Boring you again; geometry.....

[Ben]

Some of you may find this thread unbelievably patronizing. If you do, I apologize. All of you will be wondering WTF has this to do with geometry.

 

Hmm, forgotten for now; but it will come to me.........

 

Second I shall be talking a lot about dimensions. In this context, I will say exactly what I mean, but will usually suppress the word "dimensional". Thus, I will say that an n-dimensional space is an n-space. OK with you?

 

Third, this is, as far as I can judge, the way mathematicians look at this subject. So, let's get to work.........

 

Consider a line of finite length, as lines are usually understood. We will assume that this line can be infinitely sub-divided. Then we will assume that each element in this this sub-division corresponds to a real number. Thus our line is a "segment" of the real line [math]R^1[/math]

 

We now ask, how many real numbers do I need to uniquely identify a point on this line? The answer is, of course 1. Let us call this line as 1-dimensional, by virtue of this fact. Let's take our line segment, and join it head-to-tail. We instantly recognize this as a circle.

 

Obviously, the same applies; any point on this circle can be uniquely described by a single real element, and accordingly I will call this geometric object as 1-dimensional.

 

Mathmen use the symbol [math]S^1[/math] for this character, and call it the 1-sphere. Now let's try and think about the "2-dimensional line", or 2-line. What can this mean (if anything)?

 

Well, using the above, we may assume that this is the "line" that requires two numbers to uniquely describe a point. From which we infer that the "2-line" is the plane. We may also infer, from the above, that the 2-sphere [math]S^2[/math] is, in some abstract sense, a head-to-tail "joining" of a part of this plane. The implications here are profound......

 

The 2-sphere is merely some sort of jazzed up plane, that is, it knows nothing about the area/volume it may or may not enclose. The same applies to any n-sphere.

 

With a grinding of gears, let's now consider the area enclosed by the 1-sphere as defined above. Intuition tells us, in this case quite correctly, that it is part of the "2-line" i.e the plane. This part of the plane is usually referred to as the "disk" [math]D^2[/math]. It is, in fact, the 2-ball.

 

Similarly, the "area" enclosed by the 2-sphere is a part of the 3-plane and so on. Obviously, this latter "area" is the volume enclosed by the 2-sphere, from which we conclude that, provided we are allowed to think of area as a 2-volume, then, as a generalization, the n-sphere encloses an n + 1 volume.

 

But the next thing we have to think about, as we're in the mood, is whether or not, for any "n-volume", I need to have an enclosing n - 1 space.

 

Well, it largely a matter of definition; as a geometric object, the n + 1 volume enclosed by the n-sphere may or may not include the n-sphere. If it does, one says that the n + 1 ball is closed. Otherwise, they say that the n + 1 ball is open.

 

Intuition says exactly this; a set is closed iff it includes its boundary, it is open otherwise (note bene; this is not the topologist's definition)

 

So the boundary of an n-ball [math]D^n[/math] is precisely the n - 1 sphere that encloses it.