ZenoRogue | Temple of Cthulhu in 3D @ZenoRogue | Uploaded February 2019 | Updated October 2024, 1 hour ago.
HyperRogue's Temple of Cthulhu in 3D.
This is 3D hyperbolic geometry (in the 3D analog of the binary tiling). It is truly a non-Euclidean geometry, not any perspective or portal tricks (these are not non-Euclidean geometry, even though they are often called so by gamers). All blocks are of the same size, these things looking like spheres are actually horospheres, and are infinite, and there are infinitely many of them. In hyperbolic geometry, there are no parallel lines (in the Euclidean sense), because lines which should be parallel diverge. This is why, when we are inside a horosphere, it looks huge, while the horosphere inside looks small, and the next horospheres looks even smaller; even though all of them are infinite -- the light rays diverge, making the horospheres inside appear smaller.
The best way to understand what is going on (and hyperbolic geometry in general) is to play HyperRogue. Temple of Cthulhu in the 2D shifted binary tiling can be played in the browser in this demo: roguetemple.com/z/hyper/online.php?c=-geo+19+-W+Temple+-T (note: HyperRogue uses a "shifted" binary tiling, not a standard one; the 3D version uses the 3D version of the standard tiling)
The volume of a hyperbolic ball of radius r grows exponentially with r. In this video, you can see objects in distance at most 6; in distance at of 100 there would be more blocks than atoms in the known Universe. This video has not rendered in real time; in real time, the current version of this simulation can render objects in distance at most 3, which looks similar, but you do not see faraway objects. (It would be possible to render larger distances, but it would require quite a lot of work.)
HyperRogue's Temple of Cthulhu in 3D.
This is 3D hyperbolic geometry (in the 3D analog of the binary tiling). It is truly a non-Euclidean geometry, not any perspective or portal tricks (these are not non-Euclidean geometry, even though they are often called so by gamers). All blocks are of the same size, these things looking like spheres are actually horospheres, and are infinite, and there are infinitely many of them. In hyperbolic geometry, there are no parallel lines (in the Euclidean sense), because lines which should be parallel diverge. This is why, when we are inside a horosphere, it looks huge, while the horosphere inside looks small, and the next horospheres looks even smaller; even though all of them are infinite -- the light rays diverge, making the horospheres inside appear smaller.
The best way to understand what is going on (and hyperbolic geometry in general) is to play HyperRogue. Temple of Cthulhu in the 2D shifted binary tiling can be played in the browser in this demo: roguetemple.com/z/hyper/online.php?c=-geo+19+-W+Temple+-T (note: HyperRogue uses a "shifted" binary tiling, not a standard one; the 3D version uses the 3D version of the standard tiling)
The volume of a hyperbolic ball of radius r grows exponentially with r. In this video, you can see objects in distance at most 6; in distance at of 100 there would be more blocks than atoms in the known Universe. This video has not rendered in real time; in real time, the current version of this simulation can render objects in distance at most 3, which looks similar, but you do not see faraway objects. (It would be possible to render larger distances, but it would require quite a lot of work.)
