Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do.

Now I know of the video series called Dimensions which aims to help the viewer imagining the fourth dimension (besides presenting visual mathematics) and there are other sources that show possibilities to do so. This doesn’t interest me as much as articles of people describing what they do.

So my question is:

Do you know of any sources, papers, books, videos and such, in which geometers or other people describe how they visualize and work in four or higher dimensions?

Also, many such strategies include projecting four-dimensional objects onto three-dimensional space. I was wondering how this is different from imagining three-dimensional objects by projecting them onto the plane – just as is necessary to represent them on a piece of paper or the computer screen. Do we really “see” three dimensions as we see two dimensions or are we just so used to work in them that we only think we see them, whereas we really somehow “compute” three dimensional objects by projecting them onto in two dimensional space?

So another question is:

Do you know of any sources, papers, books, videos and such, concerning the difference in visualizing two, three and higher dimensinoal space and objects?

Related are the following questions:

Especially the last question contains great answers and maybe this is all there is to it, but since this question is slightly different and has an extra part, I thought: Maybe it’s worth asking it nonetheless.

• Not sure if this should be made community wiki. – k.stm May 12 '13 at 18:50
• An easy answer to part of the question: We do not see three-dimensionally. Our eyes give us two-dimensional images, and our mind interprets this as a three-dimensional reality. – Eric Stucky May 12 '13 at 18:58
• One might argue that we study topology (as a set with certain structures) or geometry (as a set with certain allowable transformations and relationships among structures) in order to avoid having to "see" in higher dimensions. I can't imagine what a 4-sphere looks like, but I can sure reason about it's properties. – Shaun Ault May 12 '13 at 19:09
• I just try to imagine a world where "throw yourself at the ground and miss" actually makes sense. Then I'm flying. – Tim May 12 '13 at 19:31
• Some addition to the comment of @EricStucky , our mind can conceptualize higher dimensional objects (including 2 dimensional manifolds that can not be embedded into $\mathbb{R}^3$), but our mind can not conceptualize the perception of such objects, for example Klein bottle. We can never imagine what Klein bottle "looks" like. – Shuhao Cao May 12 '13 at 19:43