B3D Chapter 7

Lisa Eckstein

This chapter has more beautiful illustrations than any previous chapter, but it is also much harder to understand. I am really lost - I don't know if this is because the material is tougher or if the explanations are less detailed than in earlier sections.

I am not entirely clear on what a configuration space is, or what a graph of one looks like. Is the diagram on page 133 a graph of a configuration space?

Like Irene, I am confused about the way the word "dimension" is used in this chapter. I understand that different numbers of coordinates are needed to describe different objects or events - does "dimension" refer to the dimension of the configuration space, or the dimension of the actual thing being described? I guess in this chapter we are returning to the concept of dimension that was presented in Chapter 1 with the examples of the driver in traffic and the family members recording their heights.

I had a hard time grasping the concepts of a geometry of segments and lines, though I feel that I could understand it if presented with a careful and detailed explanation.

Finally, I was completely baffled by the paragraph on page 151 containing the sentence: "The points of a 45-degree line perpendicular to a horizontal line at a point correspond to circles in the plane that are tangent to the line at the point." What is the line in a 45-degree relationship to? Surely not to the perpendicular line? What is going on?!?

There were some other smaller details which I did not understand, but these were the main points I had trouble with. I hope I will be able to comprehend these subjects with further study, since the concepts appear to be very interesting.


All I can come up with as an exercise is to experiment with string art as described on pages 142 and 143. In elementary school we did a little bit of this, but certainly our art was nothing like Naum Gabo's amazing sculpture! I am going to try to make a three-dimensional hyperbolic paraboloid like the one in the illustration.

Prof. Banchoff's Response