A video of eigenvalues of a 200x200 real matrix with entries defined by functions of time.

Click through for much bigger on Vimeo.

eigenvideo 1 from Matthew Conroy on Vimeo.

A new version of my dice collection. Problem 10 is new.

I made a similar animation over ten years ago, and have been looking for a nice way to do this since. I finally worked out how to add a java library to Processing (processing.org) which did the trick, making it really easy to experiment with this.

The idea is to take a square matrix and plot its eigenvalues (as points in the complex plane). Then vary one entry of the matrix (in this case, back and forth over a range of values), and plot the eigenvalues as this entry changes. Some of the results are quite visually interesting.

The ij-th entry of the matrix is M[i][j]=sin(cos(j)+j*cos(i+cos(2*j))).

I've been working with tikz/pgf, a system for creating figures in LaTeX. I've been encouraged to do this partly as a result of my frustration with Inkscape, which is excellent in places, but crashes often and frequently fails to be able to do exactly the thing I need.

Today, I figured out the use of variables in tikz. Here is an example:

This example shows the use of variables, nested for loops, and mathematical calculations.

Here is the output:

I have added to my dice problems collection pdf. I added a solution to the question "What is the expected value of a single player in the game of Threes?" assuming the player is playing optimally.

You can read about Threes at Wikipedia.

My collection is here. The Threes solution starts on page 31.

I've been playing around with envelopes lately. I've got a pdf of examples I'm adding to. Here is one nice one of shifted (both x and y) sine curves.

I don't know yet what the bounding curve is...

I was thinking about the motion of stationary points on a falling ladder, and then thought: what if the point moves along the ladder (at a constant speed) as the ladder falls? So I wrote this applet.

Bigger and explanation at Vimeo: just click the darned thing.

Three new videos in my series of videos which asks the question: if I make a grey scale image in which each pixel is colored as a function of the set of distances from that pixel to a set of moving particles, what kind of images/videos are possible?

These are bigger, and have explanations, on Vimeo (just click them).

I've been wanting to try some multidimensional scaling since 1998, and now I've tried it.

Multidimensional scaling is this fantastic concept whereby "objects" which you can place in a (possibly) highly dimensional space (i.e., to each object you can associate a (perhaps large) finite set of values) can be approximately placed into a lower-dimensional space for better visualization and perhaps to get a better understanding of the really important features that distinguish the objects. The crux is to generate a set of distances between your objects. From this set of distances, after a dimension is chosen (say, 2), the objects are placed into a two-dimensional space as best as possible, in the sense that the distance information is as accurately represented by the two-dimensional coordinates as possible.

I first encountered this idea in Larry Polansky's seminar on timbre at Dartmouth, in 1997 or 1998, while I was wallowing in unemployment before getting my first post-PhD job. The example I most remember is a study done in which people were played pairs of sounds and asked to rate how "similar" the sounds were (i.e., to specify a distance between pairs of sounds). Run through the MDS process, a two-dimensional model was found to be reasonably accurate. The fun part is to then figure out what, if anything, those two dimensions represent. In the case of the sounds, one of the dimensions seemed to be attack time, and I forget what the other was (I'll have to fill this in later).

Since then, I've been meaning to try MDS out. I recently found out that the horribly named statistics software R does MDS, so I thought I'd give it a whirl. But what to try it on? I decided I would measure the letter frequencies in a bunch of Dickens novels. I then defined the distance between Dickens novels to be the euclidean distance based on these frequencies (i.e. the square root of the sum of the squares of the differences in frequency for each letter). Here is the result!

Not too bad! A Christmas Carol and Pickwick Papers are definitely the outliers. I don't exactly know why, but here is a table with some data:

I created this table to show the maximum and minimum frequency of occurrence of each letter in Dickens novels.

You can see that A Christmas Carol has a number of extreme letter frequencies. This could just be due to its shortness: I imagine longer works might tend to smooth out their frequencies. Pickwick Papers also has a number of extremes; this is Dickens first novel, so perhaps he was using odd choices of letters. Or maybe it just that the work "Pickwick" appears alot, and this might explain the high occurrence of the letters c, k, and p, though why Great Expectations is more dense in 'w', I have no idea.