How to Visualize points on a high dimensional (>3) Manifold?

Are there any ways to visualize(plot/draw) points on a high dimensional (ex: dimension = 5) manifold?

A usual way to work is to take slices, CAT-scan style.

Have you seen a CAT-scan? It represents a 3-dimensional object as sequence of two-dimensional slices of the object, in a way that you can choose which directions your slices cut (after computer processing).

There are two usual ways of representing a 3-dimensional object on a plane. You can take projections (think a picture of the "outside" of the object) or slices. Similarly, to represent higher ($N$) dimensional objects on a plane (draw it on a piece of paper), you must throw away $N-2$ degrees of freedom. Each degree of freedom you can choose to throw away by projection or by slicing. So a typical way of visualizing a 4-dimensional object is to take a sequence of 3-dimensional slices, and project each slice onto a computer screen, and display this sequence one-by-one over time. This way you get the illusion of a 4 dimensional object "passing through" our three dimensional world.

Higher dimensional objects can be done similarly, you just need to take more slices.

One fun thing you can do with this is to try to play 4-dimensional tic-tac-toe. To make the game not a trivial win for the first player, you need a $5\times 5\times 5 \times 5$ grid, and first to make 5 in a row wins. So on paper you need to draw a $5\times 5$ matrix of $5\times 5$ grids, each grid represent the intersection of two orthogonal hyper-slices of the full 4-dimensional game grid.

Another fun way to develop some intuition on higher dimension spaces is to spend some time playing Magic Cube 4D, a higher dimensional analogue of Rubik's cube. (It could also get you dizzy pretty quickly).

Well, here's how I think about it.

Every time you add a dimension, you need an infinite number of the previous dimension.

1D is the x-axis. You need a line to visualize.

1D -> 2D means you need an infinite number of additional x-axes (call them y-axes). You need a plane to visualize (flatland picture).

2D -> 3D means you need an infinite number of additional xy planes. You need a space to visualize (3d picture).

3D -> 4D means you need an infinite number of additional xyz spaces. You need a movie to visualize.

4D -> 5D means you need infinitely many vhs cassettes, each with a movie on it to visualize.

So buy some VHS! They are cheap now.

(really, like Willie Wong said, to take a "slice" of the 5d space, by fixing one of the variables. Then, you can create a "movie" of the other 4 variables (use 3 for space, and 1 for time). to visualize the whole space though, you would need an infinite number of these "movies" (and would each be infinite in duration), for each value you want to the 5th variable to).