2,3 Turing Machine Type 1,2

The above image comes from Stephen Wolfram’s blog announcing proof of the simplest universal Turing machine. Believe me, I’m not about to describe a Turing machine, but I will happily describe my utter confusion when seeing the image. It initially reminded me of a Sierpinski Triangle, but obviously, something else is going on, and there’s no explanation to help! In my typical, touchy way, I got annoyed at this silly little picture, lacking any caption or descriptive text, presented as some kind of straightforward statement of the problem and its solution. Grrr…

Then I started trying to figure it out. I’ve previously blogged about the Mathematica Player and Wolfram Demonstration Project, and indeed, there’s a similar (actually, mathematically identical) example of a Turing machine in the collection. That discovery clarifies things somewhat, and yet another actually makes it quite comprehensible. In fact, reading the “New Kind of Science” prize from Wolfram’s blog, it became obvious to me that Wolfram (at least, maybe others, for all I know) has a visual shorthand that he uses in describing Turing machines—and that Mathematica evidently uses in displaying them. And I then realized that I had gotten in a huff a little hastily.

Allow me to extol a bit of personal pedagogy. In general (as I’ve mentioned in my “What Is Viz?” presentation), I divide science visualization into three basic types: 1) communication with oneself, 2) communication with a peer group, and 3) communication with public audiences. In typical astronomer parlance, I will refer to these as Type 1, Type 2, and Type 3 visualizations. In this blog, I most often comment on Type 3 visualizations (those addressed to a broad audience), and I initially mistook the above image for that type.

But it’s not. Instead, I think Wolfram has devised a Type 1 visualization of the Turing machine that, thanks to Wolfram’s influence, has transitioned to a Type 2. Presumably, there’s some small audience of “peers” out there for whom the above makes plenty of sense—perhaps they can even extract useful information from it.

Mathematica Mojo

Visiting Chicago for the annual meeting of the Astronomical Society of the Pacific, I attended a one-day session on astronomy visualization (the usual suspects). Lots of things caught my attention, but I thought I’d highlight the Wolfram Demonstrations Project, a collection of visual (non-Java) applets that utilize the free Mathematica Player.

The image above is a snapshot of the “day and night world clock” from the collection. Nothing too special about it, particularly as a snapshot, but within the player, you can fiddle with the time of year and time of day to see how the dividing line between day and night changes over the course of the year. As a kid, I was always fascinated by the Geochron clock at my local planetarium, so perhaps I’m just being nostalgic. Still, I enjoy watching the terminator’s projection on the flat surface, bending through the course of a year.

Among the other demonstrations, I like the Cepheid light curve, too, as well as the hydrogen orbitals and the inverse Hilbert matrices.

Tinkering with Polyhedra

Another terribly brief report from the Gordon Conference on Visualization. The chemistry bias comes to the fore again, with this spiffy look at polyhedral models of molecular structures. This is just one of several types of molecular model kicking around, but in the words of the supporting web documentation… “The polyhedral model, where a cation surrounded by its anions is represented by a single polyhedron, is useful for visualizing how structural components fit together. Only atoms belonging to complete polyhedra are shown.”

Hmm, well, maybe, two things won me over: first, there’s a model kit (that uses pompoms), but that’s complemented by a webpage full of links to virtual interactive structures (e.g., sodium chloride, as above).

I’ve been thinking a fair bit about these kind of complementary experiences lately (not just because of the conference). Being able to manipulate a physical model, tinker around, and assemble things with your hands stimulates the brain in one way; fiddling with an electronic model, tinkering around, and assembling things with your mind’s eye stimulates it in another. Taken together, the experiences can be powerfully reinforcing.

Just a thought, anyway.

Oh, and I like the way you can make the polyhedral surfaces partially transparent in the interactive, virtual version.