Hyperbolic Games

The Hyperbolic Games are similar in spirit to the Torus Games, but played on curved surfaces. Most people will want to start with the Torus Games instead, which offer a selection of easily playable games, designed for children ages 10 and up, all implemented in multi-connected spaces in 2 and 3 dimensions.

The Hyperbolic Games, by contrast, are for math students — advanced undergraduates and beginning graduate students. These games are more challenging than the Torus Games because they combine a multi-connected topology with a non-Euclidean geometry. Mathematically they illustrate the following:

• The hyperbolic plane, as a live scrollable object.
• The under-appreciated fact that the two traditional models of the hyperbolic plane are simply different views of the same fixed-radius surface in Minkowski space: the Beltrami-Klein model corresponds to a viewpoint at the origin (central projection) while the Poincaré disk model corresponds to a viewpoint one radian further back (stereographic projection). Players may pinch-to-zoom to pass from one to the other, or stop to view the model from any other distance.
• The strong — but also under-appreciated — correspondence between the hyperbolic plane and an ordinary sphere. In particular, central projection of the sphere corresponds to the Beltrami-Klein model of the hyperbolic plane, and stereographic projection of the sphere corresponds to the Poincaré disk model of the hyperbolic plane.
• The Klein quartic surface, viewed with its natural geometry. The sudoku puzzles take full advantage of the Klein quartic’s tremendous amount of symmetry.

On the horizon: A newer Hyperbolic Pool app for a broad audience is at the planning stage. It will focus on hyperbolic geometry itself, without the distraction of a multi-connected topology and therefore with no repeating images.

Questions?
Contact Jeff Weeks.

This material is based upon work supported by the National Science Foundation under Grant No. 1503701. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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