Hacker Remix

Why 4D geometry makes me sad [video]

100 points by surprisetalk 8 months ago | 44 comments

tobr 8 months ago

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 8 months ago

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 8 months ago

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 8 months ago

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.

rowanG077 7 months ago

I'm not so sure about that. I don't think introducing an extra dimension is elegant. There are probably more elegant proofs.

stevage 8 months ago

Nice video, incredibly well made. But what an irritatingly clickbaity title.

CaptainFever 8 months ago

Install DeArrow. I have it installed, and its de-sensationalized crowdsourced title is "Solving 2D Geometry Puzzles Using 3D Reasoning".

jasode 8 months ago

>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 8 months ago

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 8 months ago

Titles like this make me never click the videos. Why would anyone be sad by 4D shapes? (that don't exist anyway)

paulpauper 8 months ago

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.

user070223 8 months ago

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 8 months ago

I suggest thinking outside the plane.

jagged-chisel 8 months ago

A clever strategy, but a name still eludes me

ithkuil 8 months ago

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"?

jagged-chisel 7 months ago

My reply indicates that that GP was instructed to think outside the plan rather than naming the thing “thinking outside the plane.” It’s a tongue-in-cheek critique of the lack of quotes around the phrase making ambiguous the intent of the parent to my post.

YetAnotherNick 8 months ago

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.

samsartor 7 months ago

For problem #3, I think you can also prove it with perspective. The 3 circles are identical unit spheres at different distances from the viewer. The spheres are connected by 3 infinite cylinders. The cylinders form a triangle, and so lie on the same plane. Under perspective, the plane has a vanishing line on which each cylinder's vanishing point must lie.

dyingkneepad 8 months ago

This channel has the best Linear Algebra explanations I've ever seen, and it also explains the basics of AI really well.

levzettelin 8 months ago

This video makes me happy. :)

itsthecourier 8 months ago

Why though?

UltraSane 8 months ago

Because humans can't visualize more than 3 dimensions.

JadeNB 8 months ago

> 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 8 months ago

> 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 8 months ago

> 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 8 months ago

But you can still draw the concepts of a cross section, right?

JadeNB 8 months ago

> But you can still draw the concepts of a cross section, right?

I'm not sure what that means, but my inability to draw is astounding.

quuxplusone 8 months ago

> 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 8 months ago

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 8 months ago

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.

UltraSane 8 months ago

Some cool examples of 4D games

https://4dtoys.com/

https://store.steampowered.com/app/2147950/4D_Golf/

Asraelite 8 months ago

It's possible, it just takes a lot (years) of practice.

patrickthebold 8 months ago

It's actually quite easy: first, you imagine n-dimensional space, and _then_ you set n = 4.

UniverseHacker 8 months ago

I use a similar technique to swim- one merely flies, while setting their location = underwater.

amelius 8 months ago

That's not imagining; that's reasoning.

ToValueFunfetti 8 months ago

Try setting imagining = reasoning and I think it will work out

Asraelite 8 months ago

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 8 months ago

Although I (and I assume Patrick above) were joking, I will back you up here- having studied and worked with physicists. Some people can visualize, through practice and psychological “tricks” higher dimensional spaces- and different people can do this to different levels. Some people can instantly intuit approximate geometric solutions to high dimensional problems without working out the math- so I know they are either visualizing it, or doing something similar. I can generally visualize 4D with some “tricks” that compress the problem. For example if you can imagine varying configurations of the same 3D space, that itself is essentially at least a non continuous 4th dimension. If you can imagine that evolving into new configurations, you are starting to visualize or at least imagine 5 dimensions. Some people appear to be able to so this for ~8 dimensions.

NBJack 7 months ago

Depth is the third dimension for our brains. You can argue it is faked as we don't gather the depth directly, but it is still something you intuitively understand and can visualize clearly. You can rotate an object mentally, fill in the gaps, etc. We may tend to store that data in 2D, but spatial reasoning is very much achievable in 3D. We do it all the time.

While I don't dispute your method works for your purposes, would you say it allows you to visualize more than a single 4D shape side by side? What about interlocking shapes? Can you place this shape in an arbitrary 4D space among others and describe its relative position?

ks2048 8 months ago

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 8 months ago

The first step is being open minded about it, otherwise you're right, it's not possible.

paulpauper 8 months ago

Isn't the 4th dimension time? So it's like objects moving in time and interesting each other.

nulbyte 8 months ago

Time is very different from space.

https://bigthink.com/starts-with-a-bang/time-yes-dimension-n...

dbrueck 8 months ago

Not when talking about geometry - the idea is a 4th spatial dimension.