General questions and comments…
On page 58 of B3D, there are two pictures of hyperplane slices of a polynomial. What is the polynomial for the two graphics?
Chapter 3 analyzes slices of three dimensional and four dimensional objects. What do the slices of some other objects look like? For instance, in a sphere, the slices are the same no matter what part of the sphere the sphere is suspended from. But, what about an ellipsoid? In what way are these slices similar or different from the sphere?
In this chapter, it is stated that time is a fourth dimension,
and to A Square in Flatland, time is a third dimension.
Can it then be stated that time supersedes all dimensions? Or,
is time just relative to dimension? Can time truly be a mathematical
dimension? Also, when time is used as a fourth dimension, what
is the fifth dimension?
Professor Banchoff's Comments…
Good questions.
The polynomial equations for the figures on page 58 are f(X,Y,Z) = (-8X^4 + 8X^2 - 1) + (-8Y^4 + 8Y^2 - 1) + (-8Z^4 + 8Z^2 - 1) where the interesting contours occur at -3, -1, 1, and 3. You might try to look just at the x part, or the x and y surface?
What about ellipsoids? What do you suspect? There are some general principles that tell you that ellipses and ellipsoids behave rather like circles and spheres when you slice them by parallel planes, but very differently if you "slice" them by circles or spheres. Do you see how?
Madeleine L'Engle wanted to reserve dimension number 4 for time, so she used 5 to represent the new space dimension into which she wanted her characters to move. It's largely a matter of terminology. Or is there more to it?