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Mathematical Sciences Research Institute

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Summer Graduate School

Mathematical Graphics July 14, 2003 - July 25, 2003
Location: Reed College, Portland, Oregon
Organizers Bill Casselman and David Austin
Lecturer(s) No Lecturers Assigned Yet.
Speaker(s) No Speakers Assigned Yet.
Description The basic aims of this program are to teach practical techniques in producing illustrations of mathematics to accompany research papers and expository writing. Recent experience indicates that bringing out the real point in mathematical exposition through images is not straightforward; at many points in this process a number of basic computer algorithms for things such as sorting and searching must be integrated into the graphics; and, for the best style, some of the more interesting algorithms, such as those concerned with convex hulls, Voronoi partitions, and the hidden surface problem for non-convex bodies, play a role. Illustrating mathematics, as compared with more casual illustration, is a complicated and specialized subject.

In the first week, the program will present the basic techniques of graphics in PostScript and Java. In the second week we expect mostly to supervise student projects and invite a few guest lecturers to cover more relaxed topics, including a few lectures on the history of mathematics in animation movies

The PostScript lectures should be accessible to everyone attending. PostScript is almost unique in making it possible for beginners to produce interesting mathematical figures right from the beginning. The lectures on Java would be aimed at an audience with more experience at programming, and in these we would hope to emphasize the importance of a good user interface. Java applets involving mathematics are common on the internet, but good interfaces are still very rare.

The main part of the program would be for the students to produce, possibly working in teams, their own graphics project to post on the Internet. In doing the projects, the students would for the most part apply what they learn from the lectures to make up a number of web pages, images, and in some cases Java applets explaining a topic largely of their own choice. Good possibilities would include recreational mathematics, like good explanations of Rubik's cube, and perhaps an explanation of Penrose tilings. Higher level projects might discuss automatic groups, various semi-combinatorial ideas of Thurston, or the visual impressions of iterations of 2D transformations.

References

  • Jim Blinn's books of columns from his IEEE column, 'Computer Graphics and Algorithms'
  • M. de Berg et al., Computational geometry, algorithms and applications
  • http://www.math.ubc.ca/people/faculty/cass/graphics/text/www/ (recently submitted to Cambridge University Press)
  • E. Stollnitz et al., Wavelets for computer graphics
  • Any one of several books on Java programming. The on-line documentation from Sun Microsystems is also very good.