Wolfram Computation Meets Knowledge

Date Archive: 2008 December

Design & Visualization

Fun with Line Art

I’m constantly amazed by the wide variety of tasks people accomplish with Mathematica, everything from serious scientific research and development to fun games and puzzles. This one is more on the fun side. A few days ago I was trying to convert a raster image to a vector image. I remembered seeing some online service to do this in the past and I was trying to dig up the URL. In the back of my mind I thought I could probably do this with Mathematica, but it wasn’t immediately clear how. I spent a minute or two contemplating various algorithms one could use before realizing Mathematica already has a built-in visualization function that could do most of the work for me: ListContourPlot. This function was meant to handle elevation-like data, but a two-dimensional list of grayscale values is essentially the same thing. The first step is to get a suitable raster image into Mathematica 7. This is easy enough: just drag a JPEG file into the notebook window and assign it to a variable. Here is a picture of my handlebars after a muddy bike race.
Education & Academic

Mathematica 7, Johannes Kepler and Transcendental Roots

Everyone who has been through high-school mathematics knows about polynomial equations. But what about equations involving other functions? Say equations like x == 1 - Sin[x]. These are transcendental equations, and they show up in a zillion different mathematical application areas. But they’re rarely talked about—perhaps because in some sense they’ve been an embarrassment: mathematics has had very little to say about them. Polynomial equations and the algebraic numbers that represent their solutions have been one of the great success stories of pure mathematics. Over the past half millennium, a huge mathematical structure has been built on polynomial equations. But almost nothing has been done with transcendental equations. It’s not that they’re not important. In fact, what many people consider the very first computer—made of wood by Wilhelm Schickard in 1623—was built specifically to help in getting solutions to equations of the form x == 1 - e Sin[x]. Johannes Kepler was in the process of constructing his Rudolphine astronomical tables—and his killer technology for finding the longitude of a planet at a given time required solving what’s now called Kepler’s equation: essentially the transcendental equation x == 1 - e Sin[x]. With considerable effort, and probably computer support, Kepler made a table of solutions to this equation:

The Wolfram Demonstrations Project Goes 7

Just a couple of weeks ago we released Mathematica 7. This week we’ve made the Wolfram Demonstrations Project live with Mathematica 7. All the 4270 Demonstrations on the site run with Mathematica 7 (yes, we tested every single one of them, partly automatically, partly by hand). And we added 147 new Demonstrations that specifically make use of Mathematica 7’s features. Most of those Demonstrations were created internally within Wolfram Research, in a small frenzy of activity right around the actual release of Mathematica 7. I was involved in organizing this Demonstrations-fest. It’s very impressive how quickly it’s possible to create so much high-quality material with Mathematica. Of course, it helped that we were able to work directly with the key developers of much of Mathematica 7’s functionality—so people were often creating Demonstrations based on the very features they had implemented in the system. The new image processing system in Mathematica 7 was a particularly fertile source of Demonstrations. Charting, splines and vector visualization are other areas that have spawned all sorts of interesting Demonstrations. Here are a few of my personal favorites:
Design & Visualization

The Incredible Convenience of Mathematica Image Processing

It’s been possible since Version 6 of Mathematica to embed images directly into lines of code, allowing such stupid code tricks as expanding a polynomial of plots. But is this really good for anything? As with many extremely nifty technologies, this feature of Mathematica had to wait a while before the killer app for it was discovered. And that killer app is image processing. Mathematica 7 adds a suite of image processing functions from trivial to highly sophisticated. To apply them to images, you don’t need to use any form of import command or file name references. Just type the command you want to use, then drag and drop the image from your desktop or browser right into the input line.