"The summer bursary has provided me with a very useful and informative experience and insight into the world of academic research. During the project I learnt both new mathematics and also more general experiences regarding presentation of results and independent study and research."
Simon Castle spent two months in summer 2007 at Durham University working with Dr Norbert Peyerimhoff, developing programmes which would draw periodic billiard trajectories in polygons of the hyperbolic plane, and to calculate their lengths. In the first part of the project, Simon wrote the program to calculate and draw these trajectories in polygonal billiard tables.
In the image to the left, the green lines form the (polygonal) boundary at which the blue trajectory is reflected. The green lines form a polygon inside the hyperbolic plane and the hyperbolic plane is presented as a disk which is encompassed by the red circle.
In the second part of the project, Simon’s task was to compare the lengths of closed billiard trajectories of irregular polygonal billiard tables to the case of a regular billiard table. According to a recent conjecture of Dr Peyerimhoff and Professor K F Siburg of the University of Dortmund, Germany, the average length of rotated closed trajectories in regular polygonal hyperbolic billiard tables is always smaller than in any irregular billiard table. Simon’s investigations proved that this was the case for particular paths and billiard tables.
Talking about Simon’s contribution to the project, Dr Peyerimhoff said:
"Simon developed the programmes needed very effectively and professionally. The graphical images are impressive and the calculations are very useful to confirm our conjecture."