Please remember that your team should submit a short description of your planned project to me via email by Tuesday 11.12.12. This description should clearly state not only the content of the project, but also what form the project will take. It is expected that every team will produce either an interactive application or animations in digital format. These products should include documentation for the intended user/audience. In no particular order, here are some ideas for semester projects.

**Dynamical systems on orbifolds.**This is a big field, with many possible specific projects. Idea is to take some dynamical system defined on $\mathbb{R}^2$ (or $\mathbb{R}^3$ or $\mathbf{S}^2$ or $\mathbf{H}^2$ or …) and figure out how to map it onto some or any orbifold that belongs to this covering space. Example: vector fields , differential equations, difference equations, cellular automata (as in*Assignment1*). (To see how to integrate ODE’s in Java, see the example*util.ODEExample*.)**Complex functions.**There are doubly-periodic complex functions, aka elliptic functions. These can be considered functions on the torus. Interpret them for example as vector fields, calculate the integral curves.**Geometry processing on orbifolds.**For example, develop software to work with triangulations of an orbifold. How to insert new points, how to perform geometric processing on the resulting triangulations. Calculate subdivision surfaces in 2D- and 3D-orbifolds. Develop tools to study embeddings of graphs on tori and higher-genus surfaces.**Software improvement.**Improve or extend an existing application such as TriangleGroup, Wallpaper, Maniview. For example:- Provide a geometric representation for 3D isometries so that one can provide a picture of the generators of the discrete group. This has been begun in the euclidean case (in
*maniview*) but not completed. **Better painting tool.**The painting tool in the wallpaper application could be better. It would be nice for example to*fair*the input path of the mouse — this means approximate the path with a Bezier curve, as the user draws. This will provide a more uniform, smoother curve, which is also easier to store off and later edit. (Of course the user can choose whether he wants to fair or not.)

- Provide a geometric representation for 3D isometries so that one can provide a picture of the generators of the discrete group. This has been begun in the euclidean case (in
**Software applications.**Here are some ideas for self-contained applications.**3D Kaleidoscope.**Implement a general 3D Kaleidoscope in the sides of an appropriate 3D tetrahedron (one with dihedral angles which are fractions of 2Pi). Equip this kaleidoscope with a set of built-in “worlds”, each of which generates its own regular 3D tessellation. For example, with the same *236 kaleidoscope it’s possible to generate tessellations consisting of regular hexagons, regular triangles, or a combination of both. Figure out how to design these worlds so that you can*see through them*to perceive the complete tessellation. This idea is the analogy in 3D to the TriangleGroup webstart in 2D.**Designing large public sculptures with spherical symmetry.**Here the model is George Hart’s construction projects based on a single tile, typically working with the symmetry group 235. The project here would consist of an application to allow the user to design a single tile (piece of cardboard or plywood) which is so designed that it can be easily attached to other copies either via gluing tabs or thin slots. Given a suggest shape, the application should allow the user to move, scale and rotate it into place; and additionally provide some support for editing it in a 2D editor and updating the complete 3D sculpture in a separate window.**Brick patterns.**Develop a module for the wallpaper application (above) devoted to symmetry patterns based on bricks (rectangles with 2:1 aspect ratio). See the book by Peter Stephens,**Regular Patterns**, (in the math library “Semesterapparat”) for examples of such patterns. This brings one to the topic of*tilings*, which we haven’t explicitly studied. There may be several distinct tilings which all have the same symmetry group. See the web page of Craig Kaplan for more on tilings.**Color symmetry.**Using the book**The Symmetry of Things**(see below) as a guide, write an application to explore color symmetry in wallpaper groups.

- Implement some of the missing
**3D Euclidean crystallographic groups**, for example, debug the 35 irreducible ones, or begin to work on the fibered ones. References here: original paper by Thurston, Conway, Huson, and Delgado and these slides from a talk by Huson. **Content.**It’s also conceivable to author content rather than software or mathematics. That is, choose a theme and develop material to communicate this theme to others. Here the focus is not on innovation in mathematics or software, but communication. Think “movie” — without expecting the production of a finished movie (that’s too much work). Examples of possible themes:- An interesting set of animated tessellations would fall into this category
- a gallery of classical patterns from various cultures (see the book by Stephens references in 5)).
- The 59 stellations of the icosahedron.
- “The 120 Cell”

- For more ideas about symmetry-related project themes, see the following resources:
- Craig Kaplan’s web site
- George Hart’s web site.
- Vi Hart’s YouTube videos, for example this one.
- The book
**The Symmetry of Things**by John Conway (in Semesterapparat). - The book
**Regular Patterns**by Peter Stevens (in Semesterapparat).

**New for 2013**

**Tools for visualizing projective geometry.**There have been surprisingly few attempts to develop tools for visualizing projective geometry in dimensions 2 and 3. The mathematical foundation is clear and rich, the jReality system provides excellent support for the homogeneous coordinates and projective transformations. What remains to do is develop some ideas for how one can specify and render geometry and projectivities in a clear, intuitive fashion. For example, “the line segment AB” is not well-defined, one must distinguish somehow between the two possibilities.- One simple test case for the 2D case is the “guided visualization” we have been practicing in class in which 4 triangular regions change their appearance as one of the intersections points passes through the ideal line.
- Another direction for work would be to explore how far one can go in using projective transformations to represent deformations of euclidean objects — that is, take Assignment1 extra credit option and “run with it”.
- Another related application: one that allows the user to specify a 3D projective transformation by picking 5 points, and then applying it to a grid of cubes (or some other regular euclidean pattern) to obtain a linear deformation of this pattern.

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