[360° VR] Portals to Non-Euclidean Geometries @ZenoRogue

ZenoRogue | [360° VR] Portals to Non-Euclidean Geometries @ZenoRogue | Uploaded January 2024 | Updated October 2024, 1 hour ago.
This is a 360° VR version of our earlier video:youtube.com/watch?v=yqUv2JO2BCs

On this tour, portals will take us to various non-Euclidean geometries. This is not Minecraft!

A cool holonomy effect happened during this tour, but it was not explained by our guide Tehora! Have you found it? Please tell us in the comments!

Tehora Rogue's channel: youtube.com/channel/UCHRkB4HgLLReBxWJwblHsjQ

Visited geometries:

* Three-dimensional Euclidean space 𝔼³ (cyan floors, music: Icy Land by Shawn Parrotee)

* Product space ℍ²×ℝ (green floors, music: Living Caves by Shawn Parrotee)

* Three-dimensional hyperbolic space ℍ³ (yellow floors, music: R'Lyeh by Shawn Parrotte)

* Product space 𝕊²×ℝ (blue floors, music: Ocean by Will Savino)

* Three-dimensional spherical space 𝕊³ (purple floors, music: Land of Eternal Motion by Shawn Parrotte)

* Solv (brownish floors, music: Lost Mountain by Lincoln Domina)

HyperRogue soundtrack under the Creative Commons BY-SA 3.0 license

Link to RogueViz: zenorogue.itch.io/rogueviz

To learn more about non-Euclidean geometry, play HyperRogue or visit our discord: discord.gg/8G44XkR

HyperRogue Orb Strategy Mode gameplay: Palace

$Four dimensions using H3 This is hyperbolic three-dimensional space, as viewed by an Euclidean explorer magically transported to it. This is a non-Euclidean geometry where the angles of a triangle sum up to less than 180°. The parallax effects are different than in Euclidean geometry: while close objects will behave roughly similar to the Euclidean ones, distant ones will feel like zooming (because the parallel lines in hyperbolic geometry diverge, speaking very informally). The tessellation is the order-4 octahedral honeycomb [1], i.e., the hyperbolic shape is subdivided like ideal octahedra (just like the Euclidean space can be subdivided in cubes, as seen in e.g. Minecraft). While the Wikipedia page shows the edges of the octahedra, this visualization fills some of the octahedra. Note that these are *ideal* octahedra, which means that their vertices are actually infinitely far away. They have finite volume though (they get extremely narrow as you approach the ideal vertex). The video starts close to an ideal vertex. Each octahedron gets assigned coordinates in ℤ⁴ (the starting octahedron gets (0,0,0,0), adjacent octahedra get assigned the eight neighbors, in such a way that opposite octahedra get opposite neighbors). For every plane that contains faces of the octahedra, there is one coordinate that has value x on one side and x+1 on the other side. This gives an unique assignment of the cells of ℤ⁴ to all octahedra (unique for a specific assigment of cells to the neighbors of the starting octahedron).. Each octahedron gets filled or not and colored depending on its coordinates. The following four-dimensional shapes are used: tesseract (first with center in golden, then empty), 1D tunnel (red), cubes of edge length 2 with edges marked (yellow-green), 2D tunnel, 3D tunnel (green and blue), wide 3D tunnel (green and cyan), two orthogonal hyperplanes of walls, something weird that looked nice, 3-diagonal hyperplane tunnel, 4-diagonal hyperplane tunnel. VR video using ODS projection (hint: watch on a smartphone if you have no VR). Rendered using raycasting. HyperRogue does not use raycasting yet because it is difficult to use this method for rendering non-periodic, exponentially expanding world, but it should work well for the Racing mode or roughly periodic structures such as this one, where it would have advantages of huge view ranges, and rendering the mutiple geodesics in Thurston geometries correctly. Many commenters on Twitter [2] think that this is a fractal (or specifically the Mandelbrot set). This is not supposed to be a fractal fractals are shapes whose very small details are similar to the whole object, while this structure is composed of huge octahedra (their vertices are actually infinitely far away) which do not have such fine details. However, the 2D projection (which is what you actually see in the video) is a (2D) fractal. It does not seem related to the Mandelbrot set. Play HyperRogue [3] for more non-Euclidean goodness! Join us in the HyperRogue Lounge on Discord [4]! [1] https://en.wikipedia.org/wiki/Order-4_octahedral_honeycomb [2] https://twitter.com/ZenoRogue/status/1182672036499513344 [3] http://roguetemple.com/z/hyper/ [4] https://discord.gg/8G44XkR$

Cubes and dodecahedra in hyperbolic space