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If I see a circumference like the 2-d "collapse" of a function $\Bbb R^2 \to \Bbb R$ (whose graph is usually a 3-d image) I can also see the sphere like a 3-d collapse of a 4-d graph. So the "grade of diversity" between the graph of a $\Bbb R^2 \to \Bbb R$ function and the graph of a circumference can be the same as the one between a $\Bbb R^3 \to \Bbb R$ function and a sphere? With "grade of diversity" I mean the difference between the two representations as images. May this help me somehow in imagining the fourth dimension? In which way? Is there someone of you that is able to imagine the fourth dimension?

I'm neglecting the "temporal vision" of the fourth dimension because it seems to me that is possible to imagine it even in non-temporal terms. This is only a pastime, I'm not saying that I'm right in any way.

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  • $\begingroup$ the human mind really can't perceive past dimension 3 $\endgroup$ Commented Jul 27, 2017 at 18:43
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    $\begingroup$ How do you obtain a sphere from a function $\mathbb{R} \to \mathbb{R}^3$? Don't we need $\mathbb{R}^2 \to \mathbb{R}^3$? $\endgroup$
    – Adayah
    Commented Jul 27, 2017 at 18:44
  • $\begingroup$ Oh, you are right. I was reasoning with implicite function, so the sphere is $\Bbb R^3\to \Bbb R$ also for the circumference with $\Bbb R^2 \to \Bbb R$ $\endgroup$
    – pter26
    Commented Jul 27, 2017 at 19:00
  • $\begingroup$ @SakethMalyala That's wrong. Most skilled differential geometers/algebraic topologists can easily visualize $\mathbb{R}^{4}$ at least. $\endgroup$ Commented Jul 27, 2017 at 19:16
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    $\begingroup$ @MathematicsStudent1122 I don't believe that, although perhaps the disagreement is over the meaning of the word "visualize". Intuitive crutches for higher dimensional thinking have been discussed here. Note for example Scott Aaronson's comments about visualizing $\mathbb R^4$. While various techniques are available, nobody can directly visualize $\mathbb R^4$ the same way that we can visualize $\mathbb R^3$. $\endgroup$
    – littleO
    Commented Jul 27, 2017 at 19:25

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I often use a temporal dimension to try and visualize what's going on beyond the three spatial dimensions that I live in. This, combined with analogy, sometimes gives me an idea of what's going on.

As a simple example, what would a sphere passing through a plane look like? It would start out as a point, then a tiny circle, then grow to a large circle, and then shrink again to a point before disappearing. Similarly, a hyper-sphere passing through our 3-dimensional hyperplane would look like a point appearing, growing to a small sphere, to a large sphere, and then shrinking again to a point before disappearing. Each instant of that process is a 3-d cross section of the 4-d hypersphere.

Does that help at all?

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