visualization - Difficulty in visualizing more than three spatial dimension

Is there is any other way than abstract mathematics to visualize higher dimensions. Physicists working in high energy physics live in higher dimensions (pun intended), with their sophisticated mathematical tools they easily work there way out in higher dimensions. Take for a simple example of a $n$-sphere $S^N$ http://en.wikipedia.org/wiki/N-sphere.

• 0-sphere $S^0$ is a pair of points ${c − r, c + r}$, and is the boundary of a line segment (the 1-ball $B^1$).



• 1-sphere $S^1$ is a circle of radius $r$ centered at $c$, and is the boundary of a disk (the 2-ball $B^2$).


• 2-sphere $S^2$ is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (the 3-ball $B^3$).


• 3-sphere $S^3$ is a sphere in 4-dimensional Euclidean space.

The problem is how hard I think it is impossible for me to visualize a 3-sphere.