Geometry from Another Universe(falsifian.org)
falsifian.org
Geometry from Another Universe
https://www.falsifian.org/blog/2022/01/17/s3d/
90 comments
Wow, if you’re having these dreams without the aid of substances, I envy you. The most intellectual dreams I seem to be able to come up with “unenhanced” are falling off buildings and being late to college exams while having no pants on.
My favorite are the dreams I have before a semester starts: it is the end of the semester, and I've just discovered I've forgotten about an entire class for most of the semester. Then I wake up.
I'm in my 40s. For a good long while, my recurring school dream was that my high school discovered that I didn't actually finish correctly, and because that invalidates my college degrees (in dream logic land), I can't do my job until I go back and finish high school properly.
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.
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.
Wow, seeing so many comments high school/college exam dreams do seem universal. I have had these dreams like forever. To think about high school was almost 25 years ago.
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.
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.
Oof. Same here. Why does this happen all these years later? High school years must have really messed a lot kids up.
It's pretty bad that school elicits such stressful responses from us that we remember and relive them 20 years later.
edit: typo.
edit: typo.
I still have regular dreams about not passing high school exams. Worse: in my dreams I have to study to pass for high school exams, and work my day job, and take care of the family at the same time. Because of the commitments outside of studying, in the dream I worry about not being able to study. Then I wake up.
Maybe it is a bit more metaphorical - you just have to learn something.
This is extremely common, I googled a lot about it some years ago.
I have it in two versions, the first one is that I have to repeat a maths course in school, being aware about the age difference with the other students,
the other one is missing a course in university and not having graduated. The latter one feels so real that in the beginning I had to remind myself about having been at the graduation ceremony :)
holy crap, i had the ‘missing a course’ nightmare roughly every 2 months for 2 years after graduating.
I've had variations on this dream several times. I was talking to my father about it, and he still occasionally dreams about this. As a 67 year old, 5 years retired, 40 years after graduating...
This scares me, I don't want to be having these dreams in my 60s, do you know if there any kind of therapy? this is a bit like a mini PTSD
I wonder why it's common? Easiest explanation would be extreme stress/mild trauma from the stress of college...
I'm guessing that it's because (at least at my university), the college class experience is pretty anonymous. No one is taking roll and won't notice if you don't show up. You sit in class as an observer, mostly quiet, almost like a disembodied ghost. The experience is already very dream like and easy to dissociate from. This in addition to the stress of the experience makes it prime dream material.
I'm genuinely curious why this dream pattern seems to be quite distributed across many people.
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
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
My version of that dream has me unsure where the class meets (I have a syllabus but can't decipher it somehow) and not sure if I have the right textbook for the class. I keep trying different classrooms but they all just allow me to enter without a hint of whether I'm in the right one.
I have the same shared trauma response. Really does have me thinking back to my school years and trying to grapple with the two opposing feelings of "that was a lot of fun" and "gosh, why is it still giving me nightmares?"
My version is me running late for an exam and then frantically searching for the exam room the entire time as the clock ticks down.
The wave of relief and euphoria that overcomes me when I wake up is literally spiritual.
The wave of relief and euphoria that overcomes me when I wake up is literally spiritual.
Oh my, I'm not the only one! I occasionally have a dream that, at the end of a semester, I'm not going to pass because I forgot about some class! The realization, upon awakening, that I'm not in college anymore is wonderful ;)
>>it is the end of the semester, and I've just discovered I've forgotten about an entire class for most of the semester. Then I wake up.
I used to have this exact same dream for years after graduating from college.
I used to have this exact same dream for years after graduating from college.
I wish someone had told me about these dreams before I went to school/uni, so I wouldn't have ever slacked off causing me to have these worries in the first place.
If it makes you feel better, they can still happen anyway! yay!
As a student or teacher? I've had exactly this recurring dream, forgetting to teach a class. What a nightmare when they are not ready for the exam.
For some reason this is a reoccurring dream for me as well, and its been a few years since I've taken a class.
I still have those dreams -- except now I know they are going to take away my degree, and thus job.
Had those about high school recently and I'm 35 and in grad school.
I actually had a similar type of dreams as mark_l_watson when I took quantum chemistry (that's not a drug... for most people). Since I switched to performing more mundane tasks for a living, my dreams shifted to me hopelessly trying to navigate the very real, although warped and endlessly labyrinthine, dim interior landscapes of my university's library which I last visited more than 10 years ago. It was built around 1970 and it is a kind of trip during waking hours already.
> being late to college exams while having no pants on.
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.
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.
Maybe you only remember nightmares? I have many, and they obviously stand out.
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.
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.
[deleted]
I have both types. Something like what you've just described and then flying, traveling through insane and beautiful places and spaces. And it usually goes in periods. Few days of one type and then few days of the other. In between I might not see / remember nothing for a few days as well. I am 60 if it means anything.
> being late to college exams while having no pants on.
An upgrade would be 4D underwear.
An upgrade would be 4D underwear.
Have you read the Three-Body Problem trilogy? There's some stuff in the third book (Death's End) that sounds a lot like your dream. Specifically the different dimensions of space. If your dream didn't come from that book, perhaps you should consider writing some science fiction!
Personally, i really liked the series!
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.
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.
Neat!
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!
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!
> everything was happening in lower dimensional manifolds curving through that space where “all the action was.”
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.
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.
Sort of like the manifold hypothesis in deep learning.
> an infinite dimensional orange is all skin
Huh?
Huh?
As you increase the dimensionality of a sphere, an ever-increasing proportion of the sphere is within epsilon of the surface.
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.
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.
Fun facts, all major scriptures from Abrahamic religions (Jews, Christians and Islam) mentioned about multiple layers of heavens/skies where seven (7) is the top most layer. It also mentioned in the Quran [41:12] that the galaxies/stars/planets, are all residing in the lowest heaven/sky[1].
[1]https://quran.com/41/12?translations=21,20,19,17,101,22,95,1...
[1]https://quran.com/41/12?translations=21,20,19,17,101,22,95,1...
Thanks for sharing this. I love dreams like this. Flying, space travel, and alternate physics are some of the most fun. This actually makes me want to go to sleep and have a nice dream.
Does anyone know of any communities specifically for sharing awesome dreams? DreamerNews?
Does anyone know of any communities specifically for sharing awesome dreams? DreamerNews?
Very cool, but I have a question.
> 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.
> 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.
The w coordinate is there to aid the perspective transform and translations. Because in homogeneous coordinates everything becomes a matrix multiplication. At the end, before rasterisation, the coordinates are normalised by dividing by w. Yes, you can use it for whatever else you want, GPUs are just vector processors, as long as you keep in mind how the rasterisation step will happen.
Thanks. I'm aware of all those uses. But that strongly suggests that he's constructed a simulation of projective / elliptic 3-space, not spherical 3-space. The image on the screen is a perspective projection thereof.
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.
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.
The 4D coordinates are almost treated as ratios. (x,y,z,w) = (tx,ty,tz,tw) when t is positive, but if t is negative it's a different point.
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...
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...
Another thing maybe worth mentioning: the magnitudes matter when GL is doing interpolation, which is especially noticeable when there are textures.
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.
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.
As long as you normalise by only multiplying/dividing by positive numbers rather than any nonzero number, you’re really working in S^3.
If you're interested in this, you may be interested in Hyperbolica, which is a whole game about this concept:
https://store.steampowered.com/app/1256230/Hyperbolica/
https://www.youtube.com/watch?v=EMKLeS-Uq_8
https://store.steampowered.com/app/1256230/Hyperbolica/
https://www.youtube.com/watch?v=EMKLeS-Uq_8
Hyperbolica is mostly about the opposite, a hyperbolic universe, but there may be some section to the game that will be in a spherical universe. There's definitely some devlog video set in a spherical universe: https://www.youtube.com/watch?v=yY9GAyJtuJ0 The end of the video suggests this will be in the final game, at least as the time that video's creation.
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.)
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.)
And HyperRogue
http://www.roguetemple.com/z/hyper/
http://www.roguetemple.com/z/hyper/
These are great!
(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.
(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.
Hyperbolica is to have VR support too, the creator discussed [0] how he had to use a specific technique to ensure the rendering was compatible with VR optimizations.
[0]: https://youtu.be/rBr-0bHQfxc?t=357
[0]: https://youtu.be/rBr-0bHQfxc?t=357
Looks like 4D toys is VR-supported: https://store.steampowered.com/app/619210/4D_Toys/
Antichamber too
https://store.steampowered.com/app/219890/Antichamber/
Antichamber has lots of weird, cool stuff in it, but nothing to do with non-Euclidean geometry. (They just use the term incorrectly.)
'Nother 4Dgame. An old classic, just get the ball out of the (hyper)cube.
http://harmen.vanderwal.eu/hypercube/
http://harmen.vanderwal.eu/hypercube/
Specifically, the video about spherical geometry:
https://www.youtube.com/watch?v=yY9GAyJtuJ0
Sokyokuban, a non-euclidean sokoban
https://news.ycombinator.com/item?id=25892889
https://news.ycombinator.com/item?id=25892889
What happens when cats appear?
Does anyone know of a good visualization of a hypercube? Specifically, I'd like to be able to rotate it around various axes to see what its shadow looks like in 3D. On that note, a VR rendering might help since a 2D projection overcomplicates the shadow.
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.
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.
4D Toys from Marc Ten Bosch: https://4dtoys.com/
Based on the same 4D geometry concepts he's working on the game Miegakure: https://miegakure.com/
Based on the same 4D geometry concepts he's working on the game Miegakure: https://miegakure.com/
This demo is actually fairly close to showing you a 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.
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.
> rotate it around various axes
Wait, doesn't it rotate around a plane?
Wait, doesn't it rotate around a plane?
I created a Hypercube screen saver back in the day from instructions provided in a Scientific American article, which if memory serves was written by Martin Gardner.
A casual search leads me to believe the article is in his book Mathematical Carnival, though I could be wrong.
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.
http://harmen.vanderwal.eu/hypercube/
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.
http://harmen.vanderwal.eu/hypercube/
Back in the day this Java applet by the famous Ken Perlin was fun:
https://mrl.cs.nyu.edu/~perlin/demox/Hyper.html
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.
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.
You could use different projections, like gnomonic or stereographic projection. There won't be any time dimension to deal with, but it should make it easier to apply rotations and see the effects.
Stella4D free demo: https://www.software3d.com/Stella.php#stella4D
reminds me of the demonstration of hyperbolic geometry in VR by Henry Segerman (https://www.youtube.com/watch?v=ztsi0CLxmjw). there is a browser demo too where you can 'fly' through the small world (http://h3.hypernom.com/)
I thought this might have been something about a Greg Egan book (https://www.gregegan.net/DICHRONAUTS/DICHRONAUTS.html is a recent one with space-time composed of two dimensions of space and two of time), but these demos are really fun too. Especially like the connectedness of everything that's not in some sense cheated with portals.
Being able to set off a chain reaction in time 1 and then inspect that chain reaction at various moments in time 2 would be fascinating. Can you imagine a bullet hell in two dimensions of time? Or even a physics puzzle game that operates in two dimensions of time?
Egan's Diaspora also deals with this in some chapter. https://www.gregegan.net/DIASPORA/15/15.html
I don't understand anything in that page but the pictures are great.
I don't understand anything in that page but the pictures are great.
I once read a sci-fi book about vehicle that allowed the crew and it to get smaller and smaller. Nothing very special right? We've seen Antman. But when the travelers got really nucleus small they saw they were approaching a ball, and it turned out that ball was a universe and as they entered it they could then start exploring its things and as they get further smaller they got into the atom-size in that universe and then again saw a set of spheres and could go into them, and get smaller there and so on, without end. Within atoms there were universes within which there was more universes and so on ad infinitum.
Does that make sense? I've forgotten the name of the book.
Does that make sense? I've forgotten the name of the book.
Seems similar to "He Who Shrank" - I didn't read the book though.
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.
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.
After exploring this for a while, scrolling through these comments with the up and down arrows was a bit disorienting for me. The illusion of of navigating more than 2 dimensions persisted a little while on this page for me.
Would love to see this where you can toggle between a three-sphere, flat, and 3-hyperbolic space.
And arbitraily gluable diffrrentiable manifolds!
Maybe? I suggested it because these are the three possibilities for the shape of the universe and it does a great job of visualizing the 3-sphere!
That is one of my fantasies, if I had enough time. Apply different rules to different regions of space.
Reminds me when I was trying to model a 3D space and my math was off.
This is cool!
I wonder why the viewport is only a 300px square.
I wonder why the viewport is only a 300px square.
Sorry! It is because I was too lazy to figure anything else out.
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.
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 didn't know Idris could do WebGL!
I am grateful for the idris2-dom library by Stefan Höck: https://github.com/stefan-hoeck/idris2-dom/ . It covered most of the web APIs I used.
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.
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.
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!