What is the name of the geometric object that is the locus off all points in 3 dimensions called?

That's a lot of questions! Excellent! :)

First of all, I would call the set of all points in 3D space, well, space. I suppose we could come up with some other name for it, but space does the trick just fine. Perhaps the word fails to describe how we feel a set of points should be (the word "space" tends to imply emptiness), but that's just one of many imperfect words used to describe mathematical objects (I've always had trouble with the word "ring" in algebra theory). To help grasp the idea, perhaps it would be convenient to think of a worm in the ground. It can move in three basic directions, and the dirt, or "space" all around it is the set of points it can move to.

Moving to higher dimensions, we can rather simply call the set of all points in n dimensions n-space (or, granting the space coordinates, R^n, the set of all n-tuples of real numbers). As for hyperplanes, I'm unfamiliar with the terminology as well. Looking at wikipedia's article, I agree that the majority of the article isn't particularly enlightening as far as intuitive descriptions go. However, the second opening paragraph gives a bit of insight; the hyperplane for n-space is actually an (n-1)-space embedded into the n-space. That is, the hyperplane for 1-space (a line) is 0-dimensional (a point). The hyperplane for 2-space (a plane) is a line; the hyperplane for 3-space is the regular plane; and the hyperplane for 4-space (and who knows what that is) is the same as our 3-space. Some linear algebra might be helpful here, looking at bases and the like, although perhaps that's too technical and not as intuitive as we would like.

In a sense though, dimensionality is a bit like the degrees of freedom we have. On a line, we have one way to go: forward (or the negative of forward, which we might call "backward", but as far as measurements in the reals go, we don't need anything but a single number). In a plane, we have north and east (with their negatives, south and west respectively). In space, we get one more direction to travel, say out (and in). In 4-space, we have...well, not a good name, but you get the idea. As far as hyperplanes go, we pick out all but one of the directions that are available to us. Suppose we're considering the hyperplanes of 4-space, and say we want to have up, right, and out as our dimensions. Then we're left with one direction (the mysterious unnamed one) that we cannot move in. This is completely analogous to the lower dimensions, and provides us with a way to "split" the full space in half; each half is in either the positive or negative direction of our unavailable direction. [I should probably mention that the names of the directions are immaterial. In fact, they don't even really need to be perpendicular to each other, so long as no one of them can be described by some sequence of the others. If we don't bother giving a coordinate system to 3-space, you might pick three dimensions to call x, y, and z, but I may well choose three dimensions that are skewed at diagonals to your dimensions. So long as I can describe all of the points by three numbers that each represent one of my dimensions, the system is valid. In linear algebra, we call such a set of dimensions linearly independent, and a basis for the appropriate space.]

Oh, as for the solid sphere of infinite radius; it is actually the case that a plane can be considered a disc (that's what we would call the interior of a circle) of infinite radius, just as space is actually also an infinite-radius ball. Of course, that opens up the can of worms: space is also an infinite cube, a cylinder with both infinite radius and height, an infinitely large starfish, whatever we like. The more regular of these shapes, like the ball, cube, and cylinder, are the bases for our different coordinate systems (or, in the 2-d case, we have squares and discs making up the Euclidean and polar coordinate systems, respectively).

So, with all these viewpoints, which is the best? That depends on the situation. Perhaps sometimes it's best to think of any space as being a hyperplane for a larger space, although I can't imagine that's too frequent. Mostly I think it's easiest to imagine these spaces as independent of any ambient space, or embedded in 3-space if possible. As to your question of tractability of the plane as just a plane (not inside our familiar 3-space), I don't think it's too hard to imagine a 2-dimensional universe, although perhaps that's my over-familiarity with math. Picture some ridiculously tiny critter on your desktop, and then further imagine that the surface of the desk extends indefinitely. That poor little critter will never know that there's such a thing as "up" or "down"; just the direction taking him closer or further from you and to your left and right (of course he can't see anything but a line that is your frontside in the plane of the desk; otherwise he would be seeing you in the third dimension).

Perhaps if you can find it you might have a read of the book "Flatland" and its pseudo-sequels "Flatterland" and "Sphereland"...actually, wikipedia has a few others listed you might have a look at. I would recommend "Flatland" first. It won't cover all of your questions, but might help with picturing an independent 1-d and 2-d space. Besides that, it's a good read even without any mathematical questions in mind.

Whew, that turned into a bit of a rambling mess. Hope I've answered some of your questions decently.

Geometric Locus

The falowing are two dimensional objects A. triangle E. plane F. Square