimension outside of those that are the three cardinal dimensions (x, y, z), including, for example, recording several measurements of a body. He chooses foot size, height, and armspan as the three numbers, in this case. Similarly, he describes arranging the coordinates of an appointment by defining the intersection of two named streets and the number of the floor on which the appointment is to take place.

But these two examples have certain things in common: the coordinates being measured have both the same class of measurements (length) and all three coordinates are mutually perpendicular. Are we limited to using dimensions that share the same coordinate type? Must the dimensions described be perpendicular?

If the dimensions described are not perpendicular, what kind of space (what kind of graph paper?) are we describing?

If the dimensions are not the same class of units (if one cannot convert from the units of one dimension into the units of another) what does a point in the space we have described represent? Is it the same as a point in (x, y, z) space? How and how not?

Analogy and Progression

If we truly understand a theorem in plane geometry, then we should be able to find one or more analogies in solid geometry, and conversely, solid geometry theorems will often suggest new relationships among plane figures. Theorems about squares should correspond to theorems about cubes or square prisms. Theorems about circles should be analogous to theorems about spheres or cylinders or cones. (B3D, p. 8)

What do squares have in common with cubes and square prisms? How do your ideas for this question correspond to what circles have in common with spheres, cylinders, and cones? (There is obviously no one right answer.)

Are there any other kinds of shapes that should be listed in these progressions?

What are they and why should they be included?

Extra Credit: Are there any shapes that should be included in both progressions?

The Strengths of the Computer

A computer helps visualize higher dimensions. Some of the earliest computers were first designed for dimensional calculations when parabolic arc calculations, recorded in a book were needed for artillery captains and the calculations became beyond the reach of teams of human calculators. A graphics computer can show us three-dimensional images, or so we think, of nearly anything we choose. Most useful for higher-dimensional mathematics, however, can be the projections of 4-space objects onto 3-space and the rotation in 3- and 4-space for us while we watch.

But what are we really watching? What dimension does an image on a computer screen really have?

How does the computer convince us that we are observing a 3-space object if we are really watching a 2-space flat-monitor? What does this tell us about the nature of our sight?

Given the answers to the above two questions, what are we really watching when we watch a 3-space projection of a 4-space object rotating on a computer screen?