Chapter 10.7, Problem 48E

Albert Einstein's theory of general relativity is concerned with the structure, or the geometry, of the universe. In order to describe the universe, Einstein discovered that he needed four variables: three variables to locate an object in space and a fourth variable describing time. This system is known as space-time. Because we are three-dimensional beings, how can we imagine four dimensions? One interesting approach to visualizing four dimensions is to consider an analogy of a two-dimensional being struggling to understand three dimensions. This approach first appeared in a book called Flatland by Edwin Abbott, written around 1884. Flatland describes an entire civilization of beings who are two dimensional, living on a flat plane, unaware of the existence of anything outside their universe. A house in Flatland would look like a blueprint or a line drawing to us. If we were to draw a closed circle around Flatlanders, they would be imprisoned in a cell with no way to see out or escape because there is no way to move up and over the circle. For a two-dimensional being moving only on a plane, the idea of up would be incomprehensible. We could explain that up means moving in a new direction, perpendicular to the two dimensions they know, but it would be similar to telling us we can move in the fourth dimension by traveling perpendicular to our three dimensions.

Group members should obtain copies of or excerpts from Edwin Abbott's Flatland. We especially recommend The Annotated Flatland, Perseus Publishing, 2002, with fascinating commentary by mathematician and author Ian Stewart. Once all group members have read the story, the following questions are offered for group discussion.

a. How does the sphere, the visitor from the third dimension, reflect the same narrow perspective as the Flatlanders?

b. What are some of the sociological problems raised in the story?

c. What happens when we have a certain way of seeing the world that is challenged by coming into contact with something quite different? Be as specific as possible, citing either personal examples or historical examples.

d. How are A. Square's difficulties in visualizing three dimensions similar to those of a three-dimensional dweller trying to visualize four dimensions?

e. How does the author reflect the overt sexism of his time?

f. What "upward not northward" ideas do you hold that. if shared, would result in criticism, rejection, or a fate similar to that of the narrator of Flatlandl?