Hyperbolic space in Poincaré ball model, {5,3,4}, cut in half. @ZenoRogue

ZenoRogue | Hyperbolic space in Poincaré ball model, {5,3,4}, cut in half. @ZenoRogue | Uploaded November 2019 | Updated October 2024, 1 hour ago.
Hyperbolic space in Poincaré ball model, {5,3,4}, cut in half.
We go along a line connecting opposite vertices.

[360° VR] Portals to Non-Euclidean Geometries

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