Hyperbolic space (H3) is 3D the analogue of the well-known hyperbolic (or Lobachevskian) plane. It can be divided to regular polyhedra (cells) - finite or infinite - in many ways. Some tilings may be colored with a finite number of colors in periodic patterns.

Once you have a periodic pattern in H3, you may use it to make a twisty puzzle: Cut each cell with planes "parallel" to its faces, and glue the parts ("stickers") of adjacent cells together. A twist of such a puzzle is the rigid motion of the cell and all its adjacent stickers.

In the program one particular space division is used. It has Schlafli symbol {6,3,3} which means that the boundary of the cells consists of hexagons, three at each vertex, and three at each edge. The boundary of each cell is infinite. That is, it looks like the regular hexagonal tiling of Cartesian plane. Periodic patterns in H3 produce patterns on the boundaries, and it every sticker appears to be repeated an infinite number of times.

The picture above shows the puzzle in its solved state in which every cell has stickers of the same color. The view is looking through the wide gap between the central sticker of one cell (invisible from this position) and its boundary stickers. You can see a periodic pattern of the central sticker of the other cells. Small particles in the yellow cell are its own boundary stickers, and so on. The space has infinite depth.

Here is an alternative view of the same space:

Here the space between stickers are small but the cells are shrunk so you can see many of them. The cells look like balls (made entirely of hexagons!),but in reality they extend to infinity behind their visible parts.

Here are two views of the scrambled puzzle:

The puzzle in the second picture is partially solved - all 2-Color pieces are in place. To twist left or right, just left or right click a sticker. You can scale the size of cells and stickers in the control panel. You can also change the colors, hide some sticker or piece types, create and use macros, etc.

Download the program here. It is a zip archive containing the executable file. Unpack it in one folder and double-click the .exe file to run. It requires Microsoft .NET 2.0. Full instructions here.

There are 7 different puzzles implemented - from 8 to 52 colors.

Good luck!

Andrey

Magic 3D Hyperbolic Tile puzzle took many ideas from Magic Cube 4D. I would like to thank its developers Don Hatch, Melinda Green, Jay Berkenbilt and Roice Nelson for this wonderfull puzzle and for the very interesting 4D Cubing discussion group! Special thanks to Roice Nelson for his Magic Tile program which is the first twisty puzzle in the Hyperbolic Plane!

Some links:

- Magic Tile - Equivalent to Rubik's Cube in Hyperbolic plane
- MagicCube4D - Not only cubes. There are many other 4D puzzles inside one program, most of them are still unsolved.
- MagicCube5D
- Five-dimensional magic cubes from 2
^{5}to 7^{5}. - MagicCube7D
- Seven-dimensional puzzles! Cubes from 3
^{4}to 5^{7}are implemented. - Magic120Cell - 120-Cell is one of two largest regular polytopes in 4D. This puzzle is a 4D analogue of the 3D Megaminx.
- 4D Cubing Group - Mailing list for hypercubists.

1 |
Andrey Astrelin | 2010/12/24 |

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Matthew Sheerin | |

3 |
Philip Strimpel | 2013/03/20 |

4 |
Bùi Hồng Đức |

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Nobody | Not yet |

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Nobody | Not yet |

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Philip Strimpel |

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