Fourier Approximators and your Intuition

I've mentioned before that I'm a real fan of anything that improves your physical intuition. Previously, I noted a nice, compact color-coded form of the Fourier equation. Now, from Ben Grawi comes this very cool visualization tool. 

It lets you build up a few complex waveforms (square, triangle, sawtooth) with a variable number of terms. If you play with it a bit, you can quickly see just how many (or, how few) terms it takes to get a decent approximation of what you're looking for.

There are three things that I especially like about this. First,  you can isolate different terms, and see what they look like by themselves. Secondly, this provides perhaps the best intuitive insight I've ever seen into "overshoot" (if you are deeply, nerdily interested, then I suggest you follow his link about the Gibbs Phenomenon). I also really like how you can visually see the impact of phase: pick some settings, and then toggle back and forth between "square" and "sawtooth", and you'll see what I mean. 

Anyway, this thing is gorgeous, and a fantastic little intuition pump (especially if you're a visual learner for whom the math itself isn't satisfying).

Visualizing Building Vibrations

From Byron's (Course 3, '98) alma mater comes an interesting visualization technology:

They appear to be using off-the-shelf (if high-speed) cameras and feeding data to an algorithm that manages to pick out sub-pixel-sized displacements, exaggerating them in video so that you can see what is going on. Abe Davis does an excellent job describing some of the underlying technical concepts in a TED talk from March. In that talk, he claims that they can actually take advantage of the "rolling shutter" problem in consumer-grade cameras to their advantage.

The civil engineering research team is touting the structural monitoring possibilities: it's non-contact, generates global (rather than local) data, and can be executed with cheap, pre-existing hardware. 

I'm not sure that this technique (without specialized hardware) could easily be made sensitive enough to displace traditional accelerometers in the nanotech labs that we help design. (And how would we support our camera, anyway?) 

But I'm intrigued by some of the visualization possibilities. Specifically, I'm sure you could use this to show how horizontal vibrations in buildings tend to be global, whereas vertical vibrations tend to be highly localized. Or, I'm guessing you could use this to see just what it looks like when people walk on a structural floor. Or, the difference between "soft" office-style floors vs. "stiff" waffle floors like you might see in a semiconductor fab. 

Anyway, this is very cool stuff. I'm all for anything that helps train our physical intuitions.

Fourier, now in Color

This compact explanation of the Discrete Fourier Transform is a few years old, but it's new to me; from Stuart Riffle via David Smith:

Putting the DFT into a single sentence is nice, but the magic is in the color coding. It really helps connect your physical intuition to the math. I love how this draws your attention to the different mathematical machinery at work (spinning, averaging, etc).

Check out David's write-up on Revolution Analytics or Stuart's (archived?) write up on AltDevBlog for more.

If you have no idea what I'm talking about: this is the core math behind "frequency decomposition," the technique that allows you to take a recording (or time history of data) and figure out the frequency content. As you might imagine, this is pretty important. Check out the Wikipedia entry for Fourier Series; it's got some gorgeously animated illustrations.

The "invisible architecture" of sound

If you're an architect with an interest in acoustics, then I hope you've seen this TED talk from Julian Treasure. It's a few years old, but I get reminded of it periodically. Anyway, there's a growing interest in these kinds of issues in the acoustical community, and I think it is important to get architects on board. 

Hands-On Building Vibration Simulation

The Perot Museum in Dallas has a shake table that lets you experience simulated earthquakes. Having moved from Dallas to San Francisco, I have to wonder why they have this Dallas, but not at the Exploratorium. 

Anyway, for me the more interesting exhibit isn't the one that lets you stand on a shaking floor; its the one that lets you build up a 3D model of a building and see how it moves around.

You basically build up little stick structures, and then carry them over to a shake table. The table is computer-controlled with a pretty slick interface that lets you vary base excitation frequency and amplitude. And the text does a good job of making you ask the right questions:

Our firm doesn't do any seismic work, but this is a great illustration of how lateral vibrations work in real buildings. Unlike vertical vibrations -- which are highly localized within structures -- horizontal vibrations are a global phenomenon.