A little more imaging.

Now I want to talk about slightly more extravagant ways of viewing high dimensional maps.  “Extravagant” does not mean “little used”, though.  I’ve included two pictures and one video of its use at varying-levels-of-importance.  Specifically, I want to think about visualizing maps f: \mathbb{R}^3 \to \mathbb{R}, which I have done before with the fibers of those maps.  In this case, we will use color or intensity to stand in for the fourth dimension that we are trying to display: remember, 3 for the domain, one for the range.  We will use the space dimensions for our domain and the color dimension for the range.

If the pieces of plywood making this bear had pictures of bear innards in it, it would be both way more interesting for this post, and way less reasonable to use as a bookshelf.

Examples!  One immediate problem with this plan is that if you assign a color to every point inside a three dimensional space, the space becomes opaque, and you cannot see it.  This is why with these plots, one often takes “slices” instead.  Literally in the case of this video, where a chef maps the heat in an oven by placing slices of toast in it (and I love experimental math).

For a more important example, MRIs can give an idea of the inside of a person’s body by mapping slices.

Slices of a person's head, not arranged spatially like some of these other plots.

For some more traditional math plots, I want to first imitate an oven by giving a plot in the spirit of those above, colored according to distance from the origin (more realistic oven imitations will come sometime in the future with discussions of the heat equation).  Specifically, we will plot f(x,y,z) = \sqrt{x^2+y^2+z^2}, and assign an intensity of color to each point.  Recall that if we wanted to visualize this plot instead with the fibers of the map, it would be a series of nested spheres, which you can see in this picture if you sort of squint.

Slices colored according to distance from the origin.

Finally for comparison, the map of the torus from my previous post, as well as this slice plot of the torus.  Recall the the map we are visualizing takes a circle in the x-y plane, and returns the distance of a point (x,y,z) from that circle.  Hence, the level sets are tori as can be seen in either of the figures below.  I guess which method of visualization you use depends on what you are trying to do with your data:

The picture of fibers of the map.

Slices of the torus map.

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