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I try hard to "visualise" (say "imagine") a tesseract but I can't.

Why is it that I can't?


This may be a question for a scholar of some other discipline and not for a mathematician, e.g. psychology (topic: cognition?), anthropology, etc., but I am sure it is well defined and answerable as a question.

It could be answered with a definition of what I can imagine or the definition of what I can't imagine and why, for example.

There may be some fundamental property of our geometry that limits what we can represent so the question may be interpreted mathematically... anyway I think it is not a question to bounce without any thought.

Specifically: what is missing for me to be able to imagine a tesseract? Understanding? A different kind of brain, that processes information in a different way?

Can a top mathematician visualise a tesseract? I am not inviting a discussion, which would be off-topic. I am soliciting a thoughtful and articulate answer, if possible.

Note:

I already saw this question:

In what sense is a tesseract (shown) 4-dimensional?

and this video:

http://www.youtube.com/watch?NR=1&feature=endscreen&v=uP_d14zi8jk

already read this link:

http://en.wikipedia.org/wiki/Tesseract

and I have studied calculus-level maths, etc. and I found no difficulty in reasoning about imaginary numbers, infinite quantities and/or series, demonstrations ad absurdum, etc.


I would be really disappointed if this question were marked as "not constructive", or anything to that effect. I can accept it may be "off topic" because it may relate to how our brain visualises and not some mathematical property that prevents visualisation, but it really should not be considered as "not-constructive". It would actually help me so much to understand this conundrum...

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Tesseracts move through both time and space without moving in time. To quote Hermes Conrad "I can see sideways in time...". –  Asaf Karagila Oct 19 '12 at 11:55
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It's slightly off-topic, but close enough for most people, I think. –  rschwieb Oct 19 '12 at 12:03
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If you could "visualize" a tesseract, how would you convince yourself that you're correct? –  Neal Oct 19 '12 at 12:34
    
I don't know. By imagining how it would cast a 3-D shadow and compare it? Counting its constituent parts of less than 4-D? I have no idea? –  Robottinosino Oct 19 '12 at 12:37

3 Answers 3

Big surprise: our brains evolved in a three-dimensional environment, and so that is what they are best suited for thinking about. It's easy to visualize because we literally see it all the time.

Thinking in higher dimensions is harder because we have no (little?) direct experience with them, so there is not a clear prototype for most people to use as a springboard for visualizing it.

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Can you visualise it? Do you know people who can? (I don't mean: "reason about it mathematically", or "deduce from a 3-D projection which properties the 4-D entity might have"... I mean "can you imagine it"?) –  Robottinosino Oct 19 '12 at 11:56
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You say "our brains evolved in a three-dimensional environment", could it be that our brains simply evolved with 3-D perception capabilities? Are we really in a 4-D world which we can't perceive? This might not be what you are alluding to... Please explain. –  Robottinosino Oct 19 '12 at 11:59
    
@Robottinosino I wouldn't say that I can imagine it, no. The best I can do is think think of it as an analogue of squares and cubes, and think "it has cubes for sides." Maybe with more practice I could convince myself I'm successfully imagining it, but I haven't had a need to so far. "our brains simply evolved with 3-D perception capabilities" is what I meant to convey. –  rschwieb Oct 19 '12 at 11:59
    
@Robottinosino: Some people believe that we are in a 26 dimensional world, but we can only perceive 3 dimensions. Let me ask you now, can you smell with your eyes? Can you see with your hands? Each sense has a different purpose, and after all... we do have a sense of time. So in some sense we can sense four dimensions. –  Asaf Karagila Oct 19 '12 at 12:20
    
@Robottinosino After reading Asaf's answer, I'd amend what I said from "perceive" to "visualize", because I agree that pereception of time is relevant to perception. Visualization relies primarily on sight, and there are only three visually apparent dimensions. –  rschwieb Oct 19 '12 at 12:23

To get an idea of what a 4-cube could look like I started imagining being a 2 dimensional entity on a page containing the projection of a 3-D cube :

cube

When my 2D entity progresses in a (new) 'perpendicular' direction, to the 'back' of this strange 3-square, the square at the middle will become larger and larger until replacing the larger square (a new square will appear at the middle, begin to grow, and so on...). Your entity is in a precise square at every instant but other squares are 'before' and 'behind' you.


Let's try this in 3D : we consider a 3D cube like the previous one and we represent a smaller 3D cube inside the first one. Again we imagine that, as we progress in a new perpendicular direction the smaller cube will take the place of the larger one ; a new one will appear at the center, grow and so on... At any moment you are only in one of these cubes but you will cross other cubes while moving.

3D

To move in 'full-4D' you may imagine that cubes are filling the whole 3D space (generating a 3D grid).

grid

Superpose to this 3D grid a smaller and parallel one (or an infinity of smaller and larger ones if you want...) that has a junction at every intersection with the corresponding junction of the larger one. This will give you a 4D grid and as you move in 4D the smaller grid take the place of the larger one, a new small grid will get out of the mist grow and so on...

At this point you may begin to move in different directions in 3D as well as in the 4th dimension and try the more confusing rotations.

Fine explorations !

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The vertices of a tesseract may be thought of as consisting of the $16$ points of $\{0,1\}^4=\{(a,b,c,d) : a,b,c,d\in \{0,1\}\}$. Two vertices are adjacent (connected by an edge) if and only if their coordinates disagree in exactly one place. For instance, $(0,1,1,0)$ is adjacent to each of the four vertices $(1,1,1,0), (0,0,1,0), (0,1,0,0), (0,1,1,1)$.

Thus each vertex has degree $4$ pointing in mutually orthogonal directions. The direction of an edge is determined by which coordinate its vertices disagree in. This allows us to say which edges are parallel. For instance, the edge joining $(0,1,0,0)$ to $(0,1,1,0)$ is parallel to the edge joining $(1,0,0,0)$ to $(1,0,1,0)$ since in both edges, the vertices disagree in the third coordinate.

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