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One way I always liked to think of $S^n$ is in terms of suspensions. While not particularly geometrically enlightening (though it can be topologically enlightening), is still an interesting way to think of them.

Definition. The suspension $SX$ of a topological space $X$ is $(X\times[0,1])/\sim$ where $\sim$ collapses $X\times${0} to a point and $X\times${1} to a point. "Geometrically", this means we want to take $X$, and two "suspension" points, and then draw "lines" from the two points to all the points in $X$.

So we can imagine the suspension of the circle easily. $S^1\times[0,1]$ is the cylinder, and $S^1 \times${0} is the circle on the "bottom" of the cylinder, and $S^1\times${1} is the circle at the "top". We identify these circles with points, which collapses the top and bottom of the cylinder to points. This clearly gives us $S^2$!

Generally, it is not hard to show $SS^n = S^{n+1}$.

So, now let's try to imagine $S^3$, which is three dimensional, so shouldn't be too hard to think about. We start with the 2-sphere, considered embedded in $\mathbb{R}^3$, and two "suspension" points. But we can already tell this is going to look weird if we choose two points, say, above the north and south poles. So, let's pick one point inside the sphere.

The set of all lines from the point in the center of the sphere to the sphere is the solid sphere. Now, we want to deal with the second point. Say, we place this above the north pole, and connect each point on the sphere to it.

Since these lines are supposed to go to all points of the sphere, we should imagine this diagram shows the lines dense in the space around the sphere... but this is looking crowded, so let's move this extra point all the way off to infinity, and redraw this picture,

again imagining in this crudely drawn picture the lines cover the whole sphere.

But now these lines cover, in addition to the surface of the sphere and the point in the center, all of $\mathbb{R}^3$! And, all we've got left is the point at infinity.

So, we've just shown $S^3$ is $\mathbb{R}^3\cup$ {$\infty$}.

Now, how can we imagine $S^4$? We do the same thing again! Draw all lines (this time, since we can't quite imagine the bigger space to embed this in, we can instead think of formal linear combinations) from two points to every point in $\mathbb{R}^3$, and to the extra point at infinity.

I won't try to draw that one, but, thinking of it isn't too hard (although things get geometrically confusing if you try to do this process too many times!) But at the very least, you can convince yourself the spheres all have a relatively simple structure.

Generically, one can construct other spaces by suspensions, cones (suspensions over one point), joins (drawing lines between two arbitrary topological spaces), wedges ($\vee$) (quotient of disjoint unions), and smash products $X\times Y/X\vee Y$, which, with are simple enough that in some cases, with enough effort, one can use them to visualize what lots of types of higher dimensional spaces look like! For more, grab your favorite algebraic topology book.

If it helps make sense of this explanation, I'm a physicist ;).