answer this please

[Xminent]

Ok I have something I really don't get. I am only 16.

Ok I have just started reading about dimensions few days ago so i may sound dumb to u guys lol



What would you see if you looked into a fourth dimension?

In my mind there cant be a 4th dimension there is no other way we can look that we cant see.

Does anyone get what i am trying to say?

[Keiji]

the fourth dimension exists but we can't see it... like if there is a line on a plane the line can never be pointing off of the plane...

[jinydu]

What would you see if you looked into a fourth dimension?

We can't look into the 4th dimension because we are 4D beings. An analogy would be that 2D beings can't look into our 3D world. Imagine a character from a really old 2D video game trying to look out of the screen. It's impossible.

In my mind there cant be a 4th dimension there is no other way we can look that we cant see.

Again, you are right in saying we can't see it. But just because we can't see it doesn't mean it doesn't exist. For example, you can't see the computer that I'm using to type this message, but that doesn't mean the computer doesn't exist.

[RQ]

I think a very good analogy of 4D might be a video game (idea came originally from Star Wars: Rogue Squadron for N64).

Imagine they're shooting at your starship. The fire is obviously coming from behind you, but it isn't coming from off the screen is it? Please note, this isn't an actual 4th dimension, because the TV is a representation of 3D and your brain is fooled into thinking there is something behind, but to see the 4th dimension the illusion is sufficient and all there is now isn't it?

[quickfur]

[...]What would you see if you looked into a fourth dimension?

You can't. Well, at least, not with your physical eyes, which can only see 2D projections of 3D. Even if you could look into 4D, you'd only see one 3D slice of it at any one time.

BUT. Assuming (and that's a big assumption) that you somehow acquired eyes that can see 3D projections of 4D, you'd be able to see a lot of things. Assuming that your new 4D eyes operate in more-or-less the same way as your 3D eyes, you'd see 3D projections of 4D objects. For example, if you looked at a tetracube cell-on, you'd see a 3D cube, and if you looked at the tetracube vertex-on (from an angle) you'd see a rhombic dodecahedron with the vertex of the tetracube in its center. If you looked at a tetrasphere, you'd see a 3D sphere, but its center would "bulge outwards" (just like when you look at a 3D sphere with your human eyes, you see a circle but the middle of the circle "bulges out" at you).

Keep in mind that if you had such 4D eyes, you'd be able to see every point of a 3D object at the same time (both inside and outside). So if you were able to look at our 3D world with your 4D eyes at a right angle, you'd see through every object.

I recently started playing around with how a 4D face would look like, and came up with some fascinating stuff. For example, the face could turn in 3 perpendicular directions and still be right-side-up. One of these directions (corresponding with the XY plane) does not intersect with the back of the head, so the 4D person could be turning his head a full 180 degrees and you still can't see the back of his head. If he's looking at you while doing this turn, he'd be staring straight at you the whole time his head is turning. :-) Turning 180 degrees on either one of the other directions will turn the back of his head towards you, but you'd see 4 different sides of his head as he turns, 2 for each direction (XW and YW).

If his two ears lie on the X axis, then when he turns his head in the YW direction, his ears will rotate in place but won't move, until the back of his head comes into view between them. Whereas if he turns his head in the XW direction, one ear will vanish behind his head and the other will move from -X to +X with the back of his head behind it, and when it reaches the other side, the other ear reappears in -X.

The person's lips would be a 3D volume, which I assume would be something like a flat ellipsoid. When he opens his mouth, you'd see the opening inside the 3D volume (in the projected image of your 4D eyes of course).

Anyway, once you understand the basics of how 3D projections of 4D objects work, visualizing different 4D objects can be really fun. It does take a bit of imagination, since you have to "see" all points of the projected 3D volume simultaneously, but with practice it's not too hard.

I've gotta work out more details of what a 4D face might look like. Eyelids would cover a 3D region, which makes for a lot of interesting facial expressions... :-P

[PWrong]

hmm, interesting ideas. If you had vision that included the entire electromagnetic spectrum, that might be pretty close to having 4D eyes.

It wouldn't be at a right angle though, but that wouldn't really matter. You could probably train your eyes to see things differently, as I mentioned described in another thread somewhere.

I'm not sure if we can assume 4D faces would have the same features.

Here's an important question. How many eyes would a 4D being need to have proper depth (or tridth) perception?

[quickfur]

hmm, interesting ideas. If you had vision that included the entire electromagnetic spectrum, that might be pretty close to having 4D eyes.

It wouldn't be at a right angle though, but that wouldn't really matter. You could probably train your eyes to see things differently, as I mentioned described in another thread somewhere.

Well, I don't know if the range of the electromagnetic spectrum that you can see really matters that much. 'Cos to actually see 3D, you'd have to be able to somehow see every part of a 3D object simultaneously. I suppose you could do this to some extent if your eyes picked up, say, X rays that pass through most matter, but that doesn't help 'cos you'd see through the object rather than see the texture of its internals.

But even if you could see in full 3D somehow, the images you see wouldn't quite behave the same way as 4D objects projected into 3D. Unless you build a device for rendering 3D projections of 4D objects, and move them about... then you could actually see how things moved in 4D. Hmm, that's an interesting idea.

I'm not sure if we can assume 4D faces would have the same features.

I didn't think so either. But I sorta used that as a starting point in order to begin to understand what a 4D face would be like. It's kinda hard to visualize things if you start out assuming 3 eyes and 4 ears laid out in 4D coordinates, and try to view them from different angles... not as easy to understand if you don't even know what they look like in the first place.

My current working model is a hyperspherical head (for simplicity, so I don't have to worry about, say, cheek shape yet), two hyperspherical eyes with a hyperconical nose between them, and an ellipsoidal mouth underneath the nose. Granted, this may not be the best arrangement for an actual 4D being, but at least it's a starting point with familiar landmarks that I can use to explore the behaviour of a 4D face. Maybe once I understand it more, I can actually move on to a more plausible 4D facial arrangement.

Here's an important question. How many eyes would a 4D being need to have proper depth (or tridth) perception?

I've been wondering the same thing for a long time. I still don't have an answer for it, although further discussions would be welcome. I kinda lean towards 2 eyes, 'cos in 3D, our 2 eyes can only tell parallax in 1 dimension, but it seems sufficient to deduce 3D depth. The question is whether 1 dimension of parallax is enough to deduce accurate 4D depth, or whether we need another dimension.

A related question is, how many ears are needed for adequate hearing in 4D? If you assume you only need to listen in a single general direction at a time, then 2 ears ought to be enough, from what I understand. (If there are two ears placed at opposite sides of a hyperspherical head, at least one would always be "visible" no matter which way you look at the head. Which means that half the time, if I figured this correctly, at least one ear would be facing in the right general direction to hear something.) But I need to work this out more to be sure.

One thing that's important to understand is that these facial features have full 3D hyper-surfaces. In the 3D projection, they occupy a 3D volume, but the entire volume is only the surface of the feature, not the interior. This is rather interesting, since the edges of eyelids are fully 2D, and so are capable of a much wider range of eye expressions than is possible with our human eyelids, which only have 1D edges. Also, the eyeballs have 6 different planes to roll in, and can actually rotate in 2 planes simultaneously. If anything, it certainly gives you a lot more ways to express contempt. :-P Also, frown lines on your brow would be 2D rather than 3D, so when you frown you get parallel wavy planes instead of merely wavy lines. 4D faces can be very expressive, you see.

Another thing to understand about 4D projections into 3D is that the point closest to you, the observer, is actually the center of the 3D volume, not any point on its surface. So when a 4D face looks at you, the nose would be projected into the center of the projected sphere of the face, and the two eyes would be on either side of the nose inside this sphere. The tip of the nose would be projected inside the conical volume of the image of the nose (assuming the image is approximately conical). As the head turns away from you, this tip would move outwards until it emerges out one side of the conical volume, in which case you are viewing it from a 90-degree angle. If you assume the head is only turning sideways (not up/down), you'll see that there are 2 perpendicular directions in which this movement can happen, which accounts for the 2 different directions in which the head can turn sideways from you.

There's another odd sideways rotation of a 4D face, which corresponds with rotation in the XY plane in the projected image. This rotation does not change the position of the tip of the nose, for example: the nose still faces you all the time even though the head is turning! In fact, the eyes and ears still face you constantly all the while the head is turning. (This is sorta like leaning your head to/from your shoulders, you are still facing the same direction but your face is rotated sideways. In 4D, however, you can actually do this on a horizontal plane.)

4D faces are so much fun. Don't you wish you were a 4D being? ;-)

[pat]

'Cos to actually see 3D, you'd have to be able to somehow see every part of a 3D object simultaneously. I suppose you could do this to some extent if your eyes picked up, say, X rays that pass through most matter, but that doesn't help 'cos you'd see through the object rather than see the texture of its internals.

But, if the light were actually coming to your eyes from a 4D world, it wouldn't have to pass through the object to get to your retina. When you see a 3D project of a 4D object, you're only seeing the surface of the 4D object. Thus, no light had to penetrate the object, only bounce off of it. Assuming this light could bounce to your retina properly, you wouldn't need X-rays or anything like that.

[pat]

Here's an important question. How many eyes would a 4D being need to have proper depth (or tridth) perception?

I've been wondering the same thing for a long time. I still don't have an answer for it, although further discussions would be welcome. I kinda lean towards 2 eyes, 'cos in 3D, our 2 eyes can only tell parallax in 1 dimension, but it seems sufficient to deduce 3D depth. The question is whether 1 dimension of parallax is enough to deduce accurate 4D depth, or whether we need another dimension.

I believe that two eyes would be enough. As near as I can tell, we only use the parallax to gauge distance a bit. Distance is one-dimensional. A third eye wouldn't help. We may have to rotate our heads a bit to resolve some optical illusions, but... such optical illusions don't often crop up naturally in our world. I don't see any reason to believe they would crop up more often in a 4-D world.

[quickfur]

'Cos to actually see 3D, you'd have to be able to somehow see every part of a 3D object simultaneously. I suppose you could do this to some extent if your eyes picked up, say, X rays that pass through most matter, but that doesn't help 'cos you'd see through the object rather than see the texture of its internals.

But, if the light were actually coming to your eyes from a 4D world, it wouldn't have to pass through the object to get to your retina. When you see a 3D project of a 4D object, you're only seeing the surface of the 4D object. Thus, no light had to penetrate the object, only bounce off of it. Assuming this light could bounce to your retina properly, you wouldn't need X-rays or anything like that.

Correct. But I thought we were talking about how one might go about seeing 4D with our existing 2D retina, by somehow seeing in 3D with the help of X-rays (or the rest of the electromagnetic spectrum). I'm just pointing out that this doesn't really add very much to the limitation that our retinas are, after all, only 2D.

[pat]

Ah, my misunderstanding. I only skimmed the thread-thus-far.

[quickfur]

[...]I've been wondering the same thing for a long time. I still don't have an answer for it, although further discussions would be welcome. I kinda lean towards 2 eyes, 'cos in 3D, our 2 eyes can only tell parallax in 1 dimension, but it seems sufficient to deduce 3D depth. The question is whether 1 dimension of parallax is enough to deduce accurate 4D depth, or whether we need another dimension.

I believe that two eyes would be enough. As near as I can tell, we only use the parallax to gauge distance a bit. Distance is one-dimensional. A third eye wouldn't help. We may have to rotate our heads a bit to resolve some optical illusions, but... such optical illusions don't often crop up naturally in our world. I don't see any reason to believe they would crop up more often in a 4-D world.

Good point. I am unsure, though, if there might be any geometric arrangements in 4D that might require a 3rd eye in order to see parallax in, say, a perpendicular direction that lie on the same line as the other 2 eyes. Is this mathematically possible? If so, would it have a significant effect on 4D vision or would it be an acceptable part of perspective vision? Would a 3rd eye help in this case?

I'm kinda leaning on no to the last 2 questions, but I haven't explored the possibilities thoroughly enough to confidently state this yet.

[pat]

Good point. I am unsure, though, if there might be any geometric arrangements in 4D that might require a 3rd eye in order to see parallax in, say, a perpendicular direction that lie on the same line as the other 2 eyes. Is this mathematically possible? If so, would it have a significant effect on 4D vision or would it be an acceptable part of perspective vision? Would a 3rd eye help in this case?

Consider having only one eye for a moment (call it eye A). Now, either there is a straight line between the point in question and your eye or there isn't. If there is, then you can see that point. If there isn't you cannot see that point.

Assume that you can see that point. Now, add a second eye (call it eye B). Either that second eye can see that point or it cannot. If it cannot, then you know there's something between your second eye and that point... and thus is slightly closer than that point. If your second eye can also see that point, then by parallax, you can somewhat judge the distance to that point.

Assume that you can see that point with both of your eyes. Now, add a third eye (call it eye C). Either that third eye can see that point or it cannot. If it cannot, then you are in basically the same position you would have been if you had rotated your head so that eye A stayed put but eye B was where C is and then rotated your head so that eye B stayed in its original spot but eye A was where eye C is.

Basically, I think you can always end up with the same information just moving your two eyes to cover all of the pairs-of-places any two of your n eyes could cover.

I'll have to think some more to decide if there would be some evolutionary advantage to not having to turn your head so much or if those kinds of problems would be rare. I think they would be rare. In most cases, you'll be able to see the same points with either eye. The only times this isn't true is at the edges of things.

Put another way, imagine there is a big box off in the distance in our 3-D world. With your left eye you can see a sliver of the side of the box that you cannot see with your right eye. But, you can see the same points on the front of the box and top of the box in either eye. Is there any real advantage to being able to see that sliver? I don't think so. The real advantage is the points you can see with both eyes. And, you don't gain an advantage by adding more eyes except by increasing the likelihood that you can see the same point from at least two of your eyes.

[quickfur]

Good point. I am unsure, though, if there might be any geometric arrangements in 4D that might require a 3rd eye in order to see parallax in, say, a perpendicular direction that lie on the same line as the other 2 eyes. Is this mathematically possible? If so, would it have a significant effect on 4D vision or would it be an acceptable part of perspective vision? Would a 3rd eye help in this case?

Consider having only one eye for a moment (call it eye A). Now, either there is a straight line between the point in question and your eye or there isn't. If there is, then you can see that point. If there isn't you cannot see that point.

Assume that you can see that point. Now, add a second eye (call it eye B). Either that second eye can see that point or it cannot. If it cannot, then you know there's something between your second eye and that point... and thus is slightly closer than that point. If your second eye can also see that point, then by parallax, you can somewhat judge the distance to that point.

Here's where my unsureness lies. Would there be a case where the point would project onto exactly the same point in both eyes, making it impossible to judge parallax? I know this happens in 3D "at infinity", but would it cover a much larger region in 4D, and would this matter? Especially, would this happen with points that are close to the observer, which is where parallax matters the most?

Assume that you can see that point with both of your eyes. Now, add a third eye (call it eye C). Either that third eye can see that point or it cannot. If it cannot, then you are in basically the same position you would have been if you had rotated your head so that eye A stayed put but eye B was where C is and then rotated your head so that eye B stayed in its original spot but eye A was where eye C is.

Basically, I think you can always end up with the same information just moving your two eyes to cover all of the pairs-of-places any two of your n eyes could cover.

But if the 3rd eye gets rid of some of the aforementioned coincidental points, it will give you more information about parallax. Provided, of course, that there are such points, and that they occur at important enough a place in the visual image that this would be an advantage.

I'll have to think some more to decide if there would be some evolutionary advantage to not having to turn your head so much or if those kinds of problems would be rare. I think they would be rare. In most cases, you'll be able to see the same points with either eye. The only times this isn't true is at the edges of things.[...]

Yeah, I don't think turning your head around shouldn't be that big a deal. I'd expect a 4D being to have the physiology to be able to do complex neck turns easily, since that's the only way it would be able to get a good view of its surroundings.

[pat]

Assume the retina is 3-D and suppose we have a point 'a' in eye 'A' and a point 'b' in eye 'B', there is one and only one point in 4-D space that projects to those two points simultaneously.

For eye A, the set of points that map to point 'a' is a straight line of points. A point off of that line is not obscured by that line. For eye B, the set of points that map to point 'b' is a straight line of points. The intersection of these two lines is a single point.

So, the only cases that are confusing are when something obscures the intersection from one eye but not the other.

[quickfur]

Assume the retina is 3-D and suppose we have a point 'a' in eye 'A' and a point 'b' in eye 'B', there is one and only one point in 4-D space that projects to those two points simultaneously.

For eye A, the set of points that map to point 'a' is a straight line of points. A point off of that line is not obscured by that line. For eye B, the set of points that map to point 'b' is a straight line of points. The intersection of these two lines is a single point.

So, the only cases that are confusing are when something obscures the intersection from one eye but not the other.

Mmm, very good point. So I guess this settles it, that only 2 eyes are necessary to see parallax (at least for most purposes).

I figured that the 0-parallax case only happens "at infinity", i.e., at the horizon, which in the case of 4D corresponds with a plane. But a plane to a 4D being is like a line to us, so it doesn't really matter that much.

Cool, so I guess 2 eyes are enough for a 4D being. Now I wonder if this also holds for ears, arms, and legs. Especially legs. Are 2 legs enough to balance a 4D being sufficiently?

[pat]

Technically, 2 legs isn't enough to balance a 3-d being correctly. But, we've got mad skillz.

If you were building a stool in 3-d, you'd need at least 3 legs. If you were building a stool in 4-d, you'd need at least 4 legs.

But, a creature with only two legs and the skill to keep its center of gravity over them would be able to walk around just fine in a 4-D world. Again, gravity is only acting in one direction. With two legs, you'd have the ability to move your center of gravity from one to the other to effect locomotion.

[quickfur]

But, a creature with only two legs and the skill to keep its center of gravity over them would be able to walk around just fine in a 4-D world. Again, gravity is only acting in one direction. With two legs, you'd have the ability to move your center of gravity from one to the other to effect locomotion.

Well I know our having 2 legs isn't exactly a stable configuration in terms of balancing, but it is a lot more practical than walking with one leg. I'm just wondering if walking with 2 legs in 4D is as bad as walking with 1 leg in 3D.

[pat]

I can see no need for another leg. In walking, your goal is to move your center of gravity along a line (not a plane or anything). You will need to have some tridth to your feet to help you stabilize there.

But, you can stand on one leg in 3-D. You should be able to do the same in 4-D. The only thing you need for walking then is a second leg upon which to stand.

[quickfur]

I can see no need for another leg. In walking, your goal is to move your center of gravity along a line (not a plane or anything). You will need to have some tridth to your feet to help you stabilize there.

But, you can stand on one leg in 3-D. You should be able to do the same in 4-D. The only thing you need for walking then is a second leg upon which to stand.

Hmm, cool. I wonder if humanoid bodies are practical in 4D then (suitably extended to 4-space, of course). Do you think having only 2 arms would be a serious limitation, or is that a non-issue?

I do remember somebody mentioning somewhere that 7 fingers per hand would be best for a 4D person, but I can't find anything explaining why. Is there any reason for the number 7?

[pat]

I can't think of a reason one would need seven fingers. But, of course, I cannot see any reason we need five fingers. The way we use our fingers, it seems to me that three would suffice. Anything that I can grip, I can grip with my thumb, index finger, and pinky. I have more strength and more stability if I involve more fingers. But, I don't personally see any reason why we wouldn't have an opposable pinky that's as strong as the thumb to go with a ring finger in the middle. *shrug*

[PWrong]

But, you can stand on one leg in 3-D. You should be able to do the same in 4-D. The only thing you need for walking then is a second leg upon which to stand.

Still, most animals have four legs. And even though you can stand on one leg, there aren't many animals that evolved like that. So there might be very few animals with only two legs in 4D.

I can't think of a reason one would need seven fingers. But, of course, I cannot see any reason we need five fingers. The way we use our fingers, it seems to me that three would suffice. Anything that I can grip, I can grip with my thumb, index finger, and pinky. I have more strength and more stability if I involve more fingers. But, I don't personally see any reason why we wouldn't have an opposable pinky that's as strong as the thumb to go with a ring finger in the middle. *shrug*

Opposable thumbs evolved on monkeys so they could climb trees.
If you grab a horizontal branch above you, you don't want to slip off. But you only need one digit underneath the branch to keep the hand stable. The more fingers on top of the branch, the more strength.

In 3D, each finger grasps a circular portion of the cylindrical branch. In 4D, assuming a spherindrical branch, grasping it would be a bit like holding a sphere.

The most efficient way to hold a sphere would probably be three fingers, each opposable to each other. Each group of three holds a portion of the branch, but the groups might share a common finger.

For comparison, think of a 3D hand as having four sets of two opposable fingers, but each set has a common member, the thumb. The "four" is probably arbitrary, but the "two" comes from the 2D surface of the branch.

A 4D hand has an arbitrary number of sets of three fingers, and a 5D hand would probably have sets of four fingers.

Now the reason our four sets share a thumb, is that you don't need as much strength underneath the branch. In 4D, you have groups of 3 digits holding spheres. So you could have two fingers near the top and one thumb directly underneath or one finger right at the top and two thumbs near the bottom. I'm not sure which would be more stable. Either way, I think the sets would share the same thumb/pair of thumbs.

So they can have: 2k fingers sharing one thumb, or k fingers sharing two thumbs, where k is the number of "sets". Since our general goal is a humanoid, lets say 2k fingers and one thumb.

The sets extend along the branch, and each thumb has to be opposable to every finger. The further away from the thumb/s, the weaker the grip, so there must be a limit on the number of sets. so lets say four sets, like us.

This means 8 fingers and one thumb.
4 fingers for each dimension of the surface of the branch, plus one thumb for stability. You can roughly imagine having a hand with an extra row of fingers, perpendicular to the original ones. It's easy to project, because the hand is stuck to a 3D surface anyway.

......||||................||||
..\_.||||................||||._/
..\_|....|...............|....|_/
..\_|....|=>......<=|....|_/
...\_|..|.................|..|_/
......|..|.................|..|

I can't get the picture to behave. The dots might help.

[quickfur]

Still, most animals have four legs. And even though you can stand on one leg, there aren't many animals that evolved like that. So there might be very few animals with only two legs in 4D.

That's a very interesting point. Do you think 4D animals would be more likely to have 6 legs rather than 4?

Opposable thumbs evolved on monkeys so they could climb trees.
If you grab a horizontal branch above you, you don't want to slip off. But you only need one digit underneath the branch to keep the hand stable. The more fingers on top of the branch, the more strength.

In 3D, each finger grasps a circular portion of the cylindrical branch. In 4D, assuming a spherindrical branch, grasping it would be a bit like holding a sphere.

The most efficient way to hold a sphere would probably be three fingers, each opposable to each other. Each group of three holds a portion of the branch, but the groups might share a common finger.

Is there any reason not to share more than one finger?

For comparison, think of a 3D hand as having four sets of two opposable fingers, but each set has a common member, the thumb. The "four" is probably arbitrary, but the "two" comes from the 2D surface of the branch.

Good point. And I think this would apply not only to climbing branches, but to holding 4D objects in general. I can see why it could be difficult to keep a firm grasp on a 4D object if you can only hold it along a 2D region---it could rotate around that plane and roll out of your grasp.

[...]So they can have: 2k fingers sharing one thumb, or k fingers sharing two thumbs, where k is the number of "sets". Since our general goal is a humanoid, lets say 2k fingers and one thumb.

Hmm. I think it might make sense to have two thumbs + k fingers, or maybe even have sharing across two groups of perpendicular fingers.

The sets extend along the branch, and each thumb has to be opposable to every finger. The further away from the thumb/s, the weaker the grip, so there must be a limit on the number of sets. so lets say four sets, like us.

This means 8 fingers and one thumb.

Interesting. So 9 fingers per hand? Or, if we set k=3, 7 fingers per hand.

4 fingers for each dimension of the surface of the branch, plus one thumb for stability. You can roughly imagine having a hand with an extra row of fingers, perpendicular to the original ones. It's easy to project, because the hand is stuck to a 3D surface anyway.

......||||................||||
..\_.||||................||||._/
..\_|....|...............|....|_/
..\_|....|=>......<=|....|_/
...\_|..|.................|..|_/
......|..|.................|..|

I can't get the picture to behave. The dots might help.

OK, this is a lot better than the first version of it. You could also use < pre > ... < /pre > tags. Let's see here:<pre> ||||
\_ ||||
\_| |
\_| |=>
\_| |
| |</pre>

[quickfur]

Ack, the < pre > ... < /pre > tags showed up good in preview, but get screwed up after the comment is posted. Sigh...

[pat]

Yep, the only way I've found to convice the BB to leave stuff is to wrap it in the '[ code ]' tags.

[PWrong]

Ok, I've worked out what it might look like to us if the universe happened to be wrapped around a 4D tree branch. First, imagine wrapping Fred's house around a 3D tree branch. If you touch the branch, all fred can see is the part of your hand in contact with it.

Now, when you grasp the branch, four thick walls suddenly appear around Fred. Then a thicker, shorter wall appears, pointing in the opposite direction, and almost meeting the two middle walls. Because space wrapped around a cylinder is flat, not curved, the walls appear completely straight.

I don't think the surface of a spherinder is the same as this. At first I imagined a long corridor, that wraps at the walls, roof and ceiling, but I think this might be a duocylinder.

[Rkyeun]

Standing in 4D is more difficult, even if trained to not need the 'stable' number of legs.

With our 2 legs forming a line, we cancel out one direction we can fall in, and anchor ourselves from rotating around one axis.
The ground prevents our falling in another direction, and gravity prevents us from falling in the other.
This leaves only forwards and backwards to fall into.

However in 4D, you can fall forwards, backwards, ana, and kata. Your difficulties in balancing have squared. You are only anchored from rotating along a single line, but 4D objects rotate around planes. You need a third leg to form a plane in which you cannot rotate, which only leaves you balancing against the other doubly perpendicular plane.

[quickfur]

Standing in 4D is more difficult, even if trained to not need the 'stable' number of legs.

With our 2 legs forming a line, we cancel out one direction we can fall in, and anchor ourselves from rotating around one axis.
The ground prevents our falling in another direction, and gravity prevents us from falling in the other.
This leaves only forwards and backwards to fall into.

Hmm, that's a good way of looking at it. So with our 2 legs, we essentially only need to balance along a single plane of rotation, which is much easier than balancing along two (as shown by the fact that it's rather difficult to stand on one leg for extended periods of time).

However in 4D, you can fall forwards, backwards, ana, and kata. Your difficulties in balancing have squared. You are only anchored from rotating along a single line, but 4D objects rotate around planes. You need a third leg to form a plane in which you cannot rotate, which only leaves you balancing against the other doubly perpendicular plane.

OK, so assuming that the Z axis is up/down, and that gravity pulls you toward -Z, there could be rotation in the following planes as you fall: XZ, YZ, WZ. Assuming you have two legs, one on +X and one on -X, that makes rotation around XZ impossible, which leaves YZ and WZ. So you still have 2 planes of rotation to worry about, whereas if you had a 3rd leg, say in +W, then the only plane of rotation possible is YZ, which would be much easier to balance on.

This is kinda funny in projection if you think about it... if you projected the 2-legged 4D person such that XYZ map to XYZ and W is truncated, you'd see a person with two legs managing to prevent rotation in XZ and YZ, but still falls over around WZ, which looks like he just collapsed downward into a flat plane in the projected image. :-)

Of course, one ought also to keep in mind that the reason we're able to balance easily on 2 legs in 3D is because the shape of our foot already points along the last plane of rotation, so as long as we don't lean over too far we're perfectly OK. We have trouble standing on 1 leg for a long time because our foot is anchored along a line, and can easily rotate in the perpendicular plane. I'd assume a 4D person's foot would be shaped such that it anchors in 2 perpendicular directions, so that his leg wouldn't rotate along WZ and YZ. (At least not without leaning over too far in either direction.)

I think 2 legs in 4D is still feasible... you just need the right foot shape. :-) Of course, that doesn't yet take into account how convenient/inconvenient it might be to orient oneself with only 2 legs. It's not exactly trivial to orient yourself when there are 3 dimensions you can face and still be standing upright.