Hyperbolic Games vs GraphMath

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.

This graphing calculator is user friendly and powerful.  It is fully compatible with iOS 14 and 15, and mostly compatible with iOS 13 and Mac OS. Cartesian 3D Z = F (X, Y) functions:  ——————————————————— Graphs can be:  1. Drawn as surface, lines, or dots. 2. Drawn with or without perspective, and with different light and reflection settings. 3. Tilted up or down. 4. Rotated to any position. 5. Drawing is animated by either X, Y, or Z. Table of all points can be sorted by either X or Y. Cylindrical 3D Z = F (Θ, R) functions:  ——————————————————— Graphs can be:  1. Drawn as surface, lines, or dots. 2. Drawn with or without perspective, and with different light and reflection settings. 3. Tilted up or down. 4. Rotated to any position. 5. Drawing is animated by Θ, R, or Z. Table of all points can be sorted by either Θ or R. Spherical 3D R = F (Θ, Φ) functions:  ——————————————————— Graphs can be:  1. Drawn as surface, lines, or dots. 2. Drawn with or without perspective, and with different light and reflection settings. 3. Tilted up or down. 4. Rotated to any position. 5. Drawing is animated by either Θ or Φ. Table of all points can be sorted by Θ, Φ, or R. Cartesian 2D Y = F (X) functions:  ———————————————— Graph displays the following curves: 1. One or two Y = F (X) functions. 2. Derivative of the first function. 3. Integral of the first function as a shade and a curve.  Graph features:  1. Drawing is animated by X. 2. Clicking on the curves shows curve values, and finds closest point in the table. Tables: 1.  All function, integral, and derivative values. 2.  Zero intersections of first function. 3.  Intersections between first and second functions. Polar 2D R = F (Θ) functions: ———————————————  Graph displays the following curves: 1. One or two R = F (Θ) functions. 2. Integral of the first function as a shade and a curve. Graph features:  1. Drawing is animated by Θ.  2. Clicking on the curves shows curve values, and finds closest point in the table. Tables: 1.  Table of all function and integral values  2.  Table of zero intersections of first function. 3.  Table of intersections between first and second functions. Parametric 2D X = F1 (T), Y = F2 (T) functions: —————————————————————————  Graph features:  1. Drawing is animated by T.  2. Clicking on the curves shows T, X, and Y values, and finds them in the table. Table of all points. F (X, Y) = 0  equations: ———————————— Graph displays solutions as points. Clicking on the points shows X, and Y values, and finds them in the table. Table of all points. Common features: —————————— 1. List of previous entries. 2. List of preset entries. 3. Undo and redo buttons for formula entries. 4. Automatic correction of errors in function limits or steps are offered. 5. All function points can be saved to clipboard in a table format. 6. All function points can be displayed on corresponding graphs. 7. Control colors can be changed to black and white. 8. Link to Vectors and Planes 3D Geometry Guide App. 9. Link to Matrix Calculator Step by Step App. Common features of all images:  1. Animation frames can be frozen and moved back or forward. 2. Animation speed can be adjusted. 3. Zoom. 4. Scroll. 5. Save to Clipboard. 6. Save to Photos 7. Color picker. New functions and further UI improvements for our apps are coming! Developers hope to get feedback at graphmath@aol.com.