More from yesterday. Of course I see the following .gif in the course of my daily reading, which blows all my animations out of the water:

This is a sequence of 2-D slices of the human body, collected from the Visible Human Project. For those who don’t go to the wikipedia page, this data was collected in a much more… invasive manner than the MRI I described yesterday. Specifically, a cadaver was “frozen in a gelatin and water mixture”, then they grinded away 1mm, take a picture, and repeat. There is also a female data set with 0.33mm resolution. Also, apparently this guy was a Texan murderer, which is a far less pleasant tie-in than, say, Tibor Rado being at Rice before solving Plateau’s problem, or Milgram being at Syracuse before proving his theorem.

There is also a fantastic viewer for this dataset hosted here.

In any case, this sort of image is a good introduction to the next logical place I was going to go with visualizing data. That is, keeping track of five dimensions of data at once. Yesterday’s images had three spatial dimension and one color dimension. Today’s .gif has two spatial dimension, one color dimension, and one time dimension.

Long exposures and fire: another way to "see" another dimension.

This suggests (and I plan to provide examples of this) having three space dimensions, one color dimension, and one time dimension. A great example would be watching a person’s temperature change over time, or how hot an oven is over time. For today though, we have two somewhat boring examples, in that the color and time dimensions are equal to each other (i.e., I colored these by recording the height, and making that the intensity.

This first is an implementation of a variant of the heat equation with “random shocks” introduced, and Von-Neumann boundary conditions (which means: no energy escapes the system). It is meant to model how head would travel, and the “random shocks” are small points of heat introduced into the system.

The second image is an implementation of a variant of the wave equation (in both these cases, when I say “variant”, I am allowing just a little bit of energy to leave the system so that there is not too much “heat” at the end of the simulation… things tended to just white-out without this). The von Neumann boundary conditions are much more evident in this one, where you can see waves “bouncing” off the walls.

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Speaking of slicing: suppose we slice either of the two graphs by a fixed horizontal plane z=z_0. Can the number of pieces above the plane increase with time (without new sources added; i.e., for homogeneous equation)? For the wave equation the answer is clearly yes: two colliding waves produce a peak that is taller than either of them. What about the heat equation?

Polya proved in 1933 that for the one-dimensional heat equation the number of sign changes of u(x,t) does not increase with t. The paper is

http://onlinelibrary.wiley.com/doi/10.1002/zamm.19330130217/abstract

(which I can’t access right now.) I don’t know of any two-dimensional version of this result.