The Amateur Mad Scientist – Episode 2

Haha! And you thought this was gonna be another of those Life of an English Major “series” that I lose interest in two weeks later and forget about. But no! There are now at least two episodes of the Amateur Mad Scientist. In the last episode, I put five pillbugs in a nasty-ass recycled deli container and tried to force them to breed. This one’s not quite that mean, if for no other reason than no macroscopic organisms are involved. I present to you: the Super-Ghetto Biosphere.

For an enclosure, I decided to use a little glass jar that totally didn’t used to have tartar sauce in it.

To that, I added sand enriched with organic material. Sand I totally didn’t steal from my hermit crabs. And then the water. Nasty-ass water. Water, like, swimming with little critters. Paramecia ‘n’ shit, yo. Sorry…that joke was fucking stupid. But anyway: the water is also fortified with organic matter (not floating aquarium-snail poop, I promise).

And now the keystone of the entire ecosystem: a cutting of the infamously tenacious water wisteria plant (Hygrophilia difformis). Because if experience has taught me anything, there’s nothing plants like more than being sealed in jars.

So that’s the setup as of 6-22-2011. I’ll post pictures over the weeks to come detailing my resounding success (Ha!). Watch this space!

Update: As pf 7-2-2011, the plant is still (somehow) alive, and has deigned to throw down at least one root. Also, algae.

Update: As of 7-7-2011, the plant is still, in spite of my worst efforts, alive, and the algae has proliferated and started consuming all the detritus I was too damn lazy to screen out.

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Visual Numbers: The Next Generation (I)

Hey, just because I’m an English major now doesn’t mean I can’t enjoy a bit of math on the side. Therefore, I present to you Visual Numbers: The Next Generation (Part I, insert obligatory Star Wars joke).

I’ve always been a big fan of fractals. I was imagining fractal-esque recursive structures back when solving 2x + 5 = 0 seemed intimidating. Recently, in my wanderings through the hallowed halls of physics, I stumbled across the idea of generating fractals by coloring each pixel of an image according to the final state of a differential equation, with the x and y coordinates of that pixel as input parameters. Of course, being an incompetent programmer, I haven’t been able to write a stable integrator for a differential equation, but I’m just smart enough to manage simple stuff like the discrete logistic map, where every new x is computed according to the formula x = λx(1-x), where λ is a constant. The other day, I had a brainstorm and decided to create a two-dimensional version of the same sort of thing, only with x as a vector and replacing x at each step with F(x), where F is a vector function. Basically, all that means is that I now have two integer variables to play with, which allows for more interesting behavior and allows me to create pretty pictures like this:

In this image, each pixel is colored according to the following rule: all pixels start out white. Then, each pixel is turned into a vector v(x,y), where x and y are the pixel’s coordinates. The function v(x,y) = v(-x.y,x.x + floor(0.5 * x.y)) is then applied repeatedly for a fixed number of steps (in this case, 750). If at any point during iteration the vector reaches a previously-visited point, then that point forms a closed, repetitive orbit in 2-space and the pixel is colored black. If this doesn’t happen, the pixel is left white.

The shapes here are pretty nice, even if I do say so myself. From some prior experiments with similar mappings, I think that the smaller black ellipses with banding and satellite blobs represent orbits shaped like three elliptical orbits connected into a weird triangle, and that the large oval with the large banding represents the “spirograph-like” orbits.

The really neat thing about mappings like this is that they’re a (relatively) computationally-inexpensive alternative to the differential-equation mappings I discussed earlier. I’m no great mathematician (obviously), but I get the feeling that these two-dimensional mappings are discretized analogues of the mapping that generates the ever-beautiful Mandelbrot Set.

Interestingly (and here I’m playing mathematician again), you can write this type of mapping this way:

F(v) = v(ax + by,cx + dy). In the case of the above map, a = 0, b = -1 , c = 1, d = 0.5.  With a = 0.5, b = -1, c = 1, and d = 0.1, you get a beautiful spiral pattern:

It’s a little hard to see because the orbits are so dense, but this pattern is actually fractal, too (or seems to be): there are smaller spirals to the top-center and left-center of the big one, and what looks like unformed proto-spirals in between those.

And this lovely pattern is created by a = 0, b = -1, c = 1, d = 0.1. Note the eleven-pointed stars in the upper right and left and the lower right. Watch this space for more mathematical prettiness.

The Size of the Sun

Sun and Earth

In the above image, the tiny red rectangle towards the middle of the Sun represents (approximately) the surface area of the Earth. Meaning that the sunspot above it is almost big enough (approximately; some perspective effects come into play) to encompass the entire surface of the Earth. Odds are that everything you have ever done or seen has taken place in an area smaller than a sunspot. The universe is odd.

(Image courtesy of NASA’s remarkable Solar Dynamics Observatory)