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Can the fourth dimension https://en.wikipedia.org/wiki/Four-dimensional_space be visualized intuitively by the humans.

Does the professional mathematicians can do this ? If so what are the things to be learned to acquire this ability.

Talespin_Kit
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    Personally I never imagined four dimensions. I just imagine the 3D version, and it served me more than well (well enough for a masters). – 5xum Dec 12 '14 at 08:27
  • I treat them as non perpendicular in my head so its easier to imagine if it helps however after couple of semesters you get over the need for imagining more than 3D – Mr. Math Dec 12 '14 at 08:34
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    The old joke is that to visualize things in four dimensions a mathematician first visualizes an infinite-dimensional space and then cuts it down to four. – Gerry Myerson Dec 12 '14 at 08:51

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You can get pretty far by visualizing $(d+1)$-dimensional things as movies of $d$-dimensional things moving in time. These kinds of visualizations are closely related to Morse theory. For example, it's a bit tricky to visualize the $3$-dimensional sphere $S^3$, but it's not at all tricky to visualize the following movie:

  • Initially, there is nothing.
  • Then there is a point.
  • The point expands to a $2$-dimensional sphere $S^2$.
  • The sphere gets bigger.
  • The sphere gets smaller.
  • The sphere contracts to a point.
  • Then there is nothing again.

Similarly, to visualize $\mathbb{R}^4$ you can visualize a movie where you're looking at $\mathbb{R}^3$ and nothing happens. That's a bit boring, but it gets more interesting if you put stuff into $\mathbb{R}^4$. For example, you can try to visualize a knotted surface in $\mathbb{R}^4$ by visualizing a movie where you're looking at $\mathbb{R}^3$ and some knots and links appear and disappear...

Qiaochu Yuan
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Professional mathematicians can sometimes visualize simple 4 dimensional shapes and movements to a limited extent, but nowhere near the way we can visualize 3 dimensional space. To visualize four dimensional objects myself, I generally draw (or imagine) 2 planes, each with 2 axes, side by side. Any object I draw in one plane, I also mark points accordingly in the other. This helps in locating points and vectors, but not curves and shapes such as hyperspheres. Also, ability to visualize even a 3 dimensional space is a quality that some people are born with better than others.

ghosts_in_the_code
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