Many of the mathematical objects I work with are extremely hard to visualize in your head. Tools like Mathematica are certainly a great help, but sometimes the flat image projected on the screen doesn’t allow one to appreciate the full subtlety of something I’m interested in. There are a lot of possible ways to help with this: 3D printing of mathematical models, facilities like the Tufts Center for Scientific Visualization, or using a 3D display. But it would be much more convenient to have something simple to use on a laptop, and also that could be usable in the classroom without too much special equipment.
One potential solution is to use anaglyphs—images where the left and right views are encoded by color—which require a pair of glasses with complementary colors. You’ve most likely seen the Red/Cyan glasses before as they’re common for toys and available very cheaply. The image at the top of this post is an anaglyph.
I’ve made a Mathematica package that can automatically generate images (and potentially movies!) from any Graphics3D object. This includes the output of most Mathematica commands that give a 3D result such as Plot3D, ListPlot3D, ParametricPlot3D etc. The package is extremely simple to use; programs can generate anaglyphs using a single simple function call. Some examples are included on this page. You’ll need a pair of Red/Cyan glasses to view them. I found that using these in my classes resulted in students gaining a much deeper appreciation for some of the mathematical objects in question, made them far more fun and generated a great deal of excitement in the students that helped their engagement. Hope you enjoy using the package!
Installation
You can download the Mathematica package here. A notebook containing examples from my physics classes is also available.
To try the package, simply open it in Mathematica and click the “Run Package” button that appears in the top right hand corner of the file. You’ll need to do this every time you restart Mathematica and so, if you like it, you will need to install the package somewhere more permanent. Mathematica looks in a number of special folders on your computer for such files, which you can find by executing the command $Path from inside a notebook. On a Mac, a good place to do so is ~/Library/Mathematica/Autoload — the Anaglyph package will then be available for you to use whenever you need it.
Tips
• Use Manipulate to make rotatable anaglyphs as shown in the example document.
• Use a lossless file format like PNG as opposed to a lossy one like JPEG, because the lossy methods tend to mix the colors in a way that messes up the anaglyph.
• By exporting a sequence of slightly different images, it’s easy to make anaglyph movies!
Further Examples
Dupin cyclide, a mathematical surface important for smectic liquid crystals:
A spherical harmonic function, important for quantum mechanics in 3D:
The Mathematica polyhedron:
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