The comparison to children playing with blocks makes me wonder: If you spent long enough playing with these 4D toys, would you develop a deeper intuition for the 4th dimensional shape of the objects?
4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
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Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#82I am genuinely surprised for a long time that I've not seen 4D monsters in films. It has been so obvious to me there's potential there. It would of course be a flat 2D projection of a 3D slice through a 4D creature, so it would look like a smoothly morphing between different creatures, and in and out of existence so we've sort of had morphing for a long while (since Willow, C. 1985) but... there's potential. In case…
We need 4D porn, and fast!
Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#83This is super interesting but hurting my brain.
A person in 3D can see 3D, 2D and 1D. But a person in 2D can only see 1D.
How can a person living in 2D world see 2D things like Circles.
Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#84This is super interesting but hurting my brain. A person in 3D can see 3D, 2D and 1D. But a person in 2D can only see 1D. How can a person living in 2D world see 2D things like Circles.
What we (3D beings) see is a 2D projection of the world around us, 2D beings would indeed see a 1D projection of their 2D world (e.g. we only really see a circle when looking at a sphere). Just as we use depth perception to differentiate 3D objects from 2D ones, 2D beings would also be able to discern a circle from a line in a 2D world.
Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#85I'll ask any physicists out there. A 4th spatial dimension probably doesn't exist, right? Even if we couldn't perceive shapes across a 4th dimension, we would still perceive things moving through the 4th dimension like we see in 4D toys. In reality we don't ever see anything like this (spatially anyway). Is that correct?
There have been tests for other dimensions. Usually they look for energy leaking into other directions. Here's a recent one that tested gravity: https://arstechnica.com/science/2020/04/gravitys-inverse-squ...
Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#86The comparison to children playing with blocks makes me wonder: If you spent long enough playing with these 4D toys, would you develop a deeper intuition for the 4th dimensional shape of the objects?
I've played with some programs I made to explore 4D objects, and, yes, you do gain some intuition.
Interestingly, I was told one of George Boole's daughters got lessons from a guy on 4D objects by using colored wooden blocks. She really picked it up, and later in life mathematicians would consult her about intersections of 4D polytopes.
Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#87The comparison to children playing with blocks makes me wonder: If you spent long enough playing with these 4D toys, would you develop a deeper intuition for the 4th dimensional shape of the objects?
If you haven't read Mimsy Were the Borogroves, you should.
Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#88Earlier quoted context omitted.
If there were 4 spatial dimensions + 1 time dimension we would end up in an unstable universe. Ehrenfest (1917/1920) studied the hydrogen in n dimensions and concluded for n> 3 that neither classical atoms nor planetary orbits can be stable, because the inverse square law of electrostatic and gravity becomes an inverse cube law. When n > 3 there are no stable orbits to the two body problem: an incoming light body att…
Hooray! We disproved string theory!
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Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#89Wonder if there is somewhere a list or some document of "interesting" 4d shapes? For example klein bottle, which is the only one I know of.
That's a complicated question you're asking. It belongs to differential geometry, which I specialised in 15 years ago.
So, you're looking for interesting 4 dimensional manifolds. Problem is, there are many of those. Too many to count, iirc.
Contrast this with the 3 dimensional manifolds, of which there are only eight (and simple compositions of these eight).
Now, unfortunately, the Klein bottle is a 2 dimensional manifold (it's surface is the manifold, and that is locally isomorphic to a R^2). It's interesting because it does not have an embedding into 3d space, unlike most other 2d manifolds, so you need to embed it into R^4. I don't know if there is a list of these.
I can look up more details and give more pointers if there is interest.
Interesting side note: the Klein bottle was originally called "kleinsche Fläche" (Klein surface). When saying this German word with an English voice it'll sound like "Kleinsche Flasche" which translates to "Klein bottle".
Re: 4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020
#90Earlier quoted context omitted.
He would be able to see a circle the same way you can see a sphere. Do you feel like you can only see discs, not spheres?
How different would a 2d square look projected onto a 1d view, vs a circle?
Assuming you have "1d depth vision", ie computing distance to the eye using two eyes set some distance apart, you'd be able to tell them apart rather easily, in addition to shading and movement differences.
Just like our 2d eyes do in our 3d world.