The Genaster application is a complete asteroids game with a futuristic design and a challenging game play.

It is introduced with a little sci-fi story and an explanation of how the game works. Also visible in the introduction window are four different fundamental polygons representing four different ways of gluing the edges of the two dimensional playing field together. These edge gluings determine where an object like an asteroid reappears when it crosses an edge.

The leftmost fundamental polygon represents the edge gluing the classic asteroids game is using and corresponds to a torus topology.  Next comes the Klein bottle topology, the real projective plane topology and the sphere topology. Then follows a very short discussion using line nets and the euler characteristics to point out why the game can be generalized to the Klein bottle topology in a simple way.  The introduction of the game finishes with showing a nonlinear but simple point counting formula which needs to be analyzed to develop a strategy with respect to time.

This application is for people who like computer games and who want to know what kind of mathematics is hidden even in a simple classical game like asteroids.

 Suggested reading:  https://www.savoir-sans-frontieres.com/JPP/telechargeables/English/topo_…