# Dimensions?

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In Stephen Hawking's "A Brief History of Time" he states that there may(should) be as many as 10 to 26 differant dimentions to space-time but that most are either infinitely small or millionths of an inch in diameter. Will someone please explain this to someone who has no grasp of dimentions past the 4th?

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What is meant by dimension is simply the "number of numbers" need to (locally) describe a point. Technically, this means space-time is a manifold.

For example, lets think about the the manifold $\mathbb{R}^{2}$, which is just the plain. Every point $p \in \mathbb{R}^{2}$ can be described by two numbers $(x,y)$ (once you set up a coordinate system, but don't worry about that now). Thus, we say $\mathbb{R}^{2}$ is two dimensional.

Now, not all two dimensional manifolds are the same as $\mathbb{R}^{2}$, but they are locally. By this we mean (loosely) that any small part of any two dimensional manifold is the same as $\mathbb{R}^{2}$. And really all we mean by this is that any point can be locally described by two numbers.

This then continues for any dimension. Every n-dimensional manifold is locally "the same as" $\mathbb{R}^{n}$. That is, any point on an n-dimensional manifold can be described by n-numbers. This is what is meant by dimension.

Now, again not all n-dimensional manifolds are the same as $\mathbb{R}^{n}$ globally, but only locally. They can be very different topologically, meaning there global overall shape can be very different.

I hope that helps a little.

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In Stephen Hawking's "A Brief History of Time" he states that ...?

Just as a general comment, Hawking's book may not be the most up-to-date or best place to begin.

If you want to get a taste of modern ideas of space and time, including the dimensionality issue, and you want it to very accessible and non-mathematical, then you might take a look at an article in the July 2008 Scientific

American.

It's great. I invite everybody at SFN read it or at least have a look. Here is a free online version:

http://www.scribd.com/doc/3366486/SelfOrganizing-Quantum-Universe-SCIAM-June-08

My goodness Hawking's book is old! before 2000. things have really changed since then

===================

Person, questions about the nature of space and time tend to come up when looking at different SCALES

we know what space and time look like at medium (humansize) scale

but at very large scale the universe might be curved, or distances expanding etc etc. so there is a potential for us to get puzzled looking at very large scale

but even more puzzling is thinking about spacetime at very small scales

(scales even too small to be probed using currently available instruments).

even a simple question about space (not even about space time) can be puzzling. even something as simple as the relation between radius and volume! (see the SciAm for this)

at humansize scale we know that the spatial volume goes as the CUBE of radius. the volume of a ball with radius R is proportional to R3.

So if you double the radius, the volume goes up by the cube of 2, namely by a factor of 8.

In some sense that relation between radius and volume CHARACTERIZES the dimensionality of ordinary space for us, at ordinary scale.

But we havent tested the relation between radius and volume at very very small scale. Maybe volume doesn't go as the cube! Maybe if the radius and the ball are very small, doubling the radius would make the volume go up by only a factor of 4, or 5, instead of 8! Or may the volume would go up by a factor of a thousand! We have to be open to weird possibilities at very small scale because we havent checked. Also there is a lot of fantasy and speculation, which isn't always helpful.

The authors of that SciAm article actually simulate little universes in the computer. Get them to pop into existence according to quantum gravity rules. And then they go into them and explore the dimensionality at various scales. Because they are simulated universes, it is possible to examine the relation of radius to volume at arbitrarily small scale in those universes.

the results are interesting, although they are not conclusive about OUR space and time because the connection between simulation and reality still has to be tested.

what they find is that whereas some people speculate that dimensionaity might be higher at very small scale ('extra dimensions') these people found in their simulations that the dimensionality was actually LOWER.

(quantum fluctuations kind of breaking up and foaming the geometry so it got kind of feathery and not as solidly connected, fortunately the quantum fluctuations kind of average out to something smooth and familiar at our scale). well that is surprising, so it makes for interesting reading.

BTW they use a quantum gravity system that DERIVES from work that Stephen Hawking did in the 1980s and early 1990s. ("euclidean quantum gravity") but they improved on it and got over some of the problems with Hawking's original approach. In any case there is a Hawking connection, which they discuss in the article, if you are interested.

It is a great article. i will give the link again.

http://www.scribd.com/doc/3366486/SelfOrganizing-Quantum-Universe-SCIAM-June-08

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Carl Sagan once imagined a 2 dimensional world similar to a sheet of paper in which “flat-lander” inhabitants knew only of forward, backward, left and right. They had no idea that a third dimension, that was beyond their experience, touched their 2D universe at every point.

This is sort of the perception of our own universe. What you see isn’t necessarily what you get. Extra dimensions may be so narrow that no accessible frequency of electromagnetism can be reflected off them, so that they remain virtually invisible. However, there is hope of their discovery:

http://www.news.wisc.edu/13422

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I don't know if you have read this, but it is very interesting reading. For all I know, you have exhausted this topic.

http://en.wikipedia.org/wiki/String_theory

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Is an interesting video...

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Carl Sagan once imagined a 2 dimensional world...

Sagan, like many, used that as a teaching tool, but it certainly wasn't his idea originally. See Flatland: A Romance of Many Dimensions, by Edwin A. Abbott (which is what Sagan was referencing in Cosmos).

Edited by Sisyphus
multiple post merged

Thanks!

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HOLY CRAP! My brain hurts!

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