[HN Gopher] Geometry from Another Universe
___________________________________________________________________
Geometry from Another Universe
Author : dgellow
Score : 145 points
Date : 2022-01-18 12:20 UTC (2 days ago)
(HTM) web link (www.falsifian.org)
(TXT) w3m dump (www.falsifian.org)
| zackmorris wrote:
| 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.
| mariusor wrote:
| 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/
| ogogmad wrote:
| 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.
| JoeDaDude wrote:
| 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/
| enriquto wrote:
| > rotate it around various axes
|
| Wait, doesn't it rotate around a plane?
| mikewave wrote:
| 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.
| corysama wrote:
| https://apps.apple.com/de/app/the-fourth-dimension/id5042017...
|
| https://www.theverge.com/2012/3/29/2912766/the-fourth-dimens...
| p1mrx wrote:
| Stella4D free demo:
| https://www.software3d.com/Stella.php#stella4D
| ogogmad wrote:
| 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.
| Asooka wrote:
| 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.
| ogogmad wrote:
| 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.
| ericbarrett wrote:
| Would love to see this where you can toggle between a three-
| sphere, flat, and 3-hyperbolic space.
| Jeff_Brown wrote:
| And arbitraily gluable diffrrentiable manifolds!
| Jach wrote:
| 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.
| all2 wrote:
| 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?
| mark_l_watson wrote:
| I had a dream a year ago in which I was traveling at very high
| speed towards an edge of our universe. In the dream I entered a
| null space before entering the next universe in which all star
| systems obeyed solid state physics type laws: stars formed an
| outer lattice and planets in small lattices around stars.
|
| 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!
| mekal wrote:
| 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!
| melissalobos wrote:
| > 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.
| mark_l_watson wrote:
| Sort of like the manifold hypothesis in deep learning.
| Jeff_Brown wrote:
| > an infinite dimensional orange is all skin
|
| Huh?
| jerf wrote:
| 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.
| bckr wrote:
| 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?
| bopbeepboop wrote:
| ryandrake wrote:
| 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.
| all2 wrote:
| 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.
| jerf wrote:
| 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.
| geodel wrote:
| 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.
| robotnikman wrote:
| For some reason this is a reoccurring dream for me as well,
| and its been a few years since I've taken a class.
| skc wrote:
| 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.
| selimthegrim wrote:
| Had those about high school recently and I'm 35 and in grad
| school.
| ck45 wrote:
| 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 :)
| throw10920 wrote:
| I wonder why it's common? Easiest explanation would be
| extreme stress/mild trauma from the stress of college...
| mnadkvlb wrote:
| holy crap, i had the 'missing a course' nightmare roughly
| every 2 months for 2 years after graduating.
| advantager wrote:
| 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...
| unsui wrote:
| 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
| jvanderbot wrote:
| I still have those dreams -- except now I _know_ they are
| going to take away my degree, and thus job.
| kelseyfrog wrote:
| 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?"
| ballenf wrote:
| 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.
| jonsen wrote:
| 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.
| pumnikol wrote:
| 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.
| thealig wrote:
| 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/)
| pavel_lishin wrote:
| 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
| ranger207 wrote:
| And HyperRogue
|
| http://www.roguetemple.com/z/hyper/
| CyberShadow wrote:
| Specifically, the video about spherical geometry:
| https://www.youtube.com/watch?v=yY9GAyJtuJ0
| jerf wrote:
| 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.)
| jlpom wrote:
| Antichamber too
| https://store.steampowered.com/app/219890/Antichamber/
| JoeDaDude wrote:
| 'Nother 4Dgame. An old classic, just get the ball out of the
| (hyper)cube.
|
| http://harmen.vanderwal.eu/hypercube/
| echelon wrote:
| 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.
| kroltan wrote:
| 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
| asxd wrote:
| Looks like 4D toys is VR-supported:
| https://store.steampowered.com/app/619210/4D_Toys/
| maupin wrote:
| Reminds me when I was trying to model a 3D space and my math was
| off.
| rezmason wrote:
| This is cool!
|
| I wonder why the viewport is only a 300px square.
| hwers wrote:
| No css plaintext blogs is such a statement and cool signalling,
| love it.
| amelius wrote:
| Judging from the image: straight lines map to straight lines?
___________________________________________________________________
(page generated 2022-01-20 23:01 UTC)