Length of a segment with infinitesimal spikes

[apoorv]

This is a variant of a fractal. Suppose we have a ‘smooth’ segment of length 1.
Divide it into n parts, replacing alternate parts by a hat of length m/n.
_/\_/\_/\_/\_ . The length of the curve is now (1/2+m/n*n/2) = (m+1)/2.
We let n increase without limit. The height of each spike now goes to 0,
but the length of the spiked segment remains (m+1)/2. But conceptually, a segment with infinitesimal spikes (of height 0) should be indistinguishable from a smooth segment.
Further arguing in the same manner for segments of lengths ½, ¼, and so on with values of m selected suitably (m=3,7,15 etc), we can get rough segments (with infinitesimal spikes) each of length 1.
So, is there some problem with the limiting process?
Why is a 'segment of length zero' sometimes a ponit and sometimes nothing?
-apoorv

[Ross A. Finlayson]

That seems a reasonable shape to use as for example in the Dirac delta
is often used a radial basis function, or here a radial basis function
would have its concomitant form.

Then it could be any fractal bounded to the partition there, as it
were. Here the general case is considered with measuring the
shoreline. Here the features would match their scales.

So, in the asymptotic it's unbounded but in the limit it's one.

What of it?

Regards,

Ross Finlayson

[FredJeffries]

On May 23, 11:35 am, apoorv <sudhir...@hotmail.com> wrote:

> This is a variant of a fractal. Suppose we have a ‘smooth’ segment of length 1.
> Divide it into n parts, replacing alternate parts by a hat of length m/n.
> _/\_/\_/\_/\_ . The length of the curve is now (1/2+m/n*n/2) = (m+1)/2.
> We let n increase without limit. The height of each spike now goes to 0,
>  but the length of the spiked segment remains (m+1)/2. But conceptually, a segment with infinitesimal spikes (of height 0) should be indistinguishable from a smooth segment.
> Further arguing in the same manner for segments of lengths ½, ¼, and so on with values of m selected suitably (m=3,7,15 etc), we can get rough segments (with infinitesimal spikes) each of length 1.
> So, is there some problem with the limiting process?

Convergence in length is a very strong notion of convergence. See
Miriam C. Ayer and Tibor Rado "A note on convergence in length", Bull.
Amer. Math. Soc. Volume 54, Number 6 (1948), 533-539

http://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09034-6/
http://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09034-6/S0002-9904-1948-09034-6.pdf

[apoorv]

Thanks for that.So the length depends on the sequence of curves used to approximate the segment,though this is not usually recognised when we speak of length -presumably a specific method of arriving at the limit is implicit.The notion of a point as a segment of length zero does seem to have some issues.
Consider another situation. The interval [0,1] is subdivided into 2n equal sub segments
[0,1]=Union [0,1/2n),[1/2n,2/2n), ...[2n-1/2n,1]
Also let A =[0,1/2n)U(2/2n,3/2n] ... [2n-1/2n](i.e the union of alternate subsegments)
and B be [0,1]-A.
Can we take the limit as n goes to infinity and if so, what are the sets A and B
and does the point 1/2 belong to A or to B?If we cannot,as it appears,are there some specific conditions when such a limit cannot be taken?
-apoorv