In today's physics, the cutting edge theories require multiple spacial dimensions to work. Bosonic String Theory, for instance, requires 26 dimensions, while the five basic types of String Theory seem to need at least 11 dimensions. How can a person mentally visualize these extra spacial dimensions? Do they only exist as complex mathematical Calabi-Yau shapes, that only Hawking can imagine, or is there a more simple way a person can envision a sixth dimension, etc?

My short answer is that we don't need to be able to visualize higher-dimensional spaces in order to reason about them. I'd be quite astonished if Stephen Hawking could visualize 11 or 26 or even 5 dimensions. In fact, visualizing even three dimensions is not as easy as people think, as one realizes when trying to think through certain "ordinary" geometrical descriptions. But there are tricks that can sometimes give you the sense of visualizing higher dimensions, as with various diagrams of a four-dimensional "hypercube." Here's an example:

https://plus.google.com/117663015413546257905/posts/VteWm45DCff

Turns out that what I've said is more or less what the well-known physicist Sean Carroll says here:

http://www.preposterousuniverse.com/blog/2009/03/30/why-cant-we-visualiz...

though he adds some speculations that you can evaluate for yourself. But on the question you ask, it's (1) you can't, (2) you don't need to, and (3) there are all the same some tricks.

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