This chapter is very interesting, as it brings up a lot of issues of clear presentation of data. The pendulum example in particular is fun to think about...Here are some questions - try to answer them.
After having seen the 4-D representations of the pendulum oscillation data (the Clifford torus on the hypersphere), do you think you could reconstruct the actual experiments or pendulum swings based on the data?
Would you be able to recognize the pendulum swings after seeing the "patterns" pointed out by the data?
Could you explain the data or experiment to some one who was unfamiliar with the experiment?
If you answered "NO" to any of the above questions, try this one! If the data is not a primary source of concrete information that leads to reconstruction of the experiment, is it's abstract quality a distraction from the nature of the experiment?
Now, we are most familiar with data presented in two dimensions. One standard form is the X vs. Y horizontal info. vs. vertical plotted data. A version of this is on p.132, the rehabilitation charts. The ability to plot data in greater than 2 dimensions provides a great deal of freedom and as Banchoff writes, the visualization of "...more subtle relationships..."(137) What is interesting is that there is also a great deal of responsibility, as a conveyor of data, to use the given space (no matter how many dimensions) wisely! The pendulum orbit examples are very beautiful, and once the data is on the computer it can be examined from many angles. We are interested in clear and useful visualization of data that takes place in n dimensions, but with many variables, such that a many-dimensional presentation of the data is favored over a conventional 2-dimensional presentation.
To clarify this, think of a traditional pie chart in 2-D. A circle divided up into sections, with colors for each section. we can represent 2 variables here with the section size and the color. If you have one more variable, you can elevate the pie into 3 dimensions, to study another amount variable. And so on. Are there cases when many-dimensional presentation become more effective that a 2-D representation using devices such as color, size, texture, etc.? Are there cases the other way around?
There are 2 books by Edward (?) Tufte, they are called The Visual Display of Quantitative Information and Visualizing Information. These books are really neat and talk about similar subject, which is very relevant to dimensionality, the effective display of data, such that the data gets it's primary job (conveying information) done. One of them (forget which) even has a chapter in the beginning that talks about Flatland.
Is the exact definition of a configuration space: The data representations of all possible positions of the object?
Brown home page.