Hyperbolic Games 2.1.4

The Hyperbolic Games offer an intriguing departure from the familiar Torus Games, introducing players to the world of curved surfaces. Targeted at a more specialized audience, these games are tailored for math enthusiasts, encompassing advanced undergraduates and early graduate students. The complexity of the Hyperbolic Games transcends that of the Torus Games by integrating multi-connected topologies with non-Euclidean geometries, posing a stimulating challenge for players. Mathematically, these games serve as a platform to explore several key concepts:

• The visualization of the hyperbolic plane as a dynamically scrolling entity.
• An exploration of the often overlooked idea that the two conventional models of the hyperbolic plane represent distinct perspectives of a singular fixed-radius surface within Minkowski space: The Beltrami-Klein model corresponds to a viewpoint positioned at the origin (central projection), whereas the Poincaré disk model reflects a viewpoint slightly further away by one radian (stereographic projection). Players can seamlessly transition between these models via pinch-to-zoom mechanics or pause to inspect the surface from varying distances.
• An emphasis on the profound yet underappreciated parallels between the hyperbolic plane and a standard sphere. Notably, central projection of the sphere aligns with the Beltrami-Klein model of the hyperbolic plane, while stereographic projection of the sphere corresponds to the Poincaré disk model.
• An introduction to the Klein quartic surface within its inherent geometry. Sudoku puzzles within this context leverage the surface's abundance of symmetry, offering a unique gameplay experience that transcends traditional puzzles.