Actually, the most counterintuitive is 4-dimensional space. It is rather mathematically unique, often exhibiting properties no other dimension does.
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One of the most surprising is that all smooth manifolds of dimension not equal to four only have a finite number of unique smooth structures. For dimension four, there are countably infinite number of unique smooth structures. It's the only dimension with that property.
Can you give some intuition on smooth structure and manifold? I read Wikipedia articles a few times but still can't grasp them.
Most of calculus and undergraduate math, engineering, and physics takes place in Euclidean space R^n. So all the curves and surfaces directly embed into R^n, usually where n = 2 or n = 3. However, there are more abstract spaces that one would like to study and those are manifolds. To do calculus on them, they need to be smooth manifolds. A smooth structure is a collection of "patches" (normally called charts) such that each patch (chart) is homeomorphic (topologically equivalent) to an open set in R^n. Such a manifold is called an n-dimensional manifold. The smoothness criterion is a technicality such that the coordinates and transformation coordinates are smooth, i.e., infinitely differentiable. Smooth manifolds is basically the extension of calculus to more general and abstract dimensions.
For example, a circle is a 1-dimensional manifold since it locally looks like a line segment. A sphere (the shell of the sphere) is a 2-dimensional manifold because it locally looks like an open subset of R^2, i.e., it locally looks like a two dimensional plane. Take Earth for example. Locally, a Euclidean x-y coordinate system works well.
Euclidean space is a vector space and therefore pretty easy to work with in computations (especially calculus) compared to something like the surface of a sphere, but the sphere doesn't simply abandon Euclidean vector structure. We can take halves of the sphere and "flatten them out," so instead of working with the sphere we can work with two planes, keeping in mind that the flattening functions define the boundary of those planes we're allowed to work within. Then we can do computations on the plane and "unflatten" them to get the result of those computations on the sphere.
Manifolds are a generalization of this idea: you have a complicated topological structure S, but also some open subsets of S, S_i, which partition S, and smooth, invertible functions f_i: S_i -> R^n that tell you how to treat elements of S locally as if they were vectors in Euclidean space (and since the functions are invertible, it tells you how to map the vectors back to S, which is what you want).
The manifold is a pair, the space S and the smooth functions f_i. The smoothness is important because ultimately we are interested in doing calculus on S, so if the mapping functions have "sharp edges" then we're introducing sharp edges into S that are entirely a result of the mapping and not S's own geometry.