How mathematicians think in dimensions above 3 and 4

The comments at that link are beautiful. If you're the sort who likes to skip comments, I encourage you to view them in this case.<p>I'd like to highlight a point of dissonance between the title ("How mathematicians <i>think</i>...") and the actual request ("anything that makes it easier to <i>see</i>, for example, the linking of spheres") -- emphasis mine.<p>I find the identification between visualization and intuition revealing. As a rule, mathematicians must be able to reason about things they cannot even begin to visualize -- non-measurable sets, infinities so large they need special names, infinite linear combinations of orthogonal functions.<p>That's not to devalue attempts at visualization. They're useful for developing intuition. But the original joke works because the mathematician is perfectly happy reasoning in hyperspace even though he cannot see it. The fourth dimension is not particularly harder to describe than the nth.

"For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers"<p>As usual Terry Tao's comment is wonderfully illuminating (at least for a non-mathematician like me). It's common knowledge that the ratio of the volume of a n-sphere to its circumscribed n-cube goes to zero as n-&#62;0 (<a href="http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area" rel="nofollow">http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_are...</a>), but I never thought of this as another result of the law of large numbers.

Correct but unhelpful answer: only through inordinate amount of practice with solving problems.<p>A possibly helpful answer to the question "Why is it probably a good idea for most people to give up on this?":<p>A dimension R^n is only a collection of points each of which is specified by an n-tuple over R. This isn't hard to hold in your head, but it tells you very little about how objects behave in any given space. By "think in", presumably the inquirer wants to be able to predict the behaviors of objects. The trouble is that there are many more different kinds of objects and behaviors than one is naturally inclined to assume, because one doesn't normally think of his ordinary 3D intuition as an insanely complex piece of specialized hardware that it is. You can certainly learn to do in software small parts of it, one by one. You would take those n-tuples, and do some particular bit of math on them to get you where you want to go. This will be difficult, like all math. Gather a big pile of these small pieces, train them until they're fast enough, and eventually you've got something like a crude emulator. I would hazard that unless you are interested in deep abstract (as in, non-visual) problems of the relevant math branches, you won't have the discipline to carry all this out.

<a href="http://www.dimensions-math.org/Dim_reg_AM.htm" rel="nofollow">http://www.dimensions-math.org/Dim_reg_AM.htm</a><p>This video series was linked by one of the comments, parts 3 and 4 had some interesting visualizations of various the 4d platonic solids.

There were two great quotes from the math book i used last semester.<p>The first was the most mathematicians equate geometry with understanding. In otherwords, if you can describe (draw) a picture that represents what you are talking about, then you really understand it.<p>The second was that not all math is something you can visualize. In that some of the beauty of math is that it can describe things beyond humans abilities of perception, and that it is precisely in these cases that math is its most useful.

The best part of the that joke (at least in my mind) is that generally engineers are pretty good, if not the best, at visualising multiple dimensions &#62; 3.

P.S. Terry Tao posted a comment on the OP. Worth a read.

A bit offtopic (from thinking about R^n to visualizing it), but I recommend reading about parallel coordinate systems. <a href="http://en.wikipedia.org/wiki/Parallel_coordinates" rel="nofollow">http://en.wikipedia.org/wiki/Parallel_coordinates</a>

Isn't it a little <i>too convenient</i> that the Universe seems to prefer 3d for physical processes, and that almost any kind of phenomena we need to understand can be framed in a 3d model? Or is it us the whole time, unable to see beyond our own limitations? When we see grandiose contrivances in a film plot, we naturally suspend disbelief in order to enjoy the story. I think those of us who have learned university math and who think we understand higher dimension are doing likewise—telling ourselves a story so we don't have to walk out of the theater.<p>I don't think we consciously "think" in any dimensions above 3. Like all life, we are neurologically hardwired by default to think in 3d, since that's what has worked for the evolution of all life on Earth. As for formal math, I am no longer even sure there is a connection between the logical, axiomatic and cultural edifice we've built up and call "mathematics", and our day-to-day "neural navigation software" that allows each of us to get from point A to B on the surface of a rotating oblate rock tethered to a nuclear fireball that is hurtling through the infinite, continuous space we call the Universe. The map is not the territory. The mathematical deductions of our neural experience are not the same thing as the experience itself.<p>From all of the suggestions on Mathoverflow, almost all of them are variants on projecting higher dimensional objects onto 3d and 2d objects, then comparing all the different projections in a clever way to get a "feel" for how the higher dimensional object changes. Even this is completely non-intuitive, as our brain's visual apparatus is optimized to take in a total picture and immediately spot the biggest changes. i.e. there's a predator running at us from over there! That is afterall what eyeballs and visual perception evolved for! If we can't even see the whole visual field at once, but just slices of projections of it, then our finely tuned visual hardware is thwarted and unable to detect and piece together the "shape" of objects. This is why is say nobody can "think" higher than 3d—even if you are doing it, your brain is still imperceptibly and behind the scenes translating your logical construct into a 3d "sensory" construct.<p>With that said the best way I've come across for visualizing 4d objects is from complex analysis, where you can use color gradients to represent a dimension. It doesn't work so well going beyond 4d, but it's a great set of training wheels. <a href="http://www.mai.liu.se/~halun/complex/" rel="nofollow">http://www.mai.liu.se/~halun/complex/</a> <a href="http://www.nucalc.com/ComplexFunctions.html" rel="nofollow">http://www.nucalc.com/ComplexFunctions.html</a>

This video's really good for visualizing the higher dimensions:<p><a href="http://www.youtube.com/watch?v=aCQx9U6awFw" rel="nofollow">http://www.youtube.com/watch?v=aCQx9U6awFw</a>