Geometry from Another Universe
falsifian.org
falsifian.org
After traveling through the second universe in my dream, I then entered a third universe that was more compact because space was curving back onto itself in a way that was not understandable but I kept looping back to where I was before. When I woke up I thought I was dreaming of a very high dimensional space where everything was happening in lower dimensional manifolds curving through that space where “all the action was.”
Definitely one of my better dreams!
I feel like this would have to be true in a very high dimensional space, since an infinite dimensional orange is all skin.
> systems obeyed solid state physics type laws: stars formed an outer lattice and planets in small lattices around stars.
This sounds really fun, could you try explaining a bit more about it?
Thanks for sharing.
Huh?
This 3Blue1Brown video addresses enough related stuff that the rest should become reasonably comprehensible: https://www.youtube.com/watch?v=zwAD6dRSVyI This particular result is tossed in as a side note around 23:25, so it's not addressed directly, but it'll help.
For some reason in the last six months to a year it has shifted to the idea that I signed up for a full semester's worth of courses, while still trying to work my full time job, and I'm flunking all of them because I keep going to work and never attending the classes or doing the homework.
Either way, coming up on nearly 20 years since I've been in any kind of school and my subconscious is still freaked out about it, one way or another.
By contrast, my subconscious appears to not give a flip about whether I'm doing my paying job correctly. Don't tell my boss.
Another theme for me is not able to board on plane. More than missing flight it is about being stuck to places where I do not want to be.
edit: typo.
I still regularly have the dream where I realized after graduation that I had missed an entire college class or test (often varies depending on the specific dream), but this then nullifies my graduation and all my subsequent work.
Also over 20 years ago, but still a regular visitor to my dreamscape.
Taking very liberal inspiration from the idea of something like Jungian archetypes, more along the lines of there being attractors in the informational dream-space that create narrative clusters linked to common emotions (such as anxiety), I can imagine of set of emotion-to-dream-narrative mappings that would lead to some of the regularities often seen in distributed recurring dreams:
- losing one's teeth - riding naked in the subway - being late to a class or missing a class/test entirely etc.
Just wondering if there has been any research on this that doesn't try to fall on the old standbys of Jungian archetypes, collective unconscious, and other similar handwaving
The wave of relief and euphoria that overcomes me when I wake up is literally spiritual.
I used to have this exact same dream for years after graduating from college.
An upgrade would be 4D underwear.
The most intellectual nightmares my brain comes up with just involve demons chasing me. I've never had these intellectual dreams where there are exams, and deadlines, and being late.
I used to keep a dream diary and half of the entries were "Demons are chasing me. It is dark. I think maybe it's the xenomorph from Alien. I hear it breathing." so I stopped. Not remembering that stuff is much better it turns out haha.
Sometimes they are funny. I had a walk in my former school district (actually I was able to jump very high and jumped around instead) and knew there are 6 humanoid monsters. But my revolver only had 5 bullets, so I had to find the armed one to take a bullet from him first.
I often have offense/defense dreams, like when I was in a room full of people, but one of them grinned at me holding a knife. I disarmed him, pushed to the pretty high window sill and carved crosses on his knees. Everyone pretended not to notice us. I hate dreams of punching someone, because my arms feel weak and slow, but when I have a gun or a "fingersniper", all enemy men become dead quickly.
I also was beaten, burning, shot, drowning, bitten, falling, stabbed. Often it's not as terrific as it sounds. Bites usually come from my recurring friend, a dog with no body, instead it's all a large mouth on legs, which can contain your entire arm. It is ~friendly.
But the worst nightmare is sort of a poltergeist. Things subtly move on their own, body gravity changes in an "intentional" way, and space becomes filled with some sort of unpleasant numbing sound. Sometimes doors don't want you to come out of them by catching my feet, I fall down and then "fall" back up paralyzed. This one is recurring from my childhood, and I (falsely) remember to experience it awake few times.
Also, lucid dreaming techniques help with detecting nightmares and turning them into something funny, or simply moving the scene far enough.
Curiously, it seemed that the books got more and more wild with each following entry: at first it was almost like "hard science fiction" with mostly feasible technologies, whereas the later books seemed to throw that idea to the wind.
That's not to say that it was a bad thing, though.
Does anyone know of any communities specifically for sharing awesome dreams? DreamerNews?
Every now and then I recall tripped out dreams too. In the most recent one I remember, the entire universe was a discrete regular lattice whereby neighboring points would influence the "color" of each other, very much like a network of coupled oscillators.
These couplings allowed for some Game of Life-esque life-forms to exist as locally coherent "color patches" on the lattice, of which I happened to be one.
Most of the dream happened internally to this lattice, i.e. I was not some narrator looking down at it, but frustratingly that internal experience quickly faded from memory, to be replaced by the external view which ended the dream.
Fun times!
[1]https://quran.com/41/12?translations=21,20,19,17,101,22,95,1...
> This is done using WebGL, an API for drawing 3D graphics in a browser. You might think it's only designed to draw ordinary, Euclidean things, but it turns out it's perfectly suited to rendering this world as well. WebGL (and OpenGL) works with four-dimensional coordinates (x,y,z,w). Normally, you'll be advised to just set that w coordinate to 1 all the time. If you do that, the x, y and z coordinates will behave like ordinary 3-D geometry. I just ignored that rule and used the coordinates to represent points on the 3-sphere instead. It all works out. (You have to be a bit careful if you want to put textures on things without seeing seams between the triangles, but it can be done. The key is to make sure every flat surface is still a flat surface in 4-D.)
Are the 4D coordinates treated as ratios, so that (w,x,y,z) = (tw, tx, ty, tz)? Because then that would express RP^3 instead of S^3, i.e. projective 3-space, instead of the 3-sphere. Equivalently, this produces a model of 3D elliptic geometry, instead of 3D spherical geometry. My doubts are because 3D computer graphics is known to use homogeneous coordinates, which results in a model of projective 3-space or elliptic 3-space.
S^3 can be coordinatised using elements of R^4 of unit length, which might have been done here.
Additionally:
To clarify my understanding of projective vs elliptic geometry, because I keep conflating them: Projective space is a topological space, while elliptic space is projective space endowed with a certain metric, turning it into a metric space. Additionally, every elliptic isometry is a projective symmetry, while the other way round isn't true. On a purely topological level, there's no difference between elliptic geometry and projective geometry.
This happens to work with (Open/Web)GL. Here's my understanding of why. Say you tell WebGL to draw a triangle with points at clip space coordinates (i.e. after all the transforming is done) (1,0,0,1), (0,1,0,1), (0,0,0,1). If you multiply all those by a positive number, e.g. (10, 0, 0, 10), (0, 10, 0, 10), (0, 0, 0, 10), you get the same triangle. But if you multiply by a negative number, they will not be drawn at all: (-1, 0, 0, -1), (0, -1, 0, -1), (0, 0, 0, -1) is not drawn. I think this is because of the rules for clipping: only points where -w <= x,y,z <= w survive the clipping step. If w is negative, that's impossible.
Related: my code draws the universe twice on each frame. First I draw things that are more than halfway around the sphere from you, and then I reset the z buffer and draw things that are closer to you. https://git.sr.ht/~falsifian/s3d/tree/main/item/src/S3D/Draw...
That is, if you scale just one of the three points of the triangle, e.g. (1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 0, 1000), then while the triangle will appear in the same place as before, interpolation (e.g. deciding which part of the texture to draw in each pixel of the triangle) changes.
The current version of the demo doesn't have any textures but that's mostly laziness on my part. For a while I gave the rooms "ceilings" which were each a checkerboard-pattern square, which forced me to figure out how to get the interpolation thing right. So I can confirm it can be done. But I dropped the ceilings because they just got in the way.
(There's also miegakure: https://www.youtube.com/watch?v=uWsBnVtl8tA )
I can't wait for these games to start being built for VR. Maybe it'll be possible to develop a sense for higher dimensionality intuitively using our senses.
Maybe young mathematicians and physicists can explore higher dimensions in VR to get accustomed to it, which might help with their theorizing and explorations.
In the meantime, the video I link is basically a video exploration of spherical geometry. One of the better ones, in my opinion, because it has "normal" objects in it, rather than floating heads or a ton of Earths or something.
(Another amusing sidebar: As you can see in the video above, an inhabitant of that space would be naturally inclined to say the space curves in above them. It would take an Einstein to assert that it's actually flat, and Spherical Einstein would have a very hard time describing "flat" to anyone. You can only make things "flat" contingent on the observer being in a very particular place. If you look at the next devlog: https://www.youtube.com/watch?v=pXWRYpdYc7Q back in Hyperbolic space, you can also see that a resident of that space would naturally believe the space they are in is a sphere (albeit one of variable radius, which is weird, but still, you can look out in the world and see the curvature, obviously it's round), and it would again take an Einstein to say that it is flat. It's flat if you are exactly on the ground, but hyperbolic space exaggerates any degree to which you are above the ground to make the horizon look round.)
It's easy to construct, just build a point, extrude the point out to make a line, extrude the line out to make a square, extrude the square out to make a cube, then extrude the cube out to make a hypercube. It looks like two cubes with the corners connected by lines through a 4th dimension that we can't construct in 3D.
One way to visualize it is to use time, so if you translate a cube from point A to point B, the "extrusion" would be along time and you can kind of visualize that extra "dimension". But what would scaling, rotation, etc along that invisible axis look like? To fully grok it, we'd need to be able to sculpt the hypercube's shadow in 3D as easily as drawing a cube on a piece of paper.
Asking in the hopes of building a mental bridge to 4D and then possibly 5D in order to generalize to higher dimensions.
Wait, doesn't it rotate around a plane?
Based on the same 4D geometry concepts he's working on the game Miegakure: https://miegakure.com/
https://www.amazon.com/Mathematical-Carnival-Martin-Gardner/...
While you program it up, you can play this little Hypercube game. Do it in 3D, then turn on the 4D to get an intuitive feel for the 4D world.
Recommendation: Your brain actually has two depth perception systems: hardware (using your two eyes for parallax) and software (using visual cues to establish a 3D scene, the way you can with one eye, or when looking at 3d projections onto a 2d surface).
Use each depth perception mechanism for a different axis! In this old applet you could enable stereo projection and also thick-lines-mode which really helped this, and gave me a strong intuitive feeling for the motions of the hypercube.
The six rooms in the house form six cells of the hypercube. You can make one more cube by joining up the tops of the walls: that cell is always above you. And finally, there would be one more cell below you. (To be a bit more precise: the six rooms of the house would have to be extended to be twice as tall. The tops of the walls are at coordinates (+-1, 1, +-1, +-1), but right now the bottoms of the walls are at (+-1, 0, +-1, +-1). You have to remove the floor and make the walls go all the way down to (+-1, -1, +-1, +-1).
You can do this for any 4D polytope. Taking it down a dimension, the layout of the house shows how to put a cube on a 2-sphere. I could have done it for any other 3D polyhedron. Give the sphere and the polyhedron the some centre point, and then project all the lines of the polyhedron onto the sphere. Each face of the polyhedron becomes a 2-D region on the sphere (in the case of the house, each room is a square). Going a dimension up, you can centre the 3-sphere and the polytope on the same point, and do the same kind of projection. Every cell of the polytope turns into one 3-D region of the 3-sphere.
I wish I had had time to include that.
I don't understand anything in that page but the pictures are great.
There is a little bit of CSS. And I would have gone for no javascript, but unfortunately in this case that's the whole point.
Does that make sense? I've forgotten the name of the book.
https://en.wikipedia.org/wiki/He_Who_Shrank
ETA: Seems that was a popular story line from that time.
Also seems similar to Inception, with the idea that diving into deeper dream levels makes time in those levels run slower than in the real world.
I wonder why the viewport is only a 300px square.
To be honest I usually used my browser's zoom functionality to make it bigger while debugging, and you can too. Changing the aspect ratio would require some code changes, though.
I'm using the WebGL API pretty directly, though. So if it were missing I could have hacked it with some foreign function calls, but it would have been a lot more effort.