Kaku's 'Hyperspace' is currently inspiring my higher-dimensional imaginings. I'm trying to imagine what one would see in a closed, unbounded hyperspherical universe. I know of the conceot that, if you look far enough, your image can wrap around the universe so you're looking at the back of your head. It works in 3D but just as well as in 2D, but I'll start with the ant-on-the-balloon. An ant lives on the surface of a balloon universe. His movement, rays of light and everything else are constrained to follow the curvature of its surface. If the ant looks ahead, he'll see an ant way in the distance, facing away from him - it is in fact, him. If he looks to his left, he'll see an ant way in the distance looking away - again, in fact, him. Right? Right. But here's my difficulty. No matter what direction he looks, he's going to see himself. If he turns his head by ever decreasing amounts, 1 degree, half a degree, one-zillionth of a degree he's going to see himself. You quickly realize that he's not going to see an entire copy of himself at every one of those angles, lined up like a chorus line. No, he's only going to see a piece of himself - the exact opposite piece from where he's looking. Imagine he looks directly ahead; in the exact center of his vision, he will see the exact rear tip of his tail. What will be 1/10th of a degree to the right of that point? Well, if we trace the light ray to is source we see that it is coming from a spot 1/10th of a degree to the left of the tip of his tail. What he sees 2/10th of a degree to the right of the tip of his tail, is the rear-end 2/10ths of a degree to the left of his tail. You can do this across his entire vision. The net effect is that his view will be that of one *single* giant ant filling his entire vision, no matter which way he looks. His entire universe will be a shell of an (albeit concave and reversed) ant. Are you still with me? Note that it is irrelevant how *big* his balloon universe is. If it is a foot in diameter or 20 yards in diameter, he will see the same thing - while the "ant shell" will be much farther away, it will be correspondingly larger, cancelling out. Even if his balloon is a mile in diameter, the image he sees is no larger or smaller than "filling his entire vision with a single ant image". You can verify all the above by drawing the diagram on a piece of paper. An aside: The only mitigating factor is intervening objects. Obviously, large objects would block the view of the ant's universe-edge. But additionlly, if the ant's balloon universe were not perfectly transparent (say it had an atmosphere, or interstellar dust and qas), then the ant would see a correspondingly faded edge-of-universe. If his universe were large enough, the edge of his universe be lost in the distance and he would not see it at all. Now let's look at what we would see in our 3D/4D universe if it really were closed on itself. The extrapolation here is that, if we really lived in a universe that were closed on itself in a 4th dimension, we would (barring blocking objects, gas and particles) see an infinitely large backdrop of - ourselves. It would not be too small to see due to distance, no - it would be scaled up by just the right amount, so that the entire boundary of our universe would look like our own body (or planet that we're standing on). This means that the ONLY thing preventing us from seeing the edge of our universe - and seeing our ourselves - is intervening gas and dust.