Plotting functions of many variables

Question

What are the different ways to visualize functions of many variables? Is it possible to represent them as single lines like it's possible in 2d and 3d graphs?

Answer 1

If you have an n-dimensional function, you can observe the behavior of two dimensions at a time using contour lines (see https://en.wikipedia.org/wiki/Contour_line). By looking at several two-dimensional slices (aka contour graphs or contour plots) of your n-dimensional space, you may be able to integrate in your mind what the n-dimensional function really looks like.

You can use a similar approach to look at 3D volumes instead of 2D slices. A four-dimensional space requires 6 2D slices (xy, xz, xw, yz, yw, zw) but only 4 3D volumes (xyz, xyw, xzw, yzw).

Color can sometimes stand in to represent dimensional values: hue, saturation, and value can provide up to three additional dimensions of representational power, but since one cannot graph without some use of the HSV space (both for foreground and background), these dimensions are not quite as independent as contour lines.

Thus a six dimensional function can be visualized as a HSV-encoded 3D plot and a seven dimensional function can be visualized as seven six-dimensional representations.

A dimension can be varied across time to create an animation. Thus a seven dimensional function can be visualized as an animated HSV-encoded 3D plot and an eight dimensional function can be visualized as eight animated six-dimensional representations.