# The hyperbolic space $\mathbb H^3$

## What is $\mathbb H^3 $?

The hyperbolic space $\mathbb H^3$ is the three-dimensional analog of the hyperbolic plane. It is an isotropic space (all the directions play the same role). Among the eight geometries, this is probably the one which has the richest class of lattices.

Click on the button below to reveal a concrete model of the hyperbolic space.

## Some views of $\mathbb H^3$

**Warning:**
Some of the real-time simulations below requires a powerful graphic card.
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Click on the button below to reveal the fly commands.

The default controls to fly in the scene are the following.
You can choose your keyboard in the *Option controls* pannel in the top right corner of the window

Command | QWERTY keyboard | AZERTY keyboard |
---|---|---|

Yaw left | a | q |

Yaw right | d | d |

Pitch up | w | z |

Pitch down | s | s |

Roll left | q | a |

Roll right | e | e |

Move forward | arrow up | arrow up |

Move backward | arrow down | arrow down |

Move to the left | arrow left | arrow left |

Move the the right | arrow right | arrow right |

Move upwards | ' | ù |

Move downwards | / | = |

HD pictures of $\mathbb H^3$ can be found in the gallery

## Features of $\mathbb H^3 $

Some features of $\mathbb H^3$ are described in the following Bridges paper by two members of our team (Henry and Sabetta) together with Vi Hart and Andrea Hawksley.

### Cusps

### Tilings and Polytopes

### Hyperbolic Knots and Links

Have a two-way view of the same fixed hyperbolic manifold (easiest, given what I have to do Whitehead Link complement): one: give an extrinsic view, of link complement and some other reference objects in the scene in S3. Two: an intrinsic view of the same thing, with the hyperbolic metric.