Scientists get excited about the funniest things...

I feel like I need to preface this post with an explanatory cartoon:

This is me right now. *snort*

If you haven't already read about the hexaflexagon party trick, here's a youtube video explaining their history and how to make one. Basically, a hexaflexagon is a paper hexagon that you can turn inside out, and which has hidden sides within it so that new colors appear as you flex it. Watch the video for a better explanation:

Cool, huh? I made my boyfriend one the other day and he was amused for at least 30 minutes.

I've been studying for my PhD comprehensive exams recently.  Today I've been reading about linear equations. Specifically, permutation matrixes.  Matrixes are just columns and rows of numbers, like this:

This is a 4*4 matrix, but you can have matrices of all sorts of sizes, from 2* 2 to 1,587,47294 * 65 (for example. We can even visualize this matrix with colors, like this:

As you can see, 0 is dark blue, 1 is royal blue, and so on until you get to 10, which is dark red.

Now that we understand matrices, let's look at a permutation matrix.  Permutation matrices are basically just matrices that have exactly one 1 in each row and column, and all other numbers are zero.  Like this:

And here's the color representation of this matrix:

Again, zero is in dark blue, and 1 is royal blue.

The cool thing about permutation matrixes is that, when you multiply a regular matrix by a permutation matrix, it switches the rows of the regular matrix around.  It's kind of like when you flip a hexaflexagon, you get a different color.

Now I'm going to multiply my original matrix by my permutation matrix:

Every time I multiply by my permutation matrix, the rows are moved around, until on the third multiplication, we are back at the original matrix!

What happens if I use a different permutation matrix, like this one:

This time, it takes four multiplications to get back to where we started:

What about this permutation matrix?

This time, no matter how many times I multiply the matrix, nothing happens!

What's going on here?

Well, the 1 in the permutation matrix tells us how the columns will be moved around. In the first example, for row one, the 1 was in column 4, so rows four and one switched. The second 1 was in column 1, so rows two and one switched.  The third one was in column 3, so row three stayed where it was. Finally, the fourth 1 was in column 2, so rows four and two switched (remember, four had already switched with one, so row one is now in column 2.  Here it is again:

Can you figure out why, using this permutation matrix, nothing happens no matter how many times I multiply the original matrix by the permutation?

Permutation, why you no permutate?

So why is a permutation matrix like a hexaflexagon? Well, depending on how you make it, it takes you through a certain pattern a certain number of times to get back to where you started! Now you've learned two awesome math things for today: permutations and hexaflexagons!

Here's a hexaflexagon pattern to try at home.

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