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The success of these projects, I believe, rests on the fact that they are openended and visual, yet grounded by a significant mathematical component. Students regularly go beyond what is expected of them.
In the case of the parametric plots project, some students produce threedimensional plots instead of twodimensional pictures. Some have made short movies, incorporating mathematical models to reflect certain physical realities. Others have learned some linear algebra so that they could rotate a curve to produce exactly the right shape for their picture. I have had students who produce efficient MAPLE code that allows them to translate, rotate, and reflect objects quickly. They often recognize how symmetry can be used to simplify their programming.
Most common, however, is the situation in which a student (or a pair of students) produces 50100 (even 300) different parametric curves to construct an elaborate picture. The students simply enjoy creating their masterpieces, and they will work hard to make their finished products perfect. Driven by the desire to create a design that they can call their own, they also gain a deeper working knowledge of the relevant mathematical concepts.
As an added bonus, students' finished products provide tangible examples of creative work in mathematics. While it may be difficult for an outsider to see the creativity involved in a proof, the creativity involved in these projects is visible to a nonexpert. As such, the results of these projects allow students and spectators to see that mathematical work can be creative.
I have displayed student work in the hallways of our building and in the local coffee shop to better advertise this fact. Recruiting members of the administration and faculty to serve as judges of the contests also helps to make mathematics more visible across campus.

Figure 2. James Sapanski's (Kenyon College, Class of 2007 ) rendition of his dog Bernie.
Judy Holdener, "Art and Design in Mathematics  Why Do These Projects Work?," Convergence (December 2004)
Journal of Online Mathematics and its Applications