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Over the years I've come across (usually as a tangential remark in a lecture) examples of how our intuitions (derived as they are from the experience of living in 3-dimensional space) will lead us badly astray when thinking about some $n$-dimensional Euclidean space, for some $n > 3$, especially if $n \gg 3$.

Does anyone know of a compendium of these "false intuitions" (in high-dimensional Euclidean space)?


P.S. The motivation for this question is more than amusement. In my line of work, the geometrization of a problem by mapping it onto some Euclidean $n$-space is often seen as a boon to intuition, even when $n$ is huge. I suspect, however, that the net gain in intuition resulting from this maneuver may very well be negative! In any case, it seems like a good idea to be conversant with those intuitions that should be disregarded.

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Possible duplicate of math.stackexchange.com/questions/48301/… –  Chris Taylor Oct 1 '11 at 14:51
Distant relative: Klein bottle does not intersect itself in 4-D. Try to imagine that with 3-D intuition. –  user13838 Oct 1 '11 at 15:25
Another distant relative: you could untie every knot having more than 3 dimensions. –  Emanuele Natale Oct 1 '11 at 23:43
The duplicate Chris pointed out is very close, but I'm hesitating to vote for closing because it doesn't have your focus on Euclidean space. Could you comment on whether you see it as a duplicate? If not, perhaps you could focus the question more specifically on what you find missing there? –  joriki Oct 2 '11 at 5:37
@percusse: That's actually pretty intuitive by the 'Flatland' approach: It's no worse than saying that the figure-eight intersects itself in 2d but not in 3d. Just imagine 'lifting' one of the two intersecting sections into the missing dimension through the region of intersection, and then settling it back down into the (x,y,z,0) space again. –  Steven Stadnicki Oct 4 '11 at 0:03

2 Answers 2

up vote 4 down vote accepted

Intuitions fails in higher-dimensions:

  • Imagine a unit hyper-sphere within a cube with side 2. In low dimensions (2d), most of the volume (area) is within the hyper-sphere (circle) and only a small fraction of the volume is outside of the hyper-sphere, thus in the corners of the hyper-cube (square). However, for high dimensions it is the other way around. The volume of the hyper-cube is obviously $V_q = 2^n$ while the volume of the unit hyper-sphere is $V_s=\frac{\pi^{\frac{n}{2}}}{(\frac{n}{2})!}$ (for even $n$) with $\lim_{n\rightarrow \infty} \frac{\pi^{\frac{n}{2}}}{(\frac{n}{2})!}=0$. In other words: Only for low dimensions, the bounding box of a hyper-sphere is a 'fair' approximation of the volume of the sphere.

  • ...

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In 2D you rotate around a point, in 3D you rotate around a line (axis), so in 4D you rotate around a plane? Well there are also rotations in 4D which fix only a point.

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This is an incorrect generalization, but there's also a correct generalization: In 2D we rotate in a plane, in 3D we rotate in a plane, so in 4D we rotate in a plane -- and indeed rotations in any number of dimensions can be written as a product of commuting rotations in mutually orthogonal planes. –  joriki Oct 4 '11 at 10:21

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