Sierpinski gasket

[ughaibu]

Is it possible to argue that the Sierpinski gasket has an area greater than zero by using the non-standard interval?

[Qfwfq]

the non-standard interval?

Which one?

 

If by interval you mean a measure of 1-D objects and by area a measure of 2-D ones, I doubt it. You are talking about a fractal object having somewhere between 1 and 2 dimensions. It is in fact described as a curve that is self intersecting at every point, so its length is "more than infinite".

[ughaibu]

Okay, thanks. I had in mind that despite being infinitely small, the area might have some kind of hyper-real residue (I'll investigate the nature of hyper-reals further before pestering you again).

I notice you describe the Sierpinski gasket as a curve, would you say it's misleading to describe this curve as the "outline" or "circumference" as there is no area to have such a boundary? It seems to me that effectively both the area and it's outline have reduced to points and are no longer distinguishable from each other, does this sound to you like a valid way to conceive the state of being between one and two dimensions?

[Qfwfq]

The outline doesn't get reduced to a point, each recursion adds outline and removes none, only area. More and more outline, less and less area.

[ughaibu]

Yes, I was mixing up two ideas. I'll get back to you when I'm less tired.

[ughaibu]

Qfwfq: I understand how the thing mathematically has zero area and an infinite outline, and I'm quite happy with that as it stands, but I'm also considering the thing conceptually as a supertask in which one starts with an equilateral triangle and the actions are to remove the central triangle from any surviving occupied triangles. In order to increase the outline we need to remove the area represented by the central triangle, simply joining the mid-points of the sides doesn't change the length of the outline, this confirms the definition of "outline" as the border of area and demonstrates the interdependence of the two concepts, area and outline. In the maths, these two concepts are treated as if independent, which, in the supertask environment, is fine while the task is ongoing but not applicable to the end-state. In short, it can be argued that as there's no area, there's no outline, despite the outline having been infinitely long instantaneously previous to the completion of the task. Anyway, none of this is maths.

What I meant about the points is: basically, in this supertask, one has an increasing number of triangles of decreasing area. Eventually, when these triangles have no area, they will have either no outline or an outline of zero length, effectively they will be points. I find this quite interesting as the connection between these points allows them to have a total length that is "more than infinite", despite them having no individual length. Of course there's nothing unusual about this, any curve has an infinite number of points of zero length arrayed along it, what interests me is that the points historically had area to the same extent that they had length.

[Qfwfq]

Uhm, for each single triangle at the nth iteration, the perimeter is smaller. But the length of the whole entire curve includes that of the lines added at each step, multiplying by 3/2. The area is multiplied instead by 3/4 at each step.

[ughaibu]

As I say, I'm happy with the mathematicians' view of it but that doesn't preclude other views.