Is there is any other way than abstract mathematics to visualize higher dimensions. Physicists working in high energy physics live in higher dimensions (pun intended), with their sophisticated mathematical tools they easily work there way out in higher dimensions. Take for a simple example of a $n$-sphere $S^N$ http://en.wikipedia.org/wiki/N-sphere.

  • 0-sphere $S^0$ is a pair of points ${c − r, c + r}$, and is the boundary of a line segment (the 1-ball $B^1$).
  • 1-sphere $S^1$ is a circle of radius $r$ centered at $c$, and is the boundary of a disk (the 2-ball $B^2$).
  • 2-sphere $S^2$ is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (the 3-ball $B^3$).
  • 3-sphere $S^3$ is a sphere in 4-dimensional Euclidean space.

The problem is how hard I think it is impossible for me to visualize a 3-sphere.


closed as not a real question by David Z Feb 3 '12 at 8:22

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  • $\begingroup$ Is this a question? What are you asking? $\endgroup$ – Siyuan Ren Feb 3 '12 at 8:18
  • $\begingroup$ You do not need to visualize higher dimensional spaces to understand them. Let me give you an analogy (the analogy I am giving is not isomorphic). How can one understand intuitively the meaning of a vector that lives in 5 dimensional space for example? $\endgroup$ – Revo Feb 3 '12 at 8:36
  • $\begingroup$ A student that takes 5 courses this semester say, his overall performance this semester depends on how he does in the 5 courses. Hence, you can think of his performance as a vector that lives in 5 dimensional space, one axis for every course. The axes can be graduated so you can assign a mark per course. Of course in this case the axes scales run from 0 to the max possible mark. $\endgroup$ – Revo Feb 3 '12 at 8:36
  • $\begingroup$ Please understand that my example is only an analogy to give you an intuitive feeling of higher dimensions. The student performance is not a vector in the mathematical sense. $\endgroup$ – Revo Feb 3 '12 at 8:36
  • $\begingroup$ I still don't understand the question... $\endgroup$ – Debangshu Feb 3 '12 at 17:46

Unless people have some superhuman ability, I don't think they really can visualize things beyond three dimensions. There are certain concepts (for example orthogonality) which carry over to higher dimensions, so the behaviours you see in lower dimensions are a very good guide to what happens in higher dimensions. Ultimately, however, these objects are defined mathematically and any conclusions about their behaviour must come from the calculations.

  • 1
    $\begingroup$ I think the other (apart from sight) human senses may help to "visualize" additional dimensions. For example, sound. You see our 3D space in a steady state, but sound may change, so it is another "dimension" independent of our 3D perception. $\endgroup$ – Vladimir Kalitvianski Feb 3 '12 at 10:38

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