CS798 Assignment 2 extensions

What follows are some suggested non-trivial extensions to the implementation component of assignment 2. You are welcome (and indeed encouraged) to dream up your own ideas for extensions. These are presented very roughly in increasing order of when they occured to me. The harder ones make fine topics for final projects in the course.

Note that not all extensions are created equal. You don't get full marks just for having an extension. If you want to think of me as cruel, consider this as being similar to the "subjective marks" component of the cs488 final project.

• Implement a fancier rendering style for edge shapes, such as subdivision curves or splines.
• Add the ability to control the positions of tiling vertices. Different isohedral types require different numbers of parameters. Note that you can find sample code for tiling vertex parameterizations on my Escherization web page, though as I said in class these could be expressed more succinctly as matrices.
• Add support for other topology types. The biggest bang for your buck comes from 3⁶, but I find that others such as 3².4.3.4 and 3⁴.6 are fascinating.
• Implement a facility for colouring tiles, perfectly or otherwise. Because we're dealing with 4⁴, this could be pretty easy for starters, but evolve into something more interesting.
• Include some facility for including arbitrary decorations on tiles. Make the decorations contingent on the tile's colour.
• Implement a fancy, efficient method for rendering the patch of tiles visible in the viewer. Render only those tiles that intersect the view region in world coordinates.
• Implement spatial or temporal metamorphoses. This is pretty easy in the case of 4⁴ tilings with congruent tiling vertices -- you just need a morphing function for piecewise linear paths. But some experimentation could lead to many aesthetic possibilities.
• Add support for some dihedral (or more) tilings. It would be particularly interesting to support 2-isohedral tilings where there are different numbers of A and B tiles in any given translational unit.
• Add support for one or several anisohedral prototiles. You'll need to find some examples (a good start is John Berglund's page on the subject). Then you'll need to parameterize the space of tile shapes available for the combinatorial structure of that prototile.
• Add support for aperiodic prototiles, such as Penrose tiles. My thesis covers this case; other prototile sets, such as Ammann's, would be great too.
• Implement the anti-mercator projection to turn an infinite strip of tiles into a design where tiles shrink to a point at the origin (see, for example, Escher's Development 2). I won't talk about this transformation until near the end of the course. In the meantime, you can find a mention of it, together with a reference, in Chapter 5 of my thesis. Or you can come talk to me.
• Figure out a way to manipulate tiling vertices directly using constraint satisfaction rather than by adjusting abstract parameters.
• Implement Escherization.
• Use a fragment shader to sample the tiling efficiently and produce a real-time antialised vector drawing of it.
• Implement some way to cause your tiling to pass discontinuously into a different but related isohedral type if edges degenerate or tiling vertices bifurcate. Or, have a way to break symmetries in the prototile when symmetric edges are chosen to be broken. A really nice way to do this would be to use Celine Latulipe's research on bimanual input using two mice. One mouse can hold an edge, and the other can drag a symmetric copy.
• Support some 3D isohedral tilings.
• Support isohedral tilings in spherical geometry. See Yen and Séquin, Escher sphere construction kit (I3D 2001).