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4D toys (4dtoys.com)
592 points by wenderen on Aug 4, 2018 | hide | past | favorite | 110 comments

Here's a thought I had watching the bit with 2D man:

Actually, the 3D view that 2D man does not understand but which we do understand is... still 2D. My screen is flat.

You can use a 2D viewport to render a 3D scene in a way that is natural and easy to understand for us humans: A human watching the 2D scene can very quickly surmise from a glance at the viewport: Which objects are in the scene, and where are they located, in _ALL_ 3 dimensions?

This raises the question:

Can you render a 4D scene onto a 3D viewport such that us humans are pretty good at understanding where every object in the scene is, in all 4 dimensions?

I assume the answer is 'yeah, you can do that'. I wonder what that would look like.

It's complicated of course; our eyeballs are involved and they kinda work in 2D and not 3D; where we humans can casually glance at a 2D viewport rendering a 3D scene for a second and know what's happening, we'd have to walk around the 3D screen rendering a 4D scene in order to even see everything.

The problem is that while everything you see on your (2D) monitor is effectively 2D our brain is good at taking 3D cues (shadows, perspective, parallax etc...) to reconstitute 3D "data" from it. For instance look at 3D rotating cube on your monitor: https://www.youtube.com/watch?v=5E10zYKln3g

Even though there's no shading, a bad resolution and zero context anchoring it in a familiar setting you immediately see it as a 3D object when effectively that's just 3 2D quadrilaterals changing colors and shape. We really need very little to be able to extrapolate depth information from a very limited 2D-only display.

If you did that with 4D cues in 3D space (say in VR or something) I think it wouldn't work. It'd just look like a 3D object changing shape, not a projection of a higher dimension (the same way an hypothetical 2D being would see 3 quadrilaterals changing shape in the above video, not a solid 3D object that would be an abstract concept to them, not a familiar reality).

Could we teach our brain to "think in 4D" and interpret these cues differently? To a certain extent probably, maybe if you spent hours and hours and hours in a VR simulation with 4D objects you'd start getting a feel for it. I doubt you'd ever get as good as with 3D objects since we've been exposed to these literally every waking moment since birth, but maybe I underestimate the plasticity of our brain.

That actually leads me to an other question: is 3D somewhat hardcoded in our brains or is it purely learned? If we were mad scientists and took a baby brain into a 4D "plato's cave" style simulation, could it grow into being able to perceive 4D as intuitively and effectively as we do with 3D space? Also unrelated question: does anybody have a baby I could borrow?

> Could we teach our brain to "think in 4D" and interpret these cues differently? To a certain extent probably, maybe if you spent hours and hours and hours in a VR simulation with 4D objects you'd starting getting a feel for it. I doubt you'd ever get as good as with 3D objects since we've been exposed to these literally every waking moment since we're born.*

Good lord! Has anyone tried?

This reminds me of the backwards bicycle video on youtube (https://www.youtube.com/watch?v=MFzDaBzBlL0). I... really want to try what you're suggesting.

In the YouTube video he says:

"So here's why I did. It was a personal challenge. I stayed out here in this driveway and I practiced about 5 minutes every day. ... after 8 month it happened ... in two weeks he (the son) did something that took me 8 months to do".

Well, if you were serious about learning a skill you wouldn't just do one 5 minute session per day for eight months.

A more appropriate training regime would be way more intensive than one 5 minute session per day. Are we to believe he limited his son to one 5 minute session a day?

Do children really learn languages quicker than adults? By age 3, children will probably have words for almost everything. Babies might even say mama and dada by 6 months of age.

Children have several language learning advantages over adults: complete immersion; it is imperative they learn; effectively unlimited time; no responsibilities.

But, it takes them years to learn their native language to basic competency.

Compare: an English speaker can learn Afrikaans, Danish, Dutch, French, Italian, Norwegian, Portuguese, Romanian, Spanish, or Swedish to General Professional Proficiency in Speaking and Reading in 600 hours carried out over 24 weeks (25hrs per week). Cantonese, Mandarin, Japanese, Korean, or Arabic will take 2200 hours over 88 weeks (25hrs per week).[1]

Is it easier for a child? Yeah probably, they don't even have to look after themselves.

If I could live as a child in a foreign language house in a foreign language city, with only two tasks: learn to speak and read the language to basic competency, and learn to ride a backwards bicycle, I'm firmly of the believe I could out pace the 95th percentile of children at both tasks.

As an aside, he says his son is the closest person to him genetically, but aren't his parents both equally as close to him genetically as his son?

1. http://www.effectivelanguagelearning.com/language-guide/lang...

> As an aside, he says his son is the closest person to him genetically, but aren't his parents both equally as close to him genetically as his son?

And any siblings.

Yeap, https://youtu.be/oI2aMKwXXnE this guy took a more serious attempt at learning to ride the backwards and it seems it didn't take him that much time.

Thanks for finding that.

So it only took him about an hour, over three days, to get it worked out well enough to not immediately fall off, and an hour and a half over four days to get to 50 meters.

There's a comment in on the YT video from Destin saying "the RC helicopter pilot was able to learn it in about an hour but I don't think his brain is normal" - good point, the RC helicopter pilot has more experience understanding reversed input while the helicopter is flying toward him, also he had the opportunity to watch Mike learn first.

I reckon the learning process could be sped up even more by taking the pedals of, turning it in to a balance bike, learning how to ride it down slight inclines, get that sorted then putting the pedals back on.

Yes mathematicians have tried to visualize 4d for quite some time. I remember hearing about one who famously spent much time to developing a notion of 4d.

It’s possible to represent n+1-d scenes into n-d scenes.

That was a fascinating video. I wonder if the clicking in the brain is similar to the eye crossing effect of those 3d from noisy pictures.

Here is one version of 4D that our brain understands and can easily map from 4D back to 3D, and that's when the 4th dimension is time (or motion): http://www.dyscario.com/arts-and-culture/sculptures-in-motio...

I like that. It would be very interesting to have something that would use real life resolution human models and an option to control the time step.

For your question about 3D being hard coded or learned, I think I remember reading that it’s learned.

This is an old philosophy debate about wheather if I blind person could suddenly see if they’d understand that a sphere is round just by looking at it.

When some people that had had cataracts for 50 years were treated they not only couldn’t tell that a sphere was round, they couldn’t understand shadows or depth of field/distance. They thought shadows on people were black splotches and when something was moving away that it was actually getting smaller.

I suspect visual input gets trained on our neural network like anything else, though we do have some specialized hardware for it.

The article describing blind people getting vision again: https://www.google.com/amp/s/www.newyorker.com/tech/elements...

Yeah this is the one I was thinking about! Thanks for finding/linking.

I think you can't. The problem is that we don't actually see 3D, we basically see 2D twice. This gives our senses enough info to reconstruct a 3D mental image, but the information we gain isn't truly 3D.

Actual 3D-sight might include us seeing the inside of objects, what's behind them, etc.

We're aided in converting our 2x 2D sight into 3D mental models through the use of color and parallax. In projections on images, it's easy to just use color and get rid of parallax, and still get sort of close.

Suppose we tried to make a 3D projection of a 4D world, we don't actually get any significant new tools. We're still going to be limited by having to use parallax and color. This doesn't allow us to add enough information to add that 4D projection into 3D as neatly as the 3D into 2D works. At best you could try to distribute the information somehow. For example by giving information about the third dimension through brightness and the fourth dimension through hue. It certainly wouldn't work nearly as well though.

FWIW this is really not true. (The bit about stereo vision being required to see 3D.)

When I was in college I was fairly decent on the ping pong table, call it high amateur. I also have very very bad vision - -5 diopters in one eye and -6 in the other. At the time I wore contacts and I was also really really poor. At some point I lost one of my contacts and for six months I was effectively blind in one eye for the purposes of ping pong. It took me about a month, but fairly quickly I was playing at my old level without anything close to stereo vision.

Did you play with only one eye open, or just one contact lens? I have worse vision than you (-8.0, -7.50), and I know that I can still function with just one lens in of needed. You definitely still have stereo vision even if one input is blurred. Your brain does quite a bit of compensation, and I can imagine if you operated this way for months that your brain would be even better at assimilating the noisy information from your "bad" eye.

Sulam is basically correct. I was born with strabismus (my eyes don't focus on the same point). To avoid perpetual double vision, my brain compensates by basically only using one eye at a time. It switches automatically when I'm looking to the side, and I can also switch consciously. (My color perception is very slightly different in each eye, it's weird.)

It turns out that stereovision is the least important of several depth cues humans use, and is only really effective out maybe 10 feet or so--beyond that the parallax is too small to give much useful info, and we rely on apparent size, relative motion, surrounding context, etc. I can catch a ball just fine, as an adult.

It's really only an issue within arm's reach, and then only when I'm distracted. Occasionally I'll reach for something without paying attention and miss by, like, a foot. Also, I can't use 3D glasses. (With the new polarized kind I can at least wear them and see a normal 2D movie. With the old red/blue kind, everything would be either red or blue, depending on which eye I was using.)

It isn’t just depth perception that is improved with binocular vision, there is binocular summation, which helps improve acuity. Two is better than one. However as you say, stereopsis seems to be one of the least important parts of sight.

You might have an advantage with not being able to watch some 3D movies - you can tell your friends/family that a medical condition prevents you going to them.



> (My color perception is very slightly different in each eye, it's weird.)

I have comparatively uninteresting/normal vision (biggest issue shortsightedness) but I have this too. My left eye sees more blue, my right eye more red. Looks like the cone distribution wasn't perfect.

I'm not sure, but I think the sun's rays don't have much of a blue component: my left eye gets fractionally less sore from bright sunlight, so if it's really bright out I'll generally be closing my right eye.

The sun emits black body radiation with a 5800K temperature, which means its emissions peaks at yellow-greenish, but it still emits around 90% of blue compared to green

How about compared to red? That might suggest whether exikyut's experience (red-seeing eye more light-sensitive than blue-seeing eye) is coincidence or related.

I was trying to make the same point, I think? I was just trying to note that I have done the single contact lens thing as well, and even after just a few hours, your brain starts compensating for it.

I played with one contact, and if you have vision as bad as mine you know there is essentially no chance that I would be able to build a stereo image of the ball with one eye uncorrected. I know this because I had to relearn how to play, and I found myself using other cues for depth, like what the ball was occluding.

I crushed my glasses with my butt once, and went without for a year or so until I could get new ones.

Once I got the new glasses I was astonished at how flat and "PlayStation-like" everything looked. My depth perception was thrown off and everything looked much closer to me than it actually was.

I think my brain was using blurriness as an indicator of depth, in addition to binocular vision, and with my myopia corrected it lost that information and had to readjust to relying primarily on binocular vision to gauge depth.

i would argue that we don't see in any dimension, at least in terms of the physical capacity of the eye. we just take in varying wavelengths of light and those have no intrinsic dimensionality to them per se. the actual manifestation of dimensionality is, at least in my opinion, merely a construct of the brain. while I think it's accurate to represent reality as a multidimensional thing mathematically and formally, i don't know if dimensionality is an intrinsic feature of it or just a way of categorizing our concept and experience of it. either way, the fact of the matter is that the eye merely takes in raw photons and passing messages on to the brain based on the frequency and intensity of the light, and after saying that i suppose i would be willing to admit that the core nature of light and waves in general is two dimensional, since a wave is defined in terms of frequency and amplitude.

> The problem is that we don't actually see 3D, we basically see 2D twice.

Most of us do. But people who only have sight in one eye have no difficulty perceiving three dimensions. And some fairly large percentage of people with binocular vision are stereoblind, typically without even realizing it.

If anything, I'd say it's not that

> We're aided in converting our 2x 2D sight into 3D mental models through the use of color and parallax.

so much as that we're aided in converting our 2D sight into 3D mental models through the use of stuff like color and parallax through the use of stereopsis.

As you mention, binocular vision is lacking in a surprising number of people and until recently this was thought to be uncorrectable if not treated while the child was still young. That turned out to be not entirely accurate.

Those without binocular vision can have a fairly large range of non-specific symptoms and perception problems. There is an interesting discussion of the change in understanding in this link. http://www.strabismus.org/all_about_strabismus.html

Study from 45 years ago:

Heinz von Foerster’s 1970-1971 experiment at the Biological Computing Laboratory for apprehending the fourth dimension is unique ... combining four dimensional geometry, stereoscopic vision, and joystick manipulation of objects on the screen. ... The fourth dimension was chosen as the knowledge to be acquired because there was no chance that any subjects would have attempted such knowledge before the experiment. By allowing the physical “grasping” of the visual object, where one hand coordinated movement on three axes in the 3rd dimension, while the other similarly controlled three axes of movement in the 4th dimension, subjects were able to intuitively figure out that the strange succession of transforming 3D objects they were seeing (with 3D glasses) were cross-sections of a single 4D object.


Disclaimer: I studied with von Foerster years ago, and he'd mentioned this. I only today found this online reference. I want those 4D Toys. Steam, here I come.

The next thing to come from the same person, Miegakure[1], is going to be even more epic and mind-blowing. But it will take a while for it to be done.


Unfortunately, it's not as simple as that (although I wish it were). While you could generate a 3D perspective on each 2D "slice" of a 3D viewport, it wouldn't be that much more different than what's going on in here. The main issue arises from trying to render a 4-polytope in a universe where we ourselves can only sense 3 dimensions "natively."

Sad thing is that what we sense (see) is in fact just a poor projection into 2D (aka perspective) with some largely learnt concept of depth. If only we could sense space entirely with nothing "behind", not just from single point...

i think it's great that you represent it as a projection because that is exactly what it is: a representation of another, perhaps more "objective" (though I'm not sure at that level that attributes like that apply) data set. I think it's super easy for us to assume what we visually perceive has a 1:1 correspondence with the intrinsic nature of the data it is constructed from. and i think it's also important to realize that that construct is also informed by other avenues of sensory input and that extra context tends to take it even further away from the nature of the raw data it is constructed from

Perhaps it is determined by the fact senses are bound to body and body is limitary. Our 2d vision could grasp entire Flatland (if it fitted into single view), but sight of similarly finite Flatland inhabitant could grasp just his immediate surroundings, i.e. the sum of closest opaque objects in his 1D sight.

I asked the same question a few seconds before you :) found this: https://youtu.be/S-yRYmdsnGs?t=252

and even better, this: https://youtu.be/dy_MUfBuq2I

Yes, and I think this just shows how a 3d representation on 2d gives our brain the _illusion_ of 3d. This doesn't work for 4d, because our brain doesn't have an internal understanding of 4d objects.

I think once we have it interactively our brain will adapt.

It’s t least worth a shot.

>our brain doesn't have an internal understanding of 4d objects I agree and I think it will never have. We can certainly speculate but ultimately it'd be like an animal with monochromatic vision trying to see colours.

Yes. there are many ways to do 4d to 3d projections, including perspective projections. You could also make a 3d 'camera' view with 4d lens and 3d projection plane.

Here is a good intro to 4D Visualization http://eusebeia.dyndns.org/4d/vis/vis

Wow, I had the same idea a while back and thought about writing some kind of engine which does this - but I didn't even begin to research whether someone already did something similar.

Now I see this on HN :)

So I searched and found this: http://www.urticator.net/maze/

It seems to be exactly what you propose, a 4D world rendered into 3D space. It even has a kind of "stereo" mode, enabling a 3D experience on a 2D moniter.

And to expand on the idea: Once we become familiar with 4D, we can continue and use it to explore 5D, can't we?

If you start from the understandings (such as we have)of physiology and evolutionary psychology I think the answer is "no" (perhaps there's a trick someone can figure out via a different path -- I hope so).

It seems pretty clear that looking at 2D images (and in particular, still ones) is a learned task, like reading, rather than an innate one. Both appear to tie into deep structures in the brain, but both are very recent inventions.

From the perspective of developmental stages described by Piaget, children learn to view in three space primarily with objects within reach (parallax pretty much peters out around the ends of your arms. Once the child becomes mobile, she is able to use semantic understanding to estimate the size of distant objects and get a rough idea of distance. The whole human process of seeing is very different from the way, say, a NN is trained on an image: the whole thing isn't gulped in at once, but we foveate on various parts of the image and assemble / confabulate a whole. You can see this in the structure of Chinese classical painting or pre-persepctive European paintings: distant objects aren't sized in any way proportional to their apparent size. This really maps more to how much attention you pay to various objects in the scene.

You then learn to map that into a 3D model which I believe (but am not digging up refs this instance, sorry) has hardware support.

Thus the 2D->3D process exploits a lot of learned and innate knowledge and technique that you have already developed. With one exception you haven't any 4D experience. That one exception is temporal data -- we can easily extrapolate from, say, a sphere shrinking and growing. Apart from that, there isn't much to work with.

There are YouTube videos where people attempt to render 4D scenes, usually by slicing a mesh along the fourth component of its position, then drawing it as 3D. But since we lack the intuition for 4D space it just looks like a bunch of objects popping in and out of existence.

I don't think the problem is rendering but understanding. Your brain reads a cube, as eh, a cube. If I give you this [2,4,4,6]+ and tell you rzwitserloot that is a cube in the space. You can still imagine that.

So a 4D rendering is just more information in a 2D flat screen. Your brain can't process all of this information. So it "can't see it". But it is there.

+dunno if these coordinates translates to a cube or just some 2-3D object but I hope you get the point.

> Can you render a 4D scene onto a 3D viewport [...] I wonder what that would look like.

It would look like VR. 4D toys supports VR already. I tried it but it did not advance my understanding further than the 2D version, although using 3D tracked controllers is definitely a big improvement over a 2D mouse for manipulating 3D objects or 3D projections of 4D objects.

You can’t reconstruct a 3D scene from a 2D projection unless you take educated guesses.

E.g. When you look at a picture with something that looks like a chair, you assume that it’s indeed a chair, and then you can estimate its size/pose/etc. But there are infinitely many non-chair shapes that would produce the exact same projection. It’s just that you won’t encounter them in real life, except maybe in trickshots like this: https://youtu.be/SKpOKXAVjGo

Has anybody tried any of the 4d maze games?


I've tried a maze game and a 4D space shooter before and I could never wrap my head around them. I don't know if it was just poor representations or if it was just because my brain is incapable of understanding.

If you use time for 4th dimension, you can easily do a 4d-scene on a 2d-screen

This is an old video [1], but it remains the clearest visual explanation of multiple dimensions that I've ever seen. I really can't recommend it highly enough. I think anyone trying to visualize a 4D object will get something out of watching it.

I played with the 4D toys app after it showed up on /r/math a while ago. I like it and I think it's useful. My only complaint would be that it's a little too open ended. While it's nice to provide a simulated tactile experience of four dimensions, I think the app should provide a bit more visual intuition. That's one of the things I like about this video. ________________

1. https://www.youtube.com/watch?v=90olwwLdEYg

That was really great, thanks for sharing!

Wow, cool to see this here. I've been working on reproducing some of it in numpy/matplotlib for the past 2 weeks! I liked their 3D-sliced 120-cell (equivalent of the dodecahedron), and it wasn't too difficult to reproduce.

Step 1: Use Schlafli generator from here [1]. Schlafli numbers are a compact description of regular polytopes, and there is a recursive algorithm to generate vertices, edges, faces, etc. from them. The base case of the recursion is dimension 1, so you make 4 calls to get to dimension 4.

Step 2: Intersect the edges of the polytope with a hyperplane (a 3D subset of 4D).

Step 3: You get a set of 3D points out of step 2. Draw the convex hull of them, which gives you triangles.

Step 4: Render the triangles somehow. I used matplotlib's 3d facilities (mplot3d), and we are working on raytracing them.

Step 5: Animate over different hyperplanes. Take the min and max in the w plane and that will give you non-empty slices. Now you can "see" the 4D polytope using time as the 4th dimension.

I sure he is doing something more advanced (4D collision detection), but this is all we needed to reproduce something that looks kinda cool.

[1] https://github.com/aruth2/schlafli

Can someone help me out here?:

I can represent 3D quite comfortably on 2D monitors, can there be an intuitive mapping of 4D to a 3D VR view?

I know 3D mapped to 2D suffers from occlusions and heavily relies on clues like perspective, shadow etc. But given enough time even a less intuitive 4D view could become intuitive with time, too.

edit: found this: https://youtu.be/S-yRYmdsnGs?t=252

even better: https://youtu.be/dy_MUfBuq2I (turn on subtitles)

I am by no means an expert, but I have thought about trying to visualise the fourth dimension. The issue is that we don’t really see 3D. While in 2D we can view every pixel simultaneously, when we view in 3D we have the issue of aspect and concealed voxels. You can’t visualise simultaneously all voxels of a cube, as most are under others or otherwise obscured. This therefore limits how much we can use 3D to mimic 4D. Perhaps if we were to spend more time with semitransparent objects our brains could learn to visualise 3D better and then maybe 4D.

What an interesting idea! Have you ever produced anything for a representation of 4D with translucent 3D? I'd be very interested to see it...

I'm not sure that could practically work. The problem, is that translucency would allow only a partial view of concealed voxels. As more voxels cover something, the translucent effect compounds.

Additionally, most voxels would appear different with different view perspectives. Due to more or less voxels covering them.

The problem is that in 4D all voxels are visible to the viewer. So viewing a 4D apple would allow you to see the apple from all possible view points simultaneously, including interior views.

To me it just doesn't seem possible to replicate this concept in 3D VR.

What would be the benefits of the 3D VR? There will always be a 2D bottleneck on your eyes' retina. The most important perspective clue I would say is temporal sampling. For dynamical systems, Takens theorem says that a high dimensional topology can be restituted by the sequential sampling of a single variable alone. It seems likely to me that our perception generally works by this principle. That being said, I think we are quite heavily hardwired to 3D perception. I suspect these circuits cannot be wholly overridden.

What would a 3D eye even be like? Is it something like having independent depth perception per eye?

Photons travel in 3D space and may only be occluded by 2D surfaces. There is no getting around that fundamental aspect of reality. In this universe.

This is why as an armchair physicist I'm excited about the new neutrino observations and research going on, mainly because neutrinos aren't affected by gravitational lensing and other effects other particles encounter as they 'roll around' on space-time. Would it be possible to understand conceptually how other dimensions extend out of space time? I'm imagining a Y-axis perpendicular to the space-time fabric....

Gravitational lensing is a distortion of spacetime. It affects all particles, including neutrinos.

I'd say we have two eyes and compile to something 3D, which is mostly possible because of that and nothing is hardwired, just takes a lot of responsive feedback to rewire itself.

We see a 2D projection and infer some 3D information from stereo parallax and other depth cues. But it’s still a 2D projection in the end. A real 3D ”vision” would not have a ”viewpoint”, no perspective, no foreshortening. It’s difficult to even imagine what it would be like to perceive that.

Stereo parallax, great term. In other words how cross-eyed we are.

The reason one can represent 3D "quite comfortably" on a 2D monitor can best be explained using linear algebra. The 3D space in the video is "projected" (linearly) onto a 2D space the same way a single one of our eyes project 3D space onto a 2D space. This is just one choice of projection though. For instance, when he shows the 2D beings perspective, he's chosen a different projection from 3D -> 2D and it looks quite foreign to us. The reason you feel comfortable with the normal projection is because your eyes do it all the time. Your eyes don't however, project 4D -> 3D so you don't feel it as natural.

Of course you can, and it would be different from the cross-sections seen in this video. I doubt it would ever become intuitive though, I suspect our brains are hard-wired for a 3d world.

Here's a video of a rotating 4d hypercube in a 3d perspective:


It's called a tesseract, and as each face of a 3d cube is a 2d cube (a square), each face of a 4d cube is a normal 3d cube, that we see skewed by the perspective.

I'd say our brain is anything but hardwired. Sense-substitution works beautifully too and even a tool like a hammer or a mouse becomes one with your body through frequent use.

Your linked video maps the 4th dimension to time, it doesn't project to 3D. Projecting through time (especially non-interactively) lacks the immediate feedback needed for the brain to grasp it as intuition.

> Your linked video maps the 4th dimension to time, it doesn't project to 3D

No, it projects it to 3d. The movements you see are rotations, not movements of an intersection plane. You can clearly see at any point in time each of the 8 identical cubes making up the tesseract, skewed and resized by projection to a 3d perspective.

Just like how 3D objects make a 2D shadow (or a 2D projection on a screen), you can make a 3D shadow of a 4D object. https://www.fourthdimensionapp.com walks you through this.

There's also Miegakure, another beautiful game based on 4D puzzles.


Just to make it clear - the original site and your site were created by the same person. The 4d toys are an offshoot of work they did to create Miegakure.

I've been waiting for that game for more than 5 years now I think.. It'll probably stay in development for another 2 years or so.

It's getting really close, though. Puzzles, mechanics, dialogue, effects, all finished. At this point, according to the dev, all that's really left for him to do is "small things like fix collision bugs, and I will keep placing props in levels and program the occasional cool 4D thing. We still need a bunch of 3D modelling done."

I'd give a 50% shot it lands on Steam before the end of 2019. (35% it doesn't make 2019 but lands before the end of 2020, remainder that it lands later or never releases.)

90% done. It means we are halfway there :p

My gut feeling is that we are not there yet, and that 4D toys is an attempt by the author to monetize his development tools in order to be able to complete the main project. I hope it turns out well, Miegakure is definitely in my watch list.

Kinda reminds me of "Flatland": https://en.wikipedia.org/wiki/Flatland

This book is incredible, it’s hard to believe it’s 134 years old.

"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it." – Geoffrey Hinton, A geometrical view of perceptrons, https://www.coursera.org/lecture/neural-networks/a-geometric...

* but remember that going from 13D to 14-D creates as much extra complexity as going from 2-D to 3-D

There is a joke that goes like this: A mathematician and an engineer attend a lecture on the 11-dimensional M-theory. At the end of the lecture the mathematician delighted says how much he enjoyed it. The engineer puzzled asks him how anything even makes sense. The mathematician replies "It's easy. Visualize an N-dimensional space, then let N go to 11."

This is amazing and looks really fun! I remember reading a short science fiction story once, "Mimsy Were the Borogoves"[1], where small children were playing with such 4D toys, which were found to be educative in unpredictable ways.

It really makes me want to try it out. I wonder if it's really the same without VR.

[1] https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves (watch out, there are spoilers here!)

the bottom of the website includes an illustrated frame and excerpt from that very story

Wow, you're right! Didn't see that, Thanks! :)

If anyone wants to play with a 4D toy right now in their web browser, here is an interactive tesseract (4d cube) that I built. http://transdimensional.xyz It doesn't have physics but features a novel interface that I designed to help build an intuition about different rotations. (There are six axes of rotation in 4 dimensions)

Using the 4th Dimension, the two interlocking rings can be separated. However, what would be the equivalent puzzle for the 3rd dimension? What 2D system requires the 3rd dimension to be separable? Is there even an equivalent? Clearly a circle within a circle is one such system, but it corresponds to a ball wholly contained by another ball in 3D, not a pair of interlocking rings.

It would be a ring with a stick through it. In a 2D slice down the middle, it would look like a donut with a circle (the slice of the stick) in the hole. The circle appears to be stuck inside it, with no way out in 2D. In 3D, you just pull the stick out of the ring.

You can't have interlocking rings in 2D without having some kind of intersection. You have a similar problem with moebius rings and klein bottles: moebius rings are 2D objects but you have to represent them in a 3D world if you don't want them to be self-intersecting. If you attempt to represent a moebius ring in 2D it'll necessarily self-intersect.

Klein bottles are the same thing but with an added dimensions: any representation of a klein bottle in 3D makes it look like it's going through itself, even though in 4D it wouldn't: http://s3files.core77.com/blog/images/2013/06/klein-bottle-0...

It's also true if you remove a dimension: a 1D moebius strip would simply be a circle, but if you try to draw it in 1D you end up with a segment where both halves of the circle are overlapped. So every time we have a N-dimensional object that can only be properly represented in N+1 dimensions.

That's also the same reason you can't solve the problem of connected three objects to three other objects on a 2D planes without intersecting:

     A   B   C

     X   Y   Z
You can't distribute a, b and c to x, y and z on a 2D plane without intersection.

Topology is fun.

It is not possible for two ring shapes to interlock in 2D so there is no directly isomorphic trick.

How curious. What would a haptic VR experience be like? Does a 4D object still exert mass in the other 3Ds? Could you _intuit_ the other 3Ds even if they were not visible?

The VR version has haptics. It's kinda interesting. I think there is a little bit of intuition to be gained.

I would have through the he would have done a projection from 4d space to 2d space similar to how 3d graphics does a projection from 3d to 2d. Similar to http://christopheremoore.net/4d-renderer/ It I guess in that case it would be difficult to interact with it in VR.

I guess this slicing technique works but it would be a bit weird.

Some might remember this excellent treatment of higher dimensions [1] in which you first manipulate shapes to solve a puzzle in 2D, then 3D, and finally 4D. YYou develop a keen sense of 4D objects:

[1] http://harmen.vanderwal.eu/hypercube/

Agreeably playing with 4d objects is mindboggling enough, but still I would like to somehow visualize myself moving around in 4d space. I am not quite sure how a 4d room would look like and what kind of doors to other 4d rooms would be like, bit for sure navigating around a 4d house could be fun...

The video ends saying that trying to fit a 4D cube in a 4D hole is like a child playing with toy blocks. Hypothetically, if a child was raised with regular VR simulations that allowed the child to manipulate 4D objects, would their brain "learn" it and in a way unlock the 4th dimension?

I was wondering this too as I watched the video. As an adult I know how physics works and I know that what I'm seeing is not normal based on decades of experience. If you showed this to a 1 year old and let them play with 4D virtual toys for a few years, would they more intuitively understand 4D objects? Or would it still remain foreign due to the fact that they can't see things moving around the fourth dimension?

As I played around with 4D toys, I did start to gain a sense of things that were "Behind" or "In Front" of me in 4D space, that is I could start to feel their presence after they had been pushed out of sight, if I knew in what direction they were moving in the 4th dimension. If I didn't know that I could only feel that they were near.

It helps that there is a little visualization that is just a line-per-object showing where you are, and where all the objects are intersecting the 4th dimension, that you also use to move "back and forth."

One thing I found myself doing was grabbing objects at one of their edges in the 4th dimension, by moving myself to near their boundary, and then using them like brooms. It's really easy to understand with the case of a hypersphere, since at its edge it's just a smaller sphere than at the middle. So you grab that small sphere at the edge, and push in the direction towards its middle in the 4th dimension, and it will act like a bowling ball. You won't see the stuff you are pushing around because the sphere is "ahead" of you, unless they roll around the sphere, then you'll pass them. Once you reach the edge of the 4th dimension, all the stuff you kept pushing will be there.

Predicting how 3D intersections change as objects rotate about in the 4th dimension still seems like chaos to me though, except in the case of hyperspheres, which basically don't change as they rotate, but I only played around for about an hour or so. The only way I found to rotate objects in the 4th dimension was to have them collide with each other, or the walls and floor, which makes it kind of hard to carefully experiment with their rotations.

"Goundbreaking 4D+Time Physics Engine that uses new mathematics created for this project."

New mathematics no less!

This passage from Death's End by Cixin Liu really gave me pause to stop and wonder about what the experience of seeing extra dimensions might be like (here translated to English by Ken Liu): --

A person looking back upon the three-dimensional world from four-dimensional space for the first time realized this right away: He had never seen the world while he was in it. If the three-dimensional world were likened to a picture, all he had seen before was just a narrow view from the side: a line. Only from four-dimensional space could he see the picture as a whole. He would describe it this way: Nothing blocked whatever was placed behind it. Even the interiors of sealed spaces were laid open. This seemed a simple change, but when the world was displayed this way, the visual effect was utterly stunning. When all barriers and concealments were stripped away, and everything was exposed, the amount of information entering the viewer’s eyes was hundreds of millions times greater than when he was in three-dimensional space. The brain could not even process so much information right away.

In Morovich and Guan’s eyes, Blue Space was a magnificent, immense painting that had just been unrolled. They could see all the way to the stern, and all the way to the bow; they could see the inside of every cabin and every sealed container in the ship; they could see the liquid flowing through the maze of tubes, and the fiery ball of fusion in the reactor at the stern.... Of course, the rules of perspective remained in operation, and objects far away appeared indistinct, but everything was visible.

Given this description, those who had never experienced four-dimensional space might get the wrong impression that they were seeing everything “through” the hull. But no, they were not seeing “through” anything. Everything was laid out in the open, just like when we look at a circle drawn on a piece of paper, we can see the inside of the circle without looking “through” anything. This kind of openness extended to every level, and the hardest part was describing how it applied to solid objects. One could see the interior of solids, such as the bulkheads or a piece of metal or a rock—one could see all the cross sections at once! Morovich and Guan were drowning in a sea of information—all the details of the universe were gathered around them and fighting for their attention in vivid colors.

Morovich and Guan had to learn to deal with an entirely novel visual phenomenon: unlimited details. In three-dimensional space, the human visual system dealt with limited details. No matter how complicated the environment or the object, the visible elements were limited. Given enough time, it was always possible to take in most of the details one by one. But when one viewed the three-dimensional world from four-dimensional space, all concealed and hidden details were revealed simultaneously, since three-dimensional objects were laid open at every level. Take a sealed container as an example: One could see not only what was inside, but also the interiors of the objects inside. This boundless disclosure and exposure led to the unlimited details on display.

Everything in the ship lay exposed before Morovich and Guan, but even when observing some specific object, such as a cup or a pen, they saw infinite details, and the information received by their visual systems was incalculable. Even a lifetime would not be enough to take in the shape of any one of these objects in four-dimensional space. When an object was revealed at all levels in four-dimensional space, it created in the viewer a vertigo-inducing sensation of depth, like a set of Russian nesting dolls that went on without end. Bounded in a nutshell but counting oneself a king of infinite space was no longer merely a metaphor.

Wow, if this is how it’s like to see three dimensions then where would one be able to hide information to the viewer? Probably another dimension. By moving an object from one location to another the information about the position in the previous states are lost. And information about the future states remains hidden. Alas, present states conceal everything hidden behind them, or ahead of them.

Also I loved how a little later in the book, when he gets back into into normal 3D space, he finds it claustrophobic to even float outside of the spaceship, because of his experience in the fourth dimension.

This is incredibly cool. I feel this would be super useful for teaching children (like my nephew who's 11-years-old) to think about the fourth dimension as a component of the fabric of space instead of time as it's usually understood.

I'd love to know the math behind the physics engine, particularly angular momentum. Planar rotations are weird in 4d.

What tools were used to build this I wonder?

Part of it is that the developer's spent the last decade building a game engine and toolchain for 4D content while developing his 4D puzzle game Miegakure (http://miegakure.com/)

That looks incredible. Do you know if the developer has published any papers or code on the subject?

This is the best visualization of 4 dimensions I have ever seen.

Can't wait to see the 5D version. But seriously, what's the use of that? I don't get it.

It's meant to give an intuition about what a 4th space dimension might feel like, thru self experimentation.

I’m guessing from the word “toy” in the name it’s intended as entertainment. I certainly think it’s fascinating and my mind boggles as to how the developer thinks of these shapes (especially for the “full” third person game)

> But seriously, what's the use of that? I don't get it.

About the same as the use of novels, music, paintings - stuff like that...

Can this explain some of the quantum weirdness? Like some of the things quantum entanglement have is due to a forth dimension we are not aware of. Might have to do with general relativity and it is the 4th dimension in smaller scales.

Now where is my noble price?

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