4D Toys: how objects bounce and roll in 4D – paper accepted to SIGGRAPH 2020

https://scholarsarchive.byu.edu/etd/6968/

This also means such a thing should be possible for living in a 2D space and then projecting a 2D world on a 1D screen.

If you're going to play with 3D projections of 4D objects, then in order for it to really feel 4D we need to live in a 4D world. In that sense, this is quite similar to living in a 2D world while seeing a 3D projection. That's way more trippy.

To recap:

Feel 3D: 3D context -> 3D projection -> 2D screen

Feel 2D: 2D context -> 2D projection -> 1D screen

Feel 4D: 4D context -> 4D projection -> 3D screen

What we can do for 4D: 3D context -> 4D projection -> 3D screen

Compare issue with 3D: 2D context -> 3D projection -> 2D screen

I hope this makes sense (and I hope I have it right).

Man, this is tough to write about. I'm not sure if I fully understood all the nuances of your comment.

https://en.wikipedia.org/wiki/Anaglyph_3D

When projecting 4D to 2D, weird things start happening. Instead of a ray, each pixel now represents a plane in 4D space. So even while it can be done mathematically, the brain is just overwhelmed by what happens when you e.g. rotate the projection

Spoiler, obviously: https://www.youtube.com/watch?v=uBsJgceM0KI - if you haven't seen it cannot recommend it highly enough, just wait and watch the whole thing.

The antagonist is a being from another dimension who routinely shows off dominion over N dimensions, with the effect looking something similar to what is demonstrated in the siggraph video.

The story and ending are rather weak, but the physics questions posed are interesting. I like any franchise that discusses the teletransportation paradox.

[0]: https://myanimelist.net/anime/32032/Seikaisuru_Kado

     _____________________________________________________________________________________________________________________________________________________________________Interstellar kinda touched on that.

It's easy enough, albeit annoying, to brush away the inflated mediocre. It's so much harder to locate the unseen gems.

Edit: ah, I see - the video is from 2017, and there was a thread back then too: https://news.ycombinator.com/item?id=14471931. I guess we'll make an ugly hybrid of the titles.

Does such a thing exist at higher dimensions? I vaguely recall something about having complex numbers for 2D rotation, quaternions for 3D rotation, and octonions for (I'm guessing) 4D rotation, but I'm curious if the loss of associativity with octonions screws with this relationship somehow.

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.

https://en.wikipedia.org/wiki/Mimsy_Were_the_Borogoves

I’m sure most of us could learn a visceral understanding of 4D if we could push and twist 4D objects.

What Marc's done with Miegakure, from what's publicly visible, is pretty incredible. I have no idea how he's managed to seemingly create a coherent 4D world while only being able to view a slice of it at a time. I guess it's a bit like using ed instead of a modern text editor.

String theories require extra spatial dimensions (to total of 10, 11 or 26), but those dimension are so small and/or looping so that we cannot detect them.

Other approach is to treat our 3D space as slice of higher dimensional space.

https://en.wikipedia.org/wiki/String_theory#Extra_dimensions

String theories have a lot of problems (biggest ones is that they are currently unverifiable using experiments and they do not give any meaningful predictions about our 3D view of universe), but many theoretical physicists are working on them.

It is perfectly possible to formulate theories of physics in higher spatial dimensions. In fact, many high-energy theories like string theory require many spatial dimensions for mathematical consistency. It is something we actively look for signatures of in experiments.

What we have found is that our observations are incompatible with 'extended' additional dimensions. Extended here means that you can go a sizeable distance. It's not that the other dimensions would just stop, but they'd be more like Pac Man---these dimensions might have periodic boundary conditions, making them circles. If the circle's radius (or circumference, equivalently) becomes very large, such a 'compactified' dimension starts to have a lot of space in it, much like the familiar 3 dimensions. So, we can put an upper bound.

My most recent recollection. We may yet have missed dimensions as large as 1mm in [diameter, circumference, I forget]. The most sensitive probes with the fewest assumptions about the structure + particle content of the universe tend to be gravitational, and gravity is very hard to measure precisely on very short length scales.

edit to add: the latest summary of extra dimensional searches from the Particle Data Group http://pdg.lbl.gov/2019/listings/rpp2019-list-extra-dimensio...

even better, here's the PDG's review, rather than their technical summary: http://pdg.lbl.gov/2019/reviews/rpp2018-rev-extra-dimensions...

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 attracted by a heavy one would either escape to infinity or get sucked into collision.

For n = 3 we get stable elliptic orbits or non-bound parabolic and hyperbolic orbits.

Collision only occurs when the lighter body heads directly towards the heavy body within 2R (R being the heavier body's radius), ie. the impact parameter is zero [2]

[1] https://doi.org/10.1002/andp.19203660503

[2] https://en.wikipedia.org/wiki/Impact_parameter

edit: grammar

But our universe as a whole is almost entirely empty space, with visible objects appearing to make up only about 5 percent of the universe's total volume. Noticing 4D things flit in and out of our 3D universe might be difficult at that scale.

Nothing (in the classical, non-quantum) setting is solely determined by position coordinates.

[0] - https://miegakure.com/

-----

[0] https://en.wikipedia.org/wiki/Homography

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".

That this power exists in any regions and for any classes is the result of Fog.

If Fog were non-existent, all lines would appear equally and indistinguishably clear; and this is actually the case in those unhappy countries in which the atmosphere is perfectly dry and transparent. But wherever there is a rich supply of Fog, objects that are at a distance, say of three feet, are appreciably dimmer than those at a distance of two feet eleven inches; and the result is that by careful and constant experimental observation of comparative dimness and clearness, we are enabled to infer with great exactness the configuration of the object observed.

Suppose I see two individuals approaching whose rank [shape] I wish to ascertain. They are, we will suppose, a Merchant and a Physician, or in other words, an Equilateral Triangle and a Pentagon: how am I to distinguish them?

It will be obvious, to every child in Spaceland who has touched the threshold of Geometrical Studies, that if I can bring my eye so that its glance may bisect an angle of the approaching stranger, my view will lie evenly between his two sides that are next to me, so that I shall contemplate the two impartially, and both will appear of the same size.

Now in the case of the Merchant [Equilateral Triangle], what shall I see? I shall see a straight line, in which the middle point will be very bright because it is nearest to me; but on either side the line will shade away rapidly into dimness, because the sides recede rapidly into the fog; and what appear to me as the Merchant's extremities will be very dim indeed.

On the other hand in the case of the Physician [Regular Pentagon], though I shall here also see a line with a bright center, yet it will shade away less rapidly into dimness, because the sides recede less rapidly into the fog; and what appear to me the Physician's extremities will not be not so dim as the extremities of the Merchant.

The Reader will probably understand from these two instances how -- after a very long training supplemented by constant experience -- it is possible for the well-educated classes among us to discriminate with fair accuracy between the middle and lowest orders [different shapes] by the sense of sight. If my Spaceland Patrons have grasped this general conception, so far as to conceive the possibility of it and not to reject my account as altogether incredible - I shall have attained all I can reasonably expect."

Edwin A. Abbott in Flatland: A romance of many dimensions (slightly edited)