How to imagine higher dimensions?
Solution 1
In differential geometry, a space of a given number of dimensions can be curved rather than Euclidean, so for example the surface of a sphere is understood to be a 2dimensional space in spite of the fact that we can't help but visualize the sphere sitting in a higherdimensional 3D Euclidean space. This 3D space that we imagine the 2D surface sitting in is technically known as an "embedding space", but the mathematics of differential geometry allows mathematicians and physicists to describe the curvature of surfaces in purely "intrinsic" terms without the need for any embedding space, rather than in "extrinsic" terms where the surface is described by its coordinates in a higherdimensional spacesee the "Intrinsic versus extrinsic" section of the differential geometry wiki page. And all this has a practical relevance to physicists, since Einstein's theory of general relativity uses differential geometry to explain gravitation in terms of matter and energy causing spacetime to become curved (see here for a short conceptual introduction to how spacetime curvature can explain the way particle trajectories are affected by gravity).
With these ideas in mind, if you want to understand Greene's comment about higher dimensions being "curled up", picture the surface of a long cylinder or tube, like a garden hose. This surface is 2dimensional, but you only have to travel a short distance in one direction to make a circle and return to your place of originthat's the "curled up" dimensionwhile the perpendicular direction can be arbitrarily long, perhaps infinite. You could imagine 2dimensional beings that live on this surface, like those in the famous book Flatland that has introduced many people to the idea of spaces with different numbers of dimensions (and there's also a "sequel" by another author titled Sphereland which introduces the idea that a 2D universe could actually be curved). But if the circumference of the cylinder was very shortshorter even than the radius of atoms in this universethen at large scales this universe could be indistinguishable from a 1dimensional universe (like the "Lineland" that the characters in Flatland pay a visit to). So a similar idea is hypothesized in string theory to account for the fact that we only experience our space as 3dimensional even though the mathematics of string theory requires more spatial dimensionsthe extra dimensions are "curled up" into small shapes known as CalabiYau manifolds, which play a role analogous to the circular crosssections of the 2D cylinder or tube I described (although in brane theory, an extension of string theory, it's possible that one or more extra dimensions may be "large" and noncurled, but particles and forces except for gravity are confined to move in a 3dimensional "brane" sitting in this higherdimensional space, which is termed the "bulk").
Solution 2
The definition of dimension used here is that of a dimension of a manifold  essentially, how many coordinates (=real numbers) we need to describe the manifold (thought of as spacetime).
Manifolds may carry a notion of length, and one of volume. They may also be compact or noncompact, roughly^{1} corresponding to finite and infinite. E.g. a sphere of radius $R$ is compact and twodimensional  every point on it can be described by two angles, and it's volume is finite as $\frac{4}{3}\pi r^2$. Ordinary Euclidean space $\mathbb{R}^3$ is noncompact and threedimensional  every point in it is described by three real numbers (directed distance from an arbitrarily chosen origin), and you can't associate a finite volume to it.
Note that, on the sphere, you can keep increasing any one of the coordinates and, sooner or later, you will return to the point you started from. All dimensions here are "small"/compact. In Euclidean space, you never return to the origin, no matter how far you go. All dimensions are "big"/noncompact.
An infinitely long cylinder is now an example of where the two dimensions are different. Take as coordinates the obvious two  the length (how far "down"/"up" on the cylinder you are), and the angle (where on the circle that's at that length you are). The length dimension is noncompact  you never return to your starting point if you just keep increasing that coordinate. The angle coordinate is compact  you return after $2\pi$ to your starting point, and the "size" of the dimension is the radius of the circle. This is an example of a "curled up dimension". If you are far larger than the radius, you might not even notice you are on a cylinder, and instead think you are on a onedimensional line!
^{1The mathematical definition is by covering properties which are not as easily translated into intuition.}
Related videos on Youtube
Saffat Rafsan
Updated on February 06, 2020Comments

Saffat Rafsan almost 4 years
In the link below Carl Sagan described about higher dimension: http://www.youtube.com/watch?v=UnURElCzGc0
and here's a description of Brian Greene: http://www.youtube.com/watch?v=v95WjxpMIQg
Carl Sagan explains, we can not see the higher dimensions because we are limited to perceive only three dimensions. He didn't say a dimension can be small or big. This explanation completely makes sense.
But Brian Greene explains, higher dimensions can be tiny and curled up.
Isn't every dimension perpendicular to each of the other dimensions? If so, then how can a dimension be tiny or big? I want to know which is the right way to imagine higher dimensions?

Sean over 8 yearsExtra Reading/Possible Duplicates: physics.stackexchange.com/q/22542 physics.stackexchange.com/q/168071 physics.stackexchange.com/q/22238

Qmechanic over 8 years


Kyle Kanos over 8 yearsThat picture didn't work for me. I just kept picturing my library.

Hypnosifl over 8 yearsWhen people say time is the 4th dimension they are assuming there are only 3 spatial dimensions (and the dimensions don't have any intrinsic order, so you could just as easily say time is the 1st dimension and the next three are spatial dimensions, which is actually how 4vectors in relativity are usually written). But it's certainly mathematically possible to describe a universe with more than 3 spatial dimensionsin superstring theory there are 9 space dimensions and 1 time dimension (so you might say 'time is the 10th dimension' here), in Mtheory there are 10 space dimensions and 1 time.