Higher-Dimensional Spaces using Hyperbolic Geometry
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- Опубликовано: 6 июн 2024
- 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}
Paper: arxiv.org/abs/2110.00327 
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Dang, this engine of yours has so much potential.
This is what mathematicians dream about.
and i love it
finally, i put a comment at 69 likes
@@antirogue825and I ruined it because 69 isn't funny, it's just a sex position
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!
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.
This is way more interesting than mandelbrot et al. Keep going with these videos 👍
Music is f ing wild too man just great video all around
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.
I don't understand any of this. I'm just here for the psychedelic memories. Beautiful!
imagine how confused and scared the internet would be if this video didn't have a title or description
This is what you see on your way to after life, right?
We appreciate the playthrough and music selection. Nice work
This is beautiful and mesmerizing at the same time.
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
Why do parallel 3D hyperplanes in 4D, projected to 3D, look like a fractal?
Maybe because they both are fractals 😅
note that at 2:20 there is a inversion pattern where z (the complex plane) is z^-1.
this better be a single opengl fragment shader.
Thankfully every face shoes its address, so you can never get lost!
The last segment is 😻🙌🧬⛓️
This is very cool but in all honesty I don't understand how the different dimensions map to the hyperbolic space presented on screen.
3:57 I found the SLOB BOTS' hideout : )
The slices mode looks so prettty,
And also what do the numbers on the tesseracts mean?
My brain exploded even harder than just H3 hyperbolic space
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.
Indeed, this time you are projection surfaces (hyperbolic planes) to surfaces (computer screen), so it is a specific projection (while mapping a 3D space to a surface will lose some information -- usually, information about distances -- so all azimuthal projections will be fine because they only differ by how distances are mapped).
However, this loss of information in 3D space to surface mapping only happens for single static images -- but in VR you have binocular vision, and in animations you have parallax, and both can be used by humans as means of depth perception, and a way to perceive a 3D image. And that 3D projection of H3 will be also Klein-Beltrami projection (a bit stretched in case of binocular vision).
(Of course this is about perception, rendering renders each image one by one, so any azimuthal projection is fine.)
What it feels like to chew 5 gum.... Stimulate your senses.
High. On knowledge.
Wow! I don't understand this at all!
(but damn does it look cool)
Is there a way to view this in hyperrogue?
In HyperRogue you can do: special modes -> experiment with geometry -> geometry -> interesting quotient spaces -> dimensional crystal
This demo is not included with the standard HyperRogue, but it is included in RogueViz: zenorogue.itch.io/rogueviz
Ladies and gentlemen, I think my brain has committed die.
Very nice I didn't really understand but I kinda do understand if that makes sense.
i love everything about this, and the music is perfect! makes me feel nostalgic :] so glad i found this channel!
those triangle ID's seem very small, only a few thousand for quite a while
They should not be interpreted as "a few thousand" but as sequences of digits. Every digit is a coordinate. So four digits in 4D.
you ever wonder that if we had more axis's than x y and z for movement in the space that it would be easier to understand as well as far less confusing to what our senses are most commonly used to perceiving 3 dimensions.
Shepherds tone ?
honestly i would love one of this but instead of the shapes and figures its like a non euclidean higher dimensional room, like just a normal room or house but with this weird angles because that is how the world would look like if it were higher dimensional not like this this is just a representation
Unfortunately, this method of visualizing higher-dimensional spaces works only for blocky scenes; a normal room or house would be too detailed for that. (See CodeParade's Hyperbolica and 4D Golf videos, they seem to be closer to what you want.)
Imagine hide and seek in one of these spaces. Impossible to find people
Is this the divine beauty of mathematics? Cosmic horror? Por qué no l̹̗̭̤̕a͉̱̻͈̩̥ i̟͓̻̞͜n̤̹̘̭̣f͕̝̗i̪͜n̦i̘͔ṭͅa̛̹
Beauty to the eyes, horror to the mind.❤️
This is like me trying to open a box of Aussie Bites from Costco.
How did you get my colonoscopy footage????
Non-euclidean fractals, cool
Nothing is ever going to be the same . But, I've been wrong before.
would this work in 2d too?
Yes, it does work in 2D too. Using regular right-angled hexagons for 3D, right-angled octagons for 4D, etc.
@@ZenoRogue can your engine do that?
@@dadutchboy2 Yes.
@@ZenoRogue cool
@@ZenoRogue could you make a video with this in 2d?
I'll explore the sph- OMG, how the f### are there pillars?!
It just looks like a 3d fractal
3D Hyperbolic fractal, but otherwise, yeah
マンデルブロ集合とは違うのかな
most average 4 dimensional fractal(on a screen)
...wat
So in hyperbolic space, is every single angle automatically 90°?
no. You need a shape with more Than 3 sides for it to be right angled.
tho you could make a sperical hiperbólic mix geometry for that
Different music, please.
This is music he created, so he is going to use it.
lsd
@josiekins