Higher-Dimensional Spaces using Hyperbolic Geometry

A novel method of interactive visualization of higher-dimensional grids, based on hyperbolic geometry. In our approach, visualized objects are adjacent on the screen if and only if they are in adjacent cells of the grid.
Previous attempts do not show the whole higher-dimensional space at once, put close objects in distant parts of the screen, or map multiple locations to the same point on the screen; our solution lacks these disadvantages, making it applicable in data visualization, user interfaces, and game design.

Four dimensions:

0:00 a 4x4x4x4 cage with a golden point in the center
0:15 one-dimensional tunnel (bright red)
0:35 the 1-skeleton of the tessellation of ℤ⁴ with cubes of edge 2
1:12 two-dimensional tunnel
1:30 two hyperplanes in distance 2 (blue and green), i.e., three-dimensional tunnel
2:00 two hyperplanes in distance 3 (cyan and green)
2:20 two orthogonal hyperplanes (red and yellow)
2:45 four quarterspaces (red, yellow, cyan, blue)
3:00 diagonal tunnel in all coordinates except one (golden and silver)
3:45 diagonal tunnel (purple and gray)

Six dimensions:

4:10 a 4x4x4x4x4x4 cage with a golden point in the center
4:30 one-dimensional tunnel (bright red)
5:00 the 1-skeleton of the tessellation of ℤ⁶ with cubes of edge 2
5:20 two-dimensional tunnel
5:40 four-dimensional tunnel
6:00 two hyperplanes in distance 3 (cyan and green)
6:20 two orthogonal hyperplanes (red and yellow)
6:45 four quarterspaces (red, yellow, cyan, blue)
7:05 diagonal tunnel in all coordinates except one (golden and silver)
7:30 diagonal tunnel (purple and gray)

other:

7:50 time-sliced visualization of the visualization of ℤ⁴ using {3,4,4}

[einekartoffel]

Dang, this engine of yours has so much potential.

[MajikkanCat]

In Euclidean 4-space and 6-space wouldn't there still be parallax, instead of everything coming at you at once like hyperbolic space? So this is still sort of a compromise, like the more standard visualizations of loads-of-dimensions polytopes, or like choosing between stereographic, gnomonic, etc. projections. This one is super extra awesome because it lets you see everything, while projections of 3-shadows of their 1-skeletons gives you something that seems tangible... Thank you for giving us all a new window into higher dimensional spaces!

[alanhere]

Very impressive gluing together of different spaces. :) I cannot wait for this to work smoothly in first person in the main campaign of HyperRogue, and to have more lands like Palace that are more representative to contrast with the nice abstract lands. And for see more different surface appearances such as roughness/reflectivity.

[Invalid]

This is what mathematicians dream about.

[quinoa]

imagine how confused and scared the internet would be if this video didn't have a title or description

[debblez]

cant say I understand what the visuals mean or what they have to do with their names or higher dimensions, but they’re definitely pretty.

[samsungtelevision]

Music is f ing wild too man just great video all around

[christopherking]

This is what you see on your way to after life, right?

[samsungtelevision]

This is way more interesting than mandelbrot et al. Keep going with these videos 👍

[alanhere]

Why do parallel 3D hyperplanes in 4D, projected to 3D, look like a fractal?

[MartianArk]

We appreciate the playthrough and music selection. Nice work

[toimine]

The way the camera in "two hyperplanes" demonstrations flew past different planes on its way made me think, what if you translated r^3 into h^2, flew through the result, then translated the path you took back into r^3? Would you get weird curvy snake-like motion

[tsuyukurage]

This is beautiful and mesmerizing at the same time.

[aaaaaaaaaaaaaaa-rt]

This is very cool but in all honesty I don't understand how the different dimensions map to the hyperbolic space presented on screen.

[dustinhodell]

I don't understand any of this. I'm just here for the psychedelic memories. Beautiful!

[ollllj]

this better be a single opengl fragment shader.
Thankfully every face shoes its address, so you can never get lost!

[melwugon]

Is there a way to view this in hyperrogue?

[adamdickson]

Around 1.46 - if I understand correctly - you have H2 planes (look like spheres) that appear to be tiled with ideal triangles. I assume the overall space is H3. These triangles have straight edges and non zero vertex angles, which makes me think of the Klein Beltrami projection. Is that what such a tiling of a H2 plane embedded in H3 would actually look like when viewed from a distance? I am a bit puzzled - when does the K-B or Poincare projection resemble what one would actually see? I have just visited Code Parade’s channel with similar questions and his answer helped a bit - the point was made that the projection used to render the overall space was immaterial to how it looked. However I surmise this does not necessarily apply to the appearance of subspaces.

[Shyheem-B]

What it feels like to chew 5 gum.... Stimulate your senses.

[EvilStreaks]

How did you get my colonoscopy footage????