Mathematician Michael Starbird of the University of Texas at Austin refers to the fourth dimension as “eerie, mysterious, and exciting.” It didn't take him long, in a recent Distinguished Lecture simply titled “The Fourth Dimension,” to persuade a large crowd at the MAA Carriage House Conference Center that he was right.
Starbird began his presentation by noting that the process of studying the fourth dimension is an example of what is so wonderful about mathematics: the ability to create new ideas. One approach is to take what we already know or sense and expand it beyond that reality. Understanding these new realms often requires a strategy of "retreating" to the known, then extending properties elucidated for the familiar realm to the new one.
To build a picture of the fourth dimension, Starbird began by looking at properties of objects in realms familiar to us: zero-, one-, two-, and three-dimensional spaces. Beginning with a series of points, he elaborated on each realm by creating simple graphs to let members of the audience visualize the dimension in which they were working. Stating that three-dimensional space could be created by stacking two-dimensional spaces, Starbird suggested that four-dimensional spaces could be created in the same manner, by stacking three-dimensional spaces.
Starbird then posed the question of what a cube, a familiar object in three-dimensional space, might look like in four-dimensional space. Instructing the audience to once again retreat, Starbird used a homemade cardboard cube and graphed slices of a three-dimensional cube to show the audience that a four-dimensional cube would have three-dimensional cubes as its faces, much as two-dimensional cubes (or squares) serve as the faces of a three-dimensional cube.
To more clearly visualize a four-dimensional cube, Starbird once again retreated, this time to unfolding his three-dimensional cube and applying the same sequence to unfolding a four-dimensional cube. The unfolded result is similar to the shape of the cross that Salvador Dali depicted in his painting Crucifixion (Corpus Hypercubus).
Starbird then discussed other ways to construct the fourth dimension. One involved taking half of the previous dimension and rotating it around its boundary, which is two dimensions lower. Using this idea to create four-dimensional space, Starbird explained, you would sweep half of three-dimensional space around a two-dimensional plane.
To close his talk, Starbird touched on the topic of linking spherelike objects in the fourth dimension. Again retreating to lower dimensions, he discussed how, on a line, a pair of points represents a sphere, and how one pair of points links another pair of points in one-dimensional space. In two dimensions, Starbird said, a circle and a pair of points link each other. Three-dimensional space gives you the option of either a pair of rings, or a two-sphere and a pair of points that link. Using this evidence, Starbird concluded that a two-sphere and a one-sphere would link each other in four-dimensional space because the sum of their dimensions was one less than the ambient dimension, much like the instances in lower dimensions.
Michael Starbird is a University Distinguished Teaching Professor at the University of Texas at Austin. He received his B.A. degree from Pomona College and his Ph.D. in mathematics from the University of Wisconsin-Madison. He has been in the UT Department of Mathematics since 1974.
Starbird has received more than a dozen teaching awards, including the MAA's 2007 Haimo Award for Distinguished College or University Teaching of Mathematics. Starbird's books include, with co-author Edward B. Burger, the award-winning mathematics textbook for liberal arts students The Heart of Mathematics: An Invitation to Effective Thinking and the trade book Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas. With David Marshall and Edward Odell, he co-authored the MAA textbook Number Theory Through Inquiry. His Teaching Company video courses include "Change and Motion: Calculus Made Clear," "Meaning from Data: Statistics Made Clear," "What are the Chances? Probability Made Clear," and "The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas." These courses reach tens of thousands of people in the general public annually.—R. Miller
Number Theory Through Inquiry
Michael Starbird, David C. Marshall, and Edward Odell
150 pp., Hardbound, 2007
This MAA Distinguished Lecture was funded by the National Security Agency.