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If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't really know how big they are yet, but we hope to explore that in various ways.

There is a fifth dimension, beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition.

Plane geometry is sort of the key course where you learn about proving things and abstraction.

Our universe - it's three-dimensional, but we can pretend it's two-dimensional so it's like this sheet of paper - and we live in Pasadena over here and London is over there, and it's thousands of miles from Pasadena to London.

Height, width, and depth are the three phenomena which I must transfer into one plane to form the abstract surface of the picture, and thus to protect myself from the infinity of space.

We're quite into graphics that are simultaneously two- and three-dimensional. But I can't really elaborate any further because it's not something - we haven't really perfected it.

Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.

Now, what space ultimately is - I should confess, I think most physicists believe - we don't yet know.

One of the strangest features of string theory is that it requires more than the three spatial dimensions that we see directly in the world around us. That sounds like science fiction, but it is an indisputable outcome of the mathematics of string theory.

Why is there space rather than no space? Why is space three-dimensional? Why is space big? We have a lot of room to move around in. How come it's not tiny? We have no consensus about these things. We're still exploring them.