Mobile game with real 4D tesseract

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Re: Mobile game with real 4D tesseract

Postby gonegahgah » Fri Jan 10, 2014 10:49 pm

Sorry, more random sub-conscious thoughts...
I was just thinking that mentally, for me at least and maybe some others, the easiest way for me to visualise 4D bulk is utilising rotation of our 3D space.

Sadly for the 2Der they can only imagine rotating up; which we can do also but we can also imagine rotating sideways.
We can think of rotating a 2Ders cup sideways in a circle around its central vertical axis to form our cup.
One of the distinctions between 2D->3D and 3D->4D is that the 2Der has no sideways - which we introduce - and that a 4Der has a full 360° or sideways - whereas we only have left and right.

These are important conceptually I think because they effect how objects evolve from one dimensional level to another. eg. a cup, a person, a car, etc.

I find the easiest way to think about 4D space is to simply imagine that I'm in my 3D space and that I rotate into other 3D spaces around a vertical axis; just as I forced the 2Der to do (even though they can't conceive it).
By rotating around the vertical axis the axis exists in each 3D space - or 4D orientation - and forms a plane axis of 0d1 x 0d2 x 0d3 x hd4; instead of its original linear form in just my first 3D space.
Interestingly it forms a plane that starts and finishes with itself at the end of the complete 360° of rotation.

By imagining that I'm rotating 'sideways' into being in these extra 3D spaces until I am 'faux-forward' again I get a sense of the number of 3D spaces that exist.
I also understand - as for the 2Der - that there is more filler, or bulk, between each 'rotation' away from my axis of rotation. ie. the projected circumference is bigger for the outside of the cup top than its base.

But the variance from 2D->3D to 3D->4D begins here as well.
We know that a 2Der will only look forward (and maybe back too); but that in our world predator eyes are at the front of the face so that they can triangulate on prey and prey eyes are at the sides of the head so that they notice predators.
In 2D, predator and prey eyes might not be too differently placed to each other.

We might mistakenly - I think - assume that 4D animals will have eyes spaced in a triangle (that part is probably okay) and that predator and prey would again be on similar footing?
But, that might not necessarily be so... (please understand, I am thinking as I go here... I'd like to think more now but I better go get some things done. I'll come back later and finish my thoughts...)
Please feel free to comment if you have any thoughts or corrections?
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Re: Mobile game with real 4D tesseract

Postby quickfur » Sat Jan 11, 2014 3:45 am

gonegahgah wrote:Sorry, more random sub-conscious thoughts...
I was just thinking that mentally, for me at least and maybe some others, the easiest way for me to visualise 4D bulk is utilising rotation of our 3D space.

Sadly for the 2Der they can only imagine rotating up; which we can do also but we can also imagine rotating sideways.
We can think of rotating a 2Ders cup sideways in a circle around its central vertical axis to form our cup.

Be careful with this generalization, though. In going from 2D -> 3D, there is only one way to rotate a 2D cup into a 3D cup, so it's unambiguous. However, in going from 3D -> 4D, there are actually two perpendicular rotations that your cup can undergo, and infinitely many in between! This is because 4D rotation takes place not around an axis, but a plane. Since there's a whole 360° of possible vertical planes that bisect the cup through its center, there are also that many different ways you can rotate it through 4D.

Fortunately, if your cup is a perfect cylinder (no handles, no deformations, but a cylinder of a perfect circle), then all of these rotations will produce the same shape, which is, in fact, a spherinder.

But if your cup has a handle, then depending on which plane you rotate it in, you could end up with either a simple 4D handle (well, a "fattened" handle that's round in the WX plane), or you could end up with a "fat handle", that wraps around the cup in one of the dimensions (but not the other, so it's not exactly the same as the case with rotating a 2D cup with a handle that would produce a hollow ring around the 3D result -- in 4D, this "wrapping" handle has odd-shaped hole that, on the one hand, wraps around the cup, but on the other hand, you can still reach through it from outside! These are just two of the simplest cases, though, corresponding with the two perpendicular planes of rotation (assuming one of the planes is where the handle lies in). Using an oblique rotation through 4D will produce a variety of intermediate forms, with various oddly-shaped handles that have various grades of "holes". (N.B., in 4D there are two kinds of holes, a "bridging" kind of hole, and an "encircling" kind of hole. Depending on how you rotate your 3D cup, you could end up with any of the spectrum of holes between these two kinds.)

[...]I find the easiest way to think about 4D space is to simply imagine that I'm in my 3D space and that I rotate into other 3D spaces around a vertical axis; just as I forced the 2Der to do (even though they can't conceive it).
By rotating around the vertical axis the axis exists in each 3D space - or 4D orientation - and forms a plane axis of 0d1 x 0d2 x 0d3 x hd4; instead of its original linear form in just my first 3D space.
Interestingly it forms a plane that starts and finishes with itself at the end of the complete 360° of rotation.

Again, you should be aware that in 4D, rotation happens not around an axis, but around a plane. This means that there's actually 360° of possible rotations through 4D that you can undergo while standing on the same spot. Each of these rotations actually correspond with exactly one of the 360° directions that you can face in 3D.

In other words, if you turn your body (in 3D only) to face another direction, then your forward direction + your vertical direction defines a different plane of rotation through which you can rotate into 4D. So if you stand on the same spot and try to rotate into 4D, you'll discover that which rotation you end up doing actually depends on which direction you're facing! So if you look north, say, and then rotate 360° around 4D, you'll have completed one kind of rotation through 4D, but if you then turn and face east, and rotate 360° aroudn 4D again, you'll find that it's a different rotation from the first time.

[...]
We know that a 2Der will only look forward (and maybe back too); but that in our world predator eyes are at the front of the face so that they can triangulate on prey and prey eyes are at the sides of the head so that they notice predators.
In 2D, predator and prey eyes might not be too differently placed to each other.

I read an interesting article recently that describes some evidence that a rabbit's brain operates in quite a foreign way from what we may imagine: apparently it doesn't know how to reconcile the images it sees with the left eye with its right eye (there is no overlap in the field of vision for its two eyes at all, so it doesn't have any stereoscopic vision like we do), to the point that it can, for example, learn to run away when it recognizes a predator with its right eye, yet if it has never seen the same predator with its left eye before, then the first time it sees the predator with its left eye it will not know to run away.

Now, a 2D creature living on a circular planet with gravity would conceivably have two eyes, in front and behind (otherwise it would be prone to be easily killed by predators attacking from behind). Or maybe more accurately, left and right, since the two sides would be equivalent when it has two eyes. The two sides may never be reconciled in its mind, since it has no way to ever experience turning around so that its left and right eyes switch positions. So conceivably, its view of the world would be inherently dual (almost schizophrenic, to abuse a common term), and it may not even realize these two worlds are actually one and the same.

We might mistakenly - I think - assume that 4D animals will have eyes spaced in a triangle (that part is probably okay) and that predator and prey would again be on similar footing?
But, that might not necessarily be so... (please understand, I am thinking as I go here... I'd like to think more now but I better go get some things done. I'll come back later and finish my thoughts...)
Please feel free to comment if you have any thoughts or corrections?

Assuming that the triangulation theory is the basis for having 2 eyes in 3D (big assumption, but let's say it's so), in 4D certainly a triangular arrangement of 3 eyes would be ideal for the "tetrahedrization" of the location of a predator/prey/etc..

However, it may be possible to get away with only 2 eyes in 4D, because all you need to do upon spotting a predator/prey is to reorient yourself (by turning in a lateral plane) so that your 2 eyes become perpendicular to your line of sight to the target, then the usual triangulation will tell you the distance. "Tetrahedrization" is really only necessary when you can't easily change your orientation, or if doing so puts you at a disadvantage (e.g., it takes too long, and you might get attacked before you have time to figure out how far the target is). Furthermore, this is only needed part of the time when the target happens to lie exactly in the plane that is parallel to your eyes, so triangulation fails; even if you just shift slightly, the target will be out-of-plane, and triangulation becomes possible again. So 3 eyes may not be as indispensible as it may first appear (though it no doubt does give some advantage).
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Re: Mobile game with real 4D tesseract

Postby anderscolingustafson » Tue Mar 11, 2014 5:33 am

In 4d if an animal had it's eyes to the side of its head even if the field of view of every eye didn't overlap with the field of view of every other eye the fields of view of neighboring eyes would overlap and so a prey animal in 4d would be able to tell how far away something is by triangulating between the fields of view of neighboring eyes.
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Re: Mobile game with real 4D tesseract

Postby Hugh » Wed Mar 19, 2014 3:56 pm

gonegahgah wrote:I find the easiest way to think about 4D space is to simply imagine that I'm in my 3D space and that I rotate into other 3D spaces around a vertical axis; just as I forced the 2Der to do (even though they can't conceive it).
By rotating around the vertical axis the axis exists in each 3D space - or 4D orientation - and forms a plane axis of 0d1 x 0d2 x 0d3 x hd4; instead of its original linear form in just my first 3D space.
Interestingly it forms a plane that starts and finishes with itself at the end of the complete 360° of rotation.


This is how I kind of envision it possibly happening with VRIs as well gonegahgah. :)
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