Doug Dunham Computer Science
The proposed research was to add the ability to draw rays and complete hyperbolic lines to a hyperbolic pattern-drawing program, which previously did not have those capabilities. This project was carried out and completed by the PI and a Computer Science Master’s student, Ajit Marathe, who described the results in his thesis, "Incorporating Points at Infinity in a Hyperbolic Drawing Program." Unlike Euclidean and spherical geometry, there is no smooth distance-preserving embedding of hyperbolic geometry into Euclidean space. Thus we have to rely on models of hyperbolic geometry (that perforce distort distance). One such model is the Poincare circle model whose points are enclosed in a Euclidean circle. The lines of that model are represented by circular arcs that are perpendicular to the bounding circle (including diameters). In the past, my students and I developed a program to draw repeating patterns in this model. However, that program only drew finite hyperbolic line segments. The summer project extended the program to draw complete hyperbolic lines (with both endpoints at infinity - i.e. on the bounding circle) and rays or "half lines" (with one finite endpoint inside the bounding circle and one infinite endpoint on the bounding circle). Mr. Marathe’s thesis is available at: http://www.d.umn.edu/~ddunham/ajitmrpt.pdf
In addition, several other projects were either started or completed during the summer grant period. During the month of July I attended three conferences and presented papers at two of them. The first one was the fifth Mathematics and Design Conference, in Blumenau, Brazil, at which I presented a paper, "Creating Regular Repeating Hyperbolic Patterns" which was published in the conference proceedings (ISBN 978-85-7114-175-4). There isn't an online version of this paper yet.
The third conference was the 10th Bridges Conference in San Sebastian, Spain. I presented a paper titled "A Circle Limit III Calculation", which was also published in the conference proceedings (ISBN 0-9665201-8-1). An online version is at: http://www.d.umn.edu/~ddunham/dunbrid07.pdf
There were no proceedings of the second conference, Communicating Mathematics, at UMD, which honored Joe Gallian and his work with undergraduate researchers. However, the attendees were asked to submit chapters for a book based on the conference, also called Communicating Mathematics. On December 14 I submitted a chapter titled "Hamiltonian Paths and Hyperbolic Patterns" for that book.
At the Math and Design conference, we were asked to submit extended abstracts for the 13th International Conference on Geometry and Graphics by one of the organizers of the ICGG. On Dec. 15 I submitted an abstract for consideration titled "Use of Models of Hyperbolic Geometry in the Creation of Hyperbolic Patterns".
During the summer I made more progress on an ongoing research project that generalizes the hyperbolic "Circle Limit III"" pattern by the Dutch artist M.C. Escher. This built upon the work presented at the 2007 Bridges Conference, mentioned above. "Circle Limit III" is usually considered to be Escher’s most attractive hyperbolic pattern. But, in my view, it is also the most mathematically interesting. On September 20 I submitted an abstract for a talk based on this new work that was presented at the Joint Mathematics Meeting (January 6-9, 2008) in San Diego. The talk was titled "A formula for the intersection angle of backbone arcs with the bounding circle for general `Circle Limit III' patterns". I plan to submit an expanded paper based on these new results to the 2008 Bridges Conference.