24th December 2007 - 01:35 PM
To restart with a new idea:
imagine the smallest thing we can have is a straight , unbending infinitesimal piece of a line. It has a direction, but has no magnitude, and is continuous , what ever it may mean.
When looked upon from one end it is a point, a discrete thing. So this element contains both discrete and continuous elements, being an infinitesimal line and a point at the same time.
Now since it is infinitesimal, let us denote its lenght as dx. Then, as we know, a triangle made from it will have area proportional to dx^2, so in the scale of dx it will be not present- infinitely smaller. So in that scale triangle will have no area, but 1 scale lower, infinity lower, there will be an area but edges will be infinitely large.
If we construct a tetrahedron form dx, its volume will be proportional to dx^3 in that scale where dx is infinitesimal, so 0. Volume will not exist in this scale , but 2 scales lower it will be infinitesimal volume dx^3.
So we can see that in any given scale we ,starting to builkd things NOT from points , but infinitesimal vectors, directions, when reducing things to infinitely small, will be left with form only, but vanishing area and volume.
The question is:
Given the method of constructing things from infinitesimal line segments which has direction, is there any surface which will have:
a) area in the same scale as dx
volume in the same scale as dx, or at least just one, not 2 scales lower
What would be the finite ? area/volume of such a thing.
To give some idea, if we had 2 infinite lines with lenght sgrt(2) , the area of a square in the scale based on sgrt(2) would be (sgrt(2)^2)=2 or finite.
However, area of a circle built on radius sgrt(2) would be pi* r^2= 2* pi- infinite( in terms of numbers needed to describe it).
An Area of infinitesimal equilateral triangle built on from dx will be 0, of a cube =0, volume 0 as well for both.