Probably a stupid question regarding 4-dimensional space

[Mhorton91]

Okay, let me start this off by saying this, I have no idea what section this belongs in.

On break between Spring and Summer classes, I stumbled upon a tesseract, or hypercube for the first time. And it got me thinking, on a purely philosophical level (ie, I have no where near the mathematical or physics knowledge to try to approach it scientifically). What a body moving through 4 dimensional space would experience. I've had several ideas come and go, but, based on the appearance of a tesseract, at least it's representation when drawn on a 2 dimensional surface, the best assumption I've been able to make, is that, in 4 dimensional space, you would observe movement in 3 dimensions exactly as we do now, however, my visualization for 4th dimensional movement involves the object or body expanding volumetricly (I don't think that is a word) as it moves "outward" in the 4th dimension, with the opposite happening as it moves "inward" in the 4th dimension...

Now, let me apologize now if this is the most nonsensical thing you have ever read. Trust me, I have put off bring this question to all of you for fear of this.

However, as many of you probably know, when your brain gets stuck on something, there is very little you can do.In any case, I appreciate any insight into my question, because as of now I can't go 5 minutes without pondering on it.

Requ.

 

[nomadreid]

First, let's isolate your questions:

in 4 dimensional space, you would observe movement in 3 dimensions exactly as we do now, however, my visualization for 4th dimensional movement involves the object or body expanding volumetricly (I don't think that is a word) as it moves "outward" in the 4th dimension, with the opposite happening as it moves "inward" in the 4th dimension...

From your example with the tesseract, I assume you are concentrating on four-dimensional Euclidean space.
What I am not sure is whether you are referring to your perception of a 4-dimensional object from an observer in three dimensions, or from an observer who is able to perceive four dimensions.
In the former case, your assumption is correct -- in fact the object's volume would vary, including disappearing in one three-dimensional locality and re-appearing in another locality. You might like to read Flatland by Abbott.
However, in the latter case, then your second assumption is not correct unless you are referring to depth perception.
My guess is that you are referring to a 3-D observer, although that was not stated in your question. Would you care to be more precise?

 

[Mhorton91]

First, let's isolate your questions:

From your example with the tesseract, I assume you are concentrating on four-dimensional Euclidean space.
What I am not sure is whether you are referring to your perception of a 4-dimensional object from an observer in three dimensions, or from an observer who is able to perceive four dimensions.
In the former case, your assumption is correct -- in fact the object's volume would vary, including disappearing in one three-dimensional locality and re-appearing in another locality. You might like to read Flatland by Abbott.
However, in the latter case, then your second assumption is not correct unless you are referring to depth perception.
My guess is that you are referring to a 3-D observer, although that was not stated in your question. Would you care to be more precise?

Yes sorry, your assumption was correct.

Observing from our familiar 3 dimensional space, somehow being able to look across and observe a finite 4 dimensional space.

What you're saying regarding an objects volume disappearing in one 3 dimensional locality, then reappearing in another, some how I can picture that in my head, and the idea makes since, but I don't seem to grasp the basic reason as to why. Is that strictly due to the fact that in the example we are viewing 4 dimensional space, from our 3 dimensional space? Or would that behavior also occur for a being capable of observing 4 dimensions?

Thanks again!
Requ

 

[nomadreid]

Our 3-D space would be a cross section of a 4-D space, hence we would see the intersection of the 4-D object with our 3-D cross-section. To imagine an object appearing or disappearing, analogize down to a 3-D object intersecting with a 2-D cross-section. For example, a ball which moves around in space but occasionally goes through a particular plane. For concreteness, let the plane be the x-y plane in a space with x-y-z coordinates, and a ball with a north-south axis perpendicular to the x-y plane, with the north pole having a larger z-coordinate than the south pole. Now consider three types of motion:
(1) the ball moves so that its equator remains in the x-y plane. In that way, inhabitants of the 2-D "Flatland" (who could only perceive things in 2-D) would see a circle moving around their space.
(2) the ball has a multi-colored surface and it rotates so that the circumference of its intersection with the plane is still a great circle (i.e., same size as the equator): the 2-D creatures would see its size being the same but the circle changing color. Alternatively, make the sphere have jagged edges; as it rotates, the 2-D creatures would see its shape change.
(3) the ball starts with its equator in the plane, as before, but now moves upwards (in the positive z-direction) until the z-coordinate of its south pole is greater than zero; then it moves in the positive x-direction for a while, then descends so that it again intersects the x-y plane. The 2-D creatures would see it gradually shrink and disappear, stay disappeared for a while, then re-appear somewhere else, gradually getting larger.

You get even nicer effects if your plane is not flat, and you have other shapes than balls. For example, suppose your 2-D surface has (from a 3-D viewpoint) hills and valleys.

All this stays valid once you up the dimensions to a 4-D object for an observer in a 3-D cross-section.

Or, if you want to blow your mind some more, n+1 objects for an observer in n dimensions.

Oh, and no, this would not be the same for a 4-dimensional observer of a 4-D object in a 4-D space. There you would analogize from a 3-D observer in a 3-D space observing 3-D objects.

 

[Mhorton91]

Thank you for your response!

The cross sectional view makes total sense, I guess last night with the sleep deprivation weighing in I over looked that.

So essentially if an observer is in n dimensional space, viewing a body in n+1 dimensional space; for any given n value you will always lack that final dimension, and will in turn only be viewing only an n dimensional cross section of the n+1 dimensional body.

Am I grasping the basic concept properly here?

 

[nomadreid]

Yes.

 

[PhanthomJay]

Here is a projection of a 4-D hypercube projected onto a 2D planar surface. The pink lines are the usual x, y, and z dimensions, and the black lines are the 'w' axis into the 4th spatial dimension, perpendicular to the other 3.

https://www.physicsforums.com/attachment.php?attachmentid=43663&d=1328726442

Problem is that while we can easily visualize a 3D cube projected onto a 2D surface, as in an isometric drawing or photo or screen, it is difficult to visualize what the tesseract looks like when projecting 4D onto 2D, as in the above image. So.....,
I sure hope you can open the link below (I can't), which shows the 4-D hypercube projected onto 3-D. You'll need a set of 3-D glasses to view it, but not the movie theater type, they've got to be the older style with one lens red and the other blue.
The 4th dimension revealed!

(maybe...)

http://dogfeathers.com/java/hypercube2-nogl.html

 

[Mhorton91]

Yes.

Yes!

Here is a projection of a 4-D hypercube projected onto a 2D planar surface. The pink lines are the usual x, y, and z dimensions, and the black lines are the 'w' axis into the 4th spatial dimension, perpendicular to the other 3.

https://www.physicsforums.com/attachment.php?attachmentid=43663&d=1328726442

Problem is that while we can easily visualize a 3D cube projected onto a 2D surface, as in an isometric drawing or photo or screen, it is difficult to visualize what the tesseract looks like when projecting 4D onto 2D, as in the above image. So.....,
I sure hope you can open the link below (I can't), which shows the 4-D hypercube projected onto 3-D. You'll need a set of 3-D glasses to view it, but not the movie theater type, they've got to be the older style with one lens red and the other blue.
The 4th dimension revealed!

(maybe...)

http://dogfeathers.com/java/hypercube2-nogl.html

I will try to check these links out when I get home from work, my phone is wanting nothing to do with them.

At the risk of asking too many questions, when Nomadreid said this

What I am not sure is whether you are referring to your perception of a 4-dimensional object from an observer in three dimensions, or from an observer who is able to perceive four dimensions.

It made me curious about the latter case, is there a generally accepted theory as to what an observer capable of perceiving 4 dimensions would observe? Since there is no way to physically test said theory, it got me wondering if mathematicians had agreed on what such a world would be like, to someone who could perceive all dimensions.Thanks again for all the knowledge!
Requ.

 

[verty]

Here's another interesting thought experiment. Mazes in 2D are a bunch of lines, but there are also 3D mazes, for example garden mazes. The most common of these mazes are actually 2D mazes given some thickness; at any point in the maze, one can lie on the ground or look up at the sky.

Now consider the analogue of this in 4D. Consider a 3D maze of tunnels where junctions offer left, right, up and down directions. But in 4D, at any point we can look "in" at the ground or "out" at the sky. So for example you could climb "over" a wall of the maze and appear in a neighboring tunnel.One way to think about this is that a person who climbs "out" becomes translucent like a ghost, so they can move through walls. Then moving "in" to ground level, they phase in again. And 100% opaque means being at ground level.

Or imagine a maze like this tunnel maze but a 4D analogue. Junctions would offer up, down, left, right, "in", "out" directions. For a 3D observer, it would look like the previous tunnel maze but at certain places one could "phase out" one unit and the 3D visible maze would transform into an entirely different maze, quite like teleporting to a different maze. Or there could be holes so that the maze suddenly changes, sort of like teleport traps in AD&D games.

So actually, there are ways to think about 4D space if one gives it some regularity. But take a way the regularity and it becomes difficult.

 

[Mhorton91]

The maze train of thought is interesting, and much easier to visualize than a purely abstract approach.

Another random question, as an undergrad, who is just getting ready to start the Calculus sequence in the fall, what are some classes I could take that would deal with these higher dimensional spaces (once I meet the prerequisites)?

 

[Mhorton91]

You get even nicer effects if your plane is not flat, and you have other shapes than balls. For example, suppose your 2-D surface has (from a 3-D viewpoint) hills and valleys.

All this stays valid once you up the dimensions to a 4-D object for an observer in a 3-D cross-section.

Or, if you want to blow your mind some more, n+1 objects for an observer in n dimensions.

I reread this, and it gave me a couple more questions, first, my assumption of Euclidean space was that it could not contain these hills and valleys, is that assumption invalid?

And 2nd, although I have an idea, it feels too simple, what if you viewed an object in n+2 dimensional space, from an area of n dimensional space. I understand it would just be a n dimensional cross section, lacking 2 dimensions. But a visualization is giving me difficulty once we go above n=1...

 

[Bill Simpson]

There is a really old book, available used cheap now, "Experiments in Four Dimensions" by David L. Heiserman (1983).

The author begs you to actually DO all the exercises, step by step. He gradually leads you through the linear algebra of 1 and 2 and 3 and finally 4 dimensions. The claim is that if you really do all the exercises that by the time you get to 4 dimensions you will have most of the skills needed to have a feel for how this works.

There are a handful of other much more advanced books, usually in the subject of topology, who try to explain how to get a feel for, if perhaps not really being able to "see", objects in 4 dimensions. I think Steenrod wrote one of those, but it has been long enough that I'll never find either of these books buried in the boxes back there.

Ah, one more, if you have access to a good university library. Early in the last century Charles Howard Hinton was a prof at Harvard I think. He developed a set of colored cubes and a pamphlet that described some exercises. The colored cubes were to be various 3d subsets of an unseen 4d tesseract. You were supposed to do the exercises and after a while gain an ability to "see" in 4d. It is claimed that he had a truly stunning memory and could either "see in 4d" or could just do problems in 4d that nobody could imagine how you could do that quickly without being able to see it. And his daughter was even better than he. Rudy Rucker, sci fi author, claims in one of his books that he was able to reproduce this set after a century, but I've never seen a description of the exact coloring of all the faces or the booklet of exercises to practice with. In a good university library you might still find reprints of a few of Hinton's original texts that give some information on how to begin on working to see this.

This might help you get started http://www.worldcat.org/search?q=au%3Acharles+howard+hinton&qt=advanced&dblist=638
You just need to find the libraries within a hundred miles of you that have these on the shelf OR find your local librarian and be REALLY nice to them so they can get you an inter-library loan of one of these at a time.

 

[nomadreid]

Bill Simpson's suggestions are excellent, and I second them.

In the cubes he mentioned, there is the interesting remark (without citation, so unverifiable) in the Wikipedia article on Hinton that "Rumours subsequently arose that these cubes had driven more than one hopeful person insane." However, similar rumours (also unfounded) still circulate that Cantor's infinite sets drove him insane, so I wouldn't worry.

However, I need to answer the specific questions Mhorton91 posed after looking at my earlier remarks.

You get even nicer effects if your plane is not flat, and you have other shapes than balls. For example, suppose your 2-D surface has (from a 3-D viewpoint) hills and valleys.

my assumption of Euclidean space was that it could not contain these hills and valleys, is that assumption invalid?

First, I was a little sloppy. In the mathematical sense of "flat", you are right: Euclidean space is flat, so that when I referred to a plane that was not flat, I was speaking in a more everyday sense. For example, the surface of a cylinder is mathematically "locally flat", in that as the sum of the angles of a triangle is still 180 degrees, and so forth. In everyday parlance, one thinks of a cylinder, when viewed in 3-D, as curved. So I was a bit sloppy there. However, this does not stop your plane from having the hills and valleys I meant (although it will hinder many other types of hills and valleys). For example, take a piece of paper and make the same sinusoidal wave on two opposite sides. (The other two sides stay straight.) This remains mathematically flat, but it has (from a 3-D viewpoint) hills and valleys that can make intersections with 3-D objects a little more interesting. But this suggestion is definitely to the side, and doesn't contribute much to your main concern of envisioning higher-dimensional surfaces.

what if you viewed an object in n+2 dimensional space, from an area of n dimensional space. I understand it would just be a n dimensional cross section, lacking 2 dimensions. But a visualization is giving me difficulty once we go above n=1...

If the relationship between your n+2 dimensional object and the n dimensional space of the observer remains static, then there is no way the observer will be able to identify the object as anything but an n-dimensional object. That is from the definition of dimension. However, if either you look at the n+2 object from enough different n-dimensional spaces, and you know the relationship between the dimensions, (nicely shown in the first vision of the present that the robot -- Schwarzenegger -- in the first "Terminator" film, and more mundanely one of the factors used by our eyes to make a 3-D picture), then you can get a better idea of the n+2 dimensions. If the object moves through the n+2-dimensional space in such a way that it intersects the n-dimensional space at several different angles, then this can give an equivalent picture. So, if you are trying to get a feel for a 2-D view of a 4-D object as it moves through your 2-D space, then draw lots of cross-sections as you would rotate the 4-D object, and then put the pages together and flip through them, like they used to make cartoons with lots of drawings that were only different from one another by a little bit. For a 3-D view of a 5-D object, you can either make models of the different cross sections and put them next to one another, or do this in your imagination (trickier). But the human brain puts limitations on one's intuitions above spatial 3-D.

Mhorton91 also asked whether mathematicians had agreed on what an observer of "all" dimensions would look like. He/she probably meant all abstract ones (although physicists can't even agree on the number of the concrete ones!) There is no such thing as "all" abstract spatial dimensions. (Vector spaces with infinite dimensions are common -- although usually not Euclidean -- but "infinity" does not equal "all".) Perhaps Mhorton91 would like to make the question more precise?

 

[Mhorton91]

Mhorton91 also asked whether mathematicians had agreed on what an observer of "all" dimensions would look like. He/she probably meant all abstract ones (although physicists can't even agree on the number of the concrete ones!) There is no such thing as "all" abstract spatial dimensions. (Vector spaces with infinite dimensions are common -- although usually not Euclidean -- but "infinity" does not equal "all".) Perhaps Mhorton91 would like to make the question more precise?

Thank you both! My phone isn't wanting to multi quote for some reason. I am definitely going to check our library for the books mentioned tomorrow!

But to clarify the above, I was referring to an observer in 4 dimensional space, who was capable of perceiving all 4 spatial dimensions in which they reside. What would he/she see as he/she went in and out of the 4th spatial dimension. Maintaining the assumption that the addition of a 4th spatial dimension doesn't alter the behavior of re original 3 dimensions, and that movement across those planes would not be drastically different than our own observable locality.

Mainly, again, working off of a strictly theoretical 4 dimensional Euclidean space, based on a tesseract.

My question stems from some very entry level work in vectors in 3 dimensions. The x, y, and z axes, and their corresponding plans each are defined as being perpendicular (maybe orthogonal is a better term to use, not sure) to the other 2. However, with the understanding as you stated, that the human brain is limited in 4 dimensional spatial reasoning. I'm trying to grasp a 4 axis that will be orthogonal to the standard x, y, and z axes.Sorry again for so many questions! When my brain gets caught up on something, I can't let it go until I understand... No matter how far over my head it is academically speaking.

Thanks again!

It's he, by the way.

 

[nomadreid]

an observer in 4 dimensional space, who was capable of perceiving all 4 spatial dimensions in which they reside. What would he/she see as he/she went in and out of the 4th spatial dimension.

Imagine a 3-dimensional being trying to describe to a two dimensional being what the 3-dimensional being is seeing. Higher-dimensional aspects can be abstractly, mathematically, described, and projections from the higher-dimensional into the lower dimensional can be presented, and finally the two approaches can be combined into a lower-dimensional approximation to the higher-dimensional object is given, as is the case for the "images" of a tesseract which are common, or the models of a Klein bottle often found in museums. Also I am not sure what you mean by the 4-th dimensional being going "in and out of" the four dimensions. If you mean that her perception were cut off so that she could only see fewer dimensions? A good analogy there is the experience humans have when they lose one eye, so that the world becomes almost flat in their perception. (Not completely, of course, since they also rely on memory and certain cues in the environment.)

 

[Mhorton91]

Okay, so because of the strictly theoretical context of which this topic resides. We only have a mathematical description of what it could look like, with no natural way to physically "see" it, outside of those mathematical descriptions.. if that is correct, then it totally makes sense to me.

Also I am not sure what you mean by the 4-th dimensional being going "in and out of" the four dimensions.

As far as my meaning behind moving in and out of the 4th dimension, first, I intended that in and out motion to only speak to the 4th spatial dimension of the observer.

However, looking over it again, I feel like my logic underlying that question/assumption is very flawed. I was basing the shape and context of the 4th dimension on that of a tesseract, when, in real life, theoretical physics says that additional dimensions aren't necessarily oriented in a way similar to our 3 spatial dimensions. They are more often viewed as being cylindrical and rolled throughout the existing dimensions.

Now, again, that assumption again puts me further over my own head in terms of understanding, and, again, I feel the need to apologize for that potentially being ridiculous, logically or mathematically speaking.

 

[nomadreid]

First, absolutely no need to apologize. If one understood everything at the beginning, then there would be no need to pose questions. The question was not at all ridiculous.

theoretical physics says that additional dimensions aren't necessarily oriented in a way similar to our 3 spatial dimensions.

The additional dimensions of physics are not Euclidean, but be fair to yourself: you posed the question with respect to Euclidean space. So your question remains OK.