*Matt Strassler 11/07/11*

When I’m chatting with a non-physicist, and the topic turns to the possibility that space might have additional dimensions that we aren’t aware of, the most common question that I get is this one: “How do you folks think about extra dimensions? I can only imagine three and have no idea how you would go beyond that; it doesn’t make any sense to me.”

Well, what we physics folks *don’t* do (at least, no one I know claims to do this) is **visualize** extra dimensions. My brain is just as limited as yours, and while that brain effortlessly creates a three-space-dimensional image of a world that I can move around in, I can’t make it bring to mind a picture of a four- or five-dimensional world any more than yours can. My survival didn’t depend on being able to imagine anything like that, so perhaps it isn’t surprising that my brain isn’t wired to do it.

Instead, what I do (and what I am pretty sure most of my colleagues do, based on how we all exchange ideas) is develop intuition based on a combination of analogies, visualization tricks, and calculations. We’ll skip the calculations here, but a lot of the analogies and visualization tricks aren’t that hard to explain.

There are actually two parts to learning to think about extra dimensions.

- The first is easy: learning to
**represent or describe**a world with extra dimensions. You already know how to do this in several different ways, even though you may not realize it — and you can learn some more. - The second is harder: learning
**how things work**in a world with extra dimensions. How do you thread a needle in four dimensions rather than three, for instance; would planets orbit a sun with six space dimensions; would protons form, and would atoms? Here you need to learn tricks that are unfamiliar, thinking about how different a world with only one or two dimensions would be from the three-dimensional world we know, and working up by analogy.

So I’ll start with helping you learn to represent a world with extra dimensions. To answer that requires we think about how to represent any dimension of any sort. Let’s start at the beginning.

- A zero dimensional world is a point. There’s not much to say about it right now, but we’ll make use of it later.
- But a one-dimensional world is already pretty interesting. Click here to learn more.
- Here’s an article on two-dimensional worlds. A lot more going on here!
- It’s important to avoid confusion about spatial dimensions and the more general concepts of the word “dimension” as used in English and even in mathematics or statistics. A few comments about that are here.
- And then here are some examples of
dimensions, emphasizing what the “extra” really means conceptually, and how it could be that the world has spatial dimensions of which we are completely unaware.*extra*

After these articles, you can start to learn how we might figure out that extra dimensions actually exist, by clicking here.

Hi Prof. Matt Strassler,

Following along is hard……triangles on a sphere or saddle are ways off for sure, ….but yes following along systemically needs to be understood by myself. Would this sort of be like the evolution of geometry in concert with building a dimensional world??

“A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry. ”

Best,

Can I hope for a post on Hilbert space?

How does curvature show up? By geodesic convergence (attraction!), or geodesic divergence (repulsion!). Thus we can tell people who are not in the know this: each time one experiences a force, one can suspect there is/are extra dimension(s) at work.

Okay, let me see if I get this…The way in which I picture three dimensional space is LWH or the world around us. When I add a fourth dimension, I have always “pictured” a seed (a point on a line). As that seed grows, it gains width, height and lemgth. And as my imaginary tree is a Semper virens, it also gains age. Though I like the threading the needle. Can we look at the whole of threading a needle as being a 3 dimensional space, while the actual process of feeding the thread through the eye as the fourth?

Hmm… I don’t think I understand enough what you’re saying here to reply usefully…

In reference to ” …from the three-dimensional world we know…”.

I thought we lived in a 4-D world (time) it’s linear and has length.

:)

We’re both right: we live in a world with 3 spatial dimensions, 1 time dimension, and, a total of 4 space-time dimensions. Obviously one has to be clear about whether one is referring to spatial dimensions only or to the full space-time. These types of confusions have to be carefully avoided even by professionals. Here, in these articles, I have rigorously stuck to talking about the number of spatial dimensions only. If you want to talk about space-time that’s fine — just add one.

would any particle,or system existing in our known observable dimension,,,display the same properties in multiple dimensions???in other words could the same paricle appear in multiple dimensions,,,with the same Me??all at once,,,or would any system in one dimension,be missing from another…???and if subatomic particles,are multidimentional,and so far all appear as +me in our dimension,,,what would that imply about other unobservable dimensions?

You should read the following article and the articles that it links to

http://profmattstrassler.com/articles-and-posts/some-speculative-theoretical-ideas-for-the-lhc/extra-dimensions/

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