[HN Gopher] Why 4D geometry makes me sad [video]
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Why 4D geometry makes me sad [video]
Author : surprisetalk
Score : 87 points
Date : 2024-11-08 18:23 UTC (1 days ago)
(HTM) web link (www.youtube.com)
(TXT) w3m dump (www.youtube.com)
| levzettelin wrote:
| This video makes me happy. :)
| itsthecourier wrote:
| Why though?
| UltraSane wrote:
| Because humans can't visualize more than 3 dimensions.
| Asraelite wrote:
| It's possible, it just takes a lot (years) of practice.
| patrickthebold wrote:
| It's actually quite easy: first, you imagine n-dimensional
| space, and _then_ you set n = 4.
| amelius wrote:
| That's not imagining; that's reasoning.
| ToValueFunfetti wrote:
| Try setting imagining = reasoning and I think it will
| work out
| Asraelite wrote:
| I'm getting downvoted now :/
|
| I think I'll write a blog post at some point explaining
| my process for visualizing 4D. Hopefully it should make
| it clearer what I mean.
|
| I think most existing resources on the topic go about it
| in a way that makes it hard to build up a proper
| intuition. They start by assuming that humans can
| visualize 3D and then try to extend that one dimension
| higher. But humans can't actually visualize 3D, only 2D.
| We combine multiple different 2D perspectives together to
| "fake" an understanding of 3D. Our vision is also only
| stereoscopic 2D, not true 3D.
|
| If you take a similar approach with 4D, trying to project
| directly from 4D to 2D instead of going through 3D as an
| intermediate step, it's harder to visualize at first but
| better in the long run for really understanding it.
| UniverseHacker wrote:
| I use a similar technique to swim- one merely flies,
| while setting their location = underwater.
| ks2048 wrote:
| I'd say it's not possible, but that depends on your
| definition of "visualize" (I guess maybe 4d as 3d+time
| would also count)
| Asraelite wrote:
| The first step is being open minded about it, otherwise
| you're right, it's not possible.
| JadeNB wrote:
| > Because humans can't visualize more than 3 dimensions.
|
| It's awfully hard to prove that real-world things are
| impossible, especially if there's no objective measurement of
| whether they've been achieved. (For example, if I tell you
| that I can visualize more than 3 dimensions, then how could
| you verify or disprove that?)
| epidemian wrote:
| > For example, if I tell you that I can visualize more than
| 3 dimensions, then how could you verify or disprove that?
|
| I don't really know. The first thing that came to my mind
| would be to ask to draw/model different cross-sections of a
| 4D object ("cross-volumes"?).
|
| We can visualize 3D objects, and therefore can draw 2D
| cross-sections of 3D objects relatively well, and
| relatively easily. Like, sections of a human body, or a
| house. So, maybe someone who can visualize 4D objects in
| their head could also model 3D "cross-sections" of that
| object at arbitrary "cuts". And we could check if those 3D
| radiographies are accurate, because we can model those 4D
| objects on a computer, and draw their 3D cuts.
|
| Just a simple idea. I'm sure there could be other ways of
| probing this.
| JadeNB wrote:
| > We can visualize 3D objects, and therefore can draw 2D
| cross-sections of 3D objects relatively well, and
| relatively easily.
|
| Many people can draw cross-sections reasonably well, but
| I can't. Nonetheless, I believe that I can visualize 3D
| objects.
| wongogue wrote:
| But you can still draw the concepts of a cross section,
| right?
| quuxplusone wrote:
| > We can visualize 3D objects, and therefore can draw 2D
| cross-sections of 3D objects relatively well, and
| relatively easily.
|
| I don't think that's true. For example, consider a
| regular octahedron: take a parallel pair of its faces and
| bisect the octahedron between those faces. What's the
| resulting figure? What happens to the figure as you tip
| the plane?
|
| I mean, obviously the task I just set isn't _impossible_
| ; and with a little reasoning anyone can give the answer
| in a few seconds; but it feels too me like the answer is
| _not_ simply intuited merely by the virtue of our being
| 3D creatures.
|
| Sure, part of the difficulty stems from that the
| octahedron (to most folks) is both less familiar and
| slightly more complicated than the cube. But the same
| applies to the hypercube!
| UltraSane wrote:
| Humans have had tremendous evolutionary pressure to develop
| excellent 3D visualization abilities. Humans have had
| exactly ZERO evolutionary pressure to develop 4D
| visualization abilities. If someone claimed to be able to
| visualize 4D the exact same as they can visualize 3D I
| wouldn't believe them. Maybe there are some kind of tests
| that could be done to prove it?
| teemur wrote:
| My low, low priority side project has been to make a VR
| app that lets you (create and) move around 4d
| environments. I'm pretty sure that navigating even
| moderately complex 4d environment can be used as some
| kind of proof for 4d visualization/intuition
| capabilities. Level 6 on my toy maze game has all of 4
| tesseracts within the maze tesseracts and after some
| hours of playing, I am still completely lost there.
|
| If someone wants to have a look, feel free (but it is
| hard. You need time. If you do not have vr headsets but
| want to have a look, you can install browser add-ons thet
| let you simulate vr headset and controls):
|
| https://www.brainpaingames.com/Hypershack.html
|
| I have been planning to write a Show HN any day now for
| months, but maybe someday.
| paulpauper wrote:
| Isn't the 4th dimension time? So it's like objects moving in
| time and interesting each other.
| dbrueck wrote:
| Not when talking about geometry - the idea is a 4th spatial
| dimension.
| nulbyte wrote:
| Time is very different from space.
|
| https://bigthink.com/starts-with-a-bang/time-yes-
| dimension-n...
| user070223 wrote:
| I first encountered this problem solving technique of looking at
| the problem in higher dimension from lectures by tadashi
| tokieda(referenced in the video). I highly recommend any video of
| him.
|
| I dont rememer finding many examples, nor a reference to it from
| common problem solving techniques lists(terry tao, aosp? etc). I
| think it deserve it's place with a catchier name perhaps
| HappyPanacea wrote:
| I suggest thinking outside the plane.
| jagged-chisel wrote:
| A clever strategy, but a name still eludes me
| ithkuil wrote:
| You're either bantering in a higher dimension or you're
| talking past each other.
|
| Anyway, perhaps a more down to earth "thinking outside the
| hypercube"?
| YetAnotherNick wrote:
| Such a great video. Changing from sphere to cone in proving
| Monge's theorem makes the proof so much better, and way easier to
| visualize. I guess the proof hasn't caught up in other places is
| because if the proof is in writing sphere could be visualized
| first or the sphere gives more aha feeling.
| stevage wrote:
| Nice video, incredibly well made. But what an irritatingly
| clickbaity title.
| paulpauper wrote:
| yeah I find that annoying. If the video does not quickly answer
| the title yet uses the title to entice curiosity, then it's
| clickbait. I expect to find out what it makes it sad within
| minutes of clicking, not have to wait.
| CaptainFever wrote:
| Install DeArrow. I have it installed, and its de-
| sensationalized crowdsourced title is "Solving 2D Geometry
| Puzzles Using 3D Reasoning".
| jasode wrote:
| _> crowdsourced title is "Solving 2D Geometry Puzzles Using
| 3D Reasoning"._
|
| That attempted revision of the title is worse than the
| original clickbait title because it totally omits the _" 4D"_
| topic. The 4D section starts around 18m50s and is 1/3rd of
| the content. The 2D/3D section was the _prelude and
| motivation_ to prepare the viewer for the 4D section.
|
| The _" wisdom of the crowd"_ failed in this particular case.
|
| Any attempted improvement on the title still needs to have _"
| 4D" somewhere_ in the title.
| mschae23 wrote:
| It hasn't failed. Not if you do what crowdsourcing requires
| and suggest a better replacement.
|
| I agree that "Solving 2D geometry puzzles using 3D
| reasoning" does not fit the actual topic of the video, so I
| have replaced it with "Geometry puzzles with 4D analogies"
| for now - which is also not _quite_ right, since the
| largest part of the video doesn 't refer to any 4D
| concepts. I think it's, more generally, about puzzles that
| can be more easily solved by considering _any_ other
| dimension, but it 's harder to make a good title out of
| that. Do you have another idea for an improvement?
| efilife wrote:
| Titles like this make me never click the videos. Why would
| anyone be sad by 4D shapes? (that don't exist anyway)
| dyingkneepad wrote:
| This channel has the best Linear Algebra explanations I've ever
| seen, and it also explains the basics of AI really well.
| tobr wrote:
| Can someone explain why the second puzzle would need this fancy
| three-dimensional solution? The area of each strip doesn't seem
| important to solving it:
|
| We need to cover the circle as efficiently as possible. That
| means having exactly one layer of strips. Zero layers doesn't
| cover it, and two or more layers are wasted. As soon as you start
| using strips at different orientations you can't escape an
| overlap somewhere. So, clearly the optimal way to do it is to use
| some number of parallel non-overlapping strips, and their total
| width will be the diameter of the circle.
|
| Not sure if this isn't rigorous enough or something, but it seems
| perfectly clear to me.
| gjm11 wrote:
| It is absolutely not rigorous enough, I'm afraid.
|
| Sure, it feels wasteful to cover any part of the disc twice.
| But it _also_ feels wasteful to cover bits near the edge with
| strips that have only a short length of overlap with the
| circle. It 's not obvious that there isn't some clever way to
| reduce the second kind of waste that requires you to commit the
| first kind.
|
| Perhaps the following observation will help. Replace the circle
| with a "+" shape made out of five equal squares (like the cross
| on the Swiss flag). The most efficient way to cover _this_ with
| strips is (I think -- I haven 't actually tried to prove it) to
| use two strips "along" the arms of the cross. Those overlap in
| the middle, but they still do the job more efficiently than
| using (say) a single strip of 3x the width.
|
| So, how do you know that nothing like that happens with a disc
| instead of a cross?
| tobr wrote:
| That's a great explanation, thank you!
|
| I feel like it should be possible to pinpoint why this can't
| happen with a circle. Specifically, it seems to require some
| kind of protrusion in the shape, that ends up being more
| efficient to cover in a direction that creates an overlap.
| But it's clearly not as straightforward as I thought.
| jacobolus wrote:
| Any kind of "pinpointing" you can do is probably going to
| end up equivalent to the project-to-a-sphere method, except
| less clear or pretty.
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