S3 is the boundary of a four dimensional ball. S3 is a three dimensional space and so it should be possible to imagine yourself exploring S3. Here is my attempt. It is presumably the shape of the universe if one postulates a positive curvature.

I highly recommend this program which is much newer than my text below. Start the program and open Spherical/MirroredTetrahedron.gen. It is sort of the interactive movie version of my story below.

I wake up in free fall. (OK, its called micro gravity now.) I am in a room with a good deal of fog. I am in the corner of the room where three walls meet at right angles. The walls seem flat and indeed I find about a flat, thin, inflexible plate, either side of which fits snugly against the wall. I push myself along one of the room edges where two of the wall meet. Soon I come to another corner of the room. I jump to the (incorrect) conclusion that the room is a cube. Indeed I find only flat walls meeting at right angles and corners where three such walls meet.

I find a marker with which I can leave graffiti. I label each corner indicating which direction I leave and keep notes in my notebook. To make a long story short it turns out that the room has just four corners rather than the eight corners that a cubical room would have. With the corners of the room labeled and my notes I conclude that each room corner is connected to each other corner. The room is connected like a tetrahedron. This despite the walls being (locally) flat and the corners being right angles!

Gradually the fog clears and I can see the whole room. From any corner the room appears to have three flat walls, meeting at my corner. The other wall always appears to be curved like an octant of a sphere, but whenever I go to that curved wall with my plate, it turns out to be flat after all and now the distant walls appear curved.

On one of the walls I find a slit thru which I can pass. I find an identical room. These rooms now share a wall with the slit. I find that the slit propagates and soon the slitted wall has collapsed. I crumple it up and stuff it in my backpack. Where the recent wall met the other walls there is only a seam on the wall that is now twice as big.

The new room is twice as large as either room before. The seam where the recent wall was attached now divides each of the remaining three(!) walls in half. Where the older smaller room had triangular walls the new larger room has three walls each with a fading seam that bisects them. Each of the new walls meet two corners. In fact the new larger room has just two(!) corners, both of which meet each of the three walls. It seems from the center of the room that it must be shaped a bit like a banana. No matter where I go in the room the two corners are in opposite directions from me! Bizarre.

Near a seam I make a slit in another wall. On the other side is a room of the original size. I discover that the seam near which I made the slit is opposite a wall of the new room. I remove that triangular wall as I removed the first wall. At this point the two rooms look just alike. Each has three walls. Each wall is bounded by a right angle room edge where it meets another wall. Each of these three edges terminate in the two ends of the banana.

Now I remove the wall with the second slit. That wall had, of course been separating the two big banana shaped rooms. I detach it from the two walls that it meets. Now I have a room that is four times as big as the original. This room has just two walls and no corners. The two walls meet at right angles. Each of the two walls is circular; they meet like the two crusts of a pie; I am between them; they are flat; yet I am a good distance from either! Each wall retains two seams in the form of a circular diameter. The seams meet at right angles. They are where the old removed walls were attached.

Once more I slit a wall P, near a seam. On the other side I find a room of the original size. I repeat the wall removals on the other size and now there are two pie shaped rooms. With trepidation I remove wall P. Now, of course there is just one wall and no corners or edges. The remaining wall is the equator and divides S3 into two equal pieces. The equator is continuous. I slit the equator and pass thru. I remove the walls that divided the eight rooms on the other side of the equator. I leave the equator intact. I am chicken. I suppose that if I removed the equator I would just see myself off in the moderate distance, like Mr. Tompkins.