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.
24 February 2018 (Version 2.0)
(iOS) New: Hyperbolic Games for iOS
(macOS) User interface completely re-written
to eliminate the hard-to-use internal hand cursor
in favor an easy-to-use standard cursor
(macOS and iOS) Graphics re-written using Metal
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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