Area as an infinite sum of points

So here’s where I started.

If I draw two different size circles on a piece of paper, I can come up with a mapping function that says thay any point I pick in the small circle has an analogous point in the small circle.

P[sub]1/sub~P[sub]2/sub

Any point in circle 1 has a unique matching point in circle 2.

So, could I then say that the number of points in circle 1 is the same as the number of points in circle 2? The number is, of couse, technically infinite but seems to be the same infinity (slippery I know - this may be where I’m botching the idea).

Now, if I then say that the area of circle one is A[sub]1[/sub] and that this area is the infinite sum of the points within it, is the area of circle 2 also A[sub]1[/sub] since I have the same number of points in circle 2? Calculus teaches me that the sum of an infinite number of things can be finite so it seems reasonable.

Where am I going wrong here? Obviously the area of two different circles is unequal but my “proof” above makes it seem that it’s possible to say they can be equal.

If I then make the large circle increasing larger, I can contain the universe in a teaspoon by saying their areas are the same.

Yes. In cardinal set theory, this number is called Aleph-One.

Here’s where you’re going wrong. Where did you get the idea that you can think of an area as the sum of points?

It should also be noted (and this may even be related to the problem here) that in calculus/analysis, whenever you take an infinite sum, you always take the sum of Aleph-Zero terms, not Aleph-One terms. This may be a little opaque, but what it means is if you took all the points in circle 1 and tried to list them in an infinitely tall column so you could add them up, you couldn’t.

The thing is, points are zero-dimensional objects, so they have an area of zero. While it’s true that a circle has an infinite number of points in it, adding up the areas of all those points isn’t going to tell you the area, since you are just adding a bunch of zeros. You have to get the area by the familiar pi*r[sup]2[/sup], or by taking the limit of the areas of inscribed polygons. Basically, you can work with finite quantities and show that there are infinities lurking about, but if you try to tackle the infinities directly you aren’t likely to get anywhere.

It turns out that “area” is a suprisingly tricky word to define mathematically, and you just now discovered one of the reasons why. The standard mathematical definition of area involves something called “Lesbesgue measure”, and one of it’s properties is that the area of the sum is not always equal to the sum of the areas. This is especially true when there are an uncountably infinite number of terms in the sum, an in your example.

It was the inscribed polygons that lead me to this idea.

If I start filling my circle with little squares with sides equal to “X”, then as X approaches zero, their sum appraches pi*r[sup]2[/sup].

As X approaches zero, the square’s area approaches zero. So at the point where their sum is actually pi*r[sup]2[/sup], then X[sup]2[/sup]=0 and it’s, nearly by definition, a point. So, the area of a circle can be expressed as an infinite sum of points.

But only under the continuum hypothesis. Without the continuum hypothesis, this cardinality might be something else entirely.

Pardon me for demonstrating my absolute ignorance of these issues (all but a vague memory)…
Is this equivalent to saying that the points in a 2-d surface cannot be mapped to the points on a line?

Wasn’t there some trick where you could give points x-y coordinates and then interleave the digits, always using odd digits for X and even digits for Y (or vice-versa)?

I thought that to get to the next level beyond “points in a line” (Aleph zero?), one had to enumerate the number of possible curves in a plane.

Again, please correct my blunders.

No it’s not equivalent. A mapping between two sets is not the same as taking a infinite sum.

And besides, it is possible to map the points on a 2-d surface to the points on a line.

Cabbage: You’re right, sorry. It doesn’t change the rest of my post, right? Just change “Aleph-One” to “Continuum”.

minor7flat5: “Is this equivalent to saying that the points in a 2-d surface cannot be mapped to the points on a line?”
No, the number of points in a line is also Continuum. However, the points in a line cannot be mapped to the items in a list.

The problem is that a point is not just a polygon of area zero. And even if it were, the limit of something is not always the same as the value at that point. So for instance, the limit of f(x) as x approaches 0 is not necessarily the same thing as f(0).

A line and a plane have the same number of points. Furthermore, the number of curves in the plane (i.e., graphs of continuous functions from R to R) is the same as the number of points on a line.

Under GCH, aleph-2 = 2[sup]aleph-1[/sub].

Orbifold
Member

Registered: Oct 2000
Location: Princeton, NJ, USA

Cool! I could probably just walk across the street and ask you to draw me a picture :).

“Approaches”.
“Approaches”.
“Approaches”.

Not “reaches”, but “approaches”.

Very different.

At what point does a curve cross its asymptote? Never, it infinitely approaches it.

At what point does the area of an inscribed polygon become zero? Never, it infinitely approaches it.

Divide 1 by 2. Repeat. Stop only when your result is equal to zero, not equal to zero within so many decimal places but completely equal to zero.

I understand that we’re dealing with limits & asymptopes and such, but, like I said before, calculus allows that an infinite sum of something can have a finite result.

If it didn’t, all the inscribed polygons in the world would never allow a circle’s area to be pir[sup]2[/sup]. It’d be almost-but-not-quite-but-really-really-close-to pir[sup]2[/sup].

I can write:
pi=4*(sum (-1[sup]n[/sup]/(2n+1)))
as n from 0 to infinity (Leibniz series)

and be writing something true, because while the sum approaches pi for n less than infinity, it is pi for n=inifinity.

Can’t I then say that if my inscribed boxes of size X[sup]2[/sup] are nearly zero as area approaches pir[sup]2[/sup] (ie: X approaches zero), then they are actually of size zero when area is pir[sup]2[/sup]? Can’t I then say that the area is the sum of zero-sized polygons?

Calculus allows that a countably infinite sum can have a value. You’re looking at a sum of uncountably many points. To do that, you need to use integration.

Furthermore, “the sum…is pi for n=inifinity” is false. Go get an analysis textbook and look up summation. There’s a subtlety you’re missing that may help you understand.

Off-topic: If you want a picture, minor7flat5, you can save yourself the walk and look here. The mapping is something called the “Peano curve” (technically, it’s defined as a map from the line to the plane, not vice-versa, but it can be shown that it’s a bijective map.)

Gah! That’ll teach me not to preview my posts. Here’s a fixed link.

You might want to wait a week or two, as I’m currently on vacation and am posting from Toronto :slight_smile:

Interesting responses. A whole different tack from my approach to such questions which is “Look at the units.” A point is not a unit of area. (This has nothing to do with the area of a point.)

The OP’s question is little different than “How many meters in a second?”

(Note: this is also my official answer to the dreaded sum equals 1 question.)

I don’t understand. Integration is a part of calculus. And every sum used in integration is a finite sum. There are no actual infinities in calculus. Infinity in calculus is a formal symbol – it does not represent an actual infinity. If you look to the definition of the definite integral, you will see that there are only finite sums; you can get a finite sum as close as you want to the limit by making the norm of the net smaller than some value, and I think it helps to think of the integral as the sum of an infinite number of infinitesimals, but you are never adding an actual infinite number of terms.