Unlike the Torus Games, which are enjoyed by kids ages 10 on up, the Hyperbolic Games are intended for

• undergraduate geometry classes, to let students build up their intuition for the hyperbolic plane by playing games on a live, scrollable, hyperbolic game board, and
• undergraduate or graduate topology classes, to deepen students' grasp of the Geometrization Theorem for Surfaces by experiencing first hand the connection between a surface's topology and its constant-curvature geometry.

Using a mouse in the hyperbolic plane is tricky!

In the Euclidean plane, mouse motion is straightforward:

when you move your mouse northeastward on your mousepad,	all the hand cursors on the gameboard move northeastward in unison.

In the hyperbolic plane, mouse motion is trickier:

when you move your mouse northeastward on your mousepad,	all the hand cursors on the gameboard move in different directions!

Clearly those hand cursors can’t all move northeastward at the same time, so how do they decide which direction to move? The answer is that they move relative to each hand’s local coordinate system. That is, when your physical hand on your mousepad moves towards its pinky (towards your little finger), all those hand cursors on the game board move towards their pinkies as well. Because the hand cursors are all oriented differently on the board, with their pinkies located in different directions from their centers, they all move in different directions on your computer monitor. Fortunately they all move in the same direction relative to their own surroundings (for example, relative to nearby pool balls), which is of course exactly as it must be, because they are images in the universal cover of a single well-defined hand in the (finite) surface itself, and that single well-defined hand sits in a well-defined position relative to the eight pool balls.

The idea that all those images move in a consistent way relative to their own local coordinate system is fine in theory. In practice, though, when you want to get the hand cursor to a specific destination, say you want to move it to the right a bit, but to get it there you need to move your own physical hand downward and to the left, well… let’s just say that this places greater demands on your hand-eye coordination than most people feel comfortable with. Fortunately you don’t need to re-train your brain to interpret motions at weird angles. For a much easier solution to the problem, please read on…

To keep mouse use easy, rotate your physical mouse to align with one of the hand cursors on the game board. For example:

if you’re focusing on a hand cursor that’s rotated 90° counterclockwise,	then rotate your own physical mouse 90° counterclockwise as well.

With your physical mouse aligned with the hand cursor, your own hand motions on your mousepad will correspond naturally to the cursor’s motions on the screen. The hand-eye coordination will feel natural, and you’ll save a lot of unnecessary frustration.

If you later shift your attention to a different hand cursor,	be sure to rotate your physical mouse to match.

Tip for lefties: If you’re left handed, go to the Hyperbolic Games’ menu bar and choose Options : Cursor : Left Hand. The cursor will change from a right hand to a left hand, so you can more easily align your own physical left hand with the hand cursor on the board. Now when you move your own left hand (the hand holding the mouse) in the direction of its thumb, the game’s hand cursor also moves in the direction of its thumb, and similarly when you move your hand in the direction of your wrist, your pinky or your fingertips, the game’s cursor does the same.

Challenge for experts: If you’re looking for a challenge, you can train your eye-brain-hand system to think in terms of your hand’s local coordinates — so you can play hyperbolic games with no need to rotate your mouse on its mousepad — but this is an advanced skill, and is not required.

When you click on the game board, the usual system cursor (the arrow cursor) disappears and you get the hand cursor instead. The hand cursor lives in the game board (the instructions above explain how to use it). When you want to recover the system cursor, for example to access the menu bar, you may either

• double-click with the mouse, or
• press the esc key at the upper left-hand corner of your keyboard.

Fill in the empty cells so that the numbers 1 through 8 appear in every “row”. Unlike in standard square Sudoku, where each cell belongs to one row, one column and one 3×3 block, in hyperbolic Sudoku each cell belongs to three different rows, and there are no 3×3 blocks at all. For example, in the puzzle shown below, the green cell belongs to the blue row (5 ∙ 2 1 ∙ 4 ∙ 8), the yellow row (5 2 6 ∙ ∙ 1 ∙ 4) and the red row (∙ ∙ ∙ ∙ ∙ 4 3 ∙); the only number missing from those three rows is 7, so we can write a 7 into the green cell.

You might notice that the perspective view of the sphere seems a little odd, particularly if you rotate the sphere while you’re inside it (at projection distance d = 0 or d = 1). This is because the Torus Games give you a wide-angle view. The advantage of the wide-angle view is that you see more of the surface. The disadvantage is that to see a perspectively correct image, you need to shut one eye and position your other eye very close to your computer monitor. Specifically, the Torus Games show a 90° field of view (±45° from center). So for example if your display is 36cm wide, and you position your eye 18cm in front of the display’s center point, then you’ll see a perpectively perfect image: when you drag with the mouse to scroll the image, the sphere’s rotation will look natural and correct. This same 90° field-of-view rule applies no matter whether the Torus Games’ projection distance is set to d = 0 or d = 1 or anything other value.

The Study Questions accompanying this software introduce the concepts of holonomy and curvature, leading to the simple proof of the beautiful Gauss-Bonnet Theorem.