Strange objcet...
Daniel Janzon
0 iterations: ______
1 iteration: __ / \ __
Etc. Also, let K be defined as K(n) when n -> Infinity.
Then Length(K(n)) -> Infinity as n -> Infinity.With length I mean ordinary
euclidian length. Now suppose that I keep Length(K(n)) fixed to the value of
1. So after each iteration, the length is still 1 and K shrinks a little bit
(75 % if I'm guessing right). As n -> Infinity, K(n) seems to approach a
point.
Here is my reasoning: What do I mean when I say that a point belongs to K?
Well, if a point belongs to K(n) for an arbitrarily large n, then it also
belongs to K. But as K shrinks with each iteration, I guess only a point
will be left in the end. But the length of a point is not 1.
What's a good solution to this problem? Is it satisfactory to say that the
length of a point isn't defined so there's no problem? I think not. (Am I
wrong?) Is it possible to conclude that we don't want points with length
1, thus it wasn't possible to keep the length of K fixed at 1?
Any help is appreciated, Daniel Janzon
Joona I Palaste
> 0 iterations: ______
Warning: I know only very little about fractal geometry.
I guess the problem here is that K(inf) is not a normal Euclidean
shape, because it consists of infinitely many line segments.
For any finite n, K(n) is bigger than a point, if its length is
greater than 1. This is because all K(n) where n is finite are normal
Euclidean shapes.
If you zoom a K(n) far enough you eventually get to a single line
segment. But if you zoom K(inf) you never get to a single line segment,
new ones always appear in the middle.
The length, area, volume, etc. of a point are all defined to be
exactly 0, in Euclidean geometry. But since K(inf) is not Euclidean we
have to invent new definitions, and those are unfortunately beyond my
comprehension (at least for now).
--
/-- Joona Palaste (pal...@cc.helsinki.fi) ---------------------------\
| Kingpriest of "The Flying Lemon Tree" G++ FR FW+ M- #108 D+ ADA N+++|
| http://www.helsinki.fi/~palaste W++ B OP+ |
\----------------------------------------- Finland rules! ------------/
"The truth is out there, man! Way out there!"
- Professor Ashfield
David C. Ullrich
<bert...@hotmail.com> wrote:
>Consider a Koch fractal K(n), where n is the number of iterations:
>
>0 iterations: ______
>
>1 iteration: __ / \ __
>
>Etc. Also, let K be defined as K(n) when n -> Infinity.
>
>Then Length(K(n)) -> Infinity as n -> Infinity.With length I mean ordinary
>euclidian length. Now suppose that I keep Length(K(n)) fixed to the value of
>1.
Meaning you perform the iteration and then rescale the result so the
length is 1?
>So after each iteration, the length is still 1 and K shrinks a little bit
>(75 % if I'm guessing right). As n -> Infinity, K(n) seems to approach a
>point.
That's correct.
>Here is my reasoning: What do I mean when I say that a point belongs to K?
>Well, if a point belongs to K(n) for an arbitrarily large n, then it also
>belongs to K. But as K shrinks with each iteration, I guess only a point
>will be left in the end. But the length of a point is not 1.
>
>What's a good solution to this problem?
What problem?
>Is it satisfactory to say that the
>length of a point isn't defined so there's no problem? I think not. (Am I
>wrong?) Is it possible to conclude that we don't want points with length
>1, thus it wasn't possible to keep the length of K fixed at 1?
Oh, the problem that a bunch of curves of length 1 converge to a
point, which does not have length 1. That's not a problem - there's
simply no reason the length of the limiting curve _should_ be the
limit of the lengths of the curves approaching it. You've just given
a simple example showing this.
>Any help is appreciated, Daniel Janzon
>
David C. Ullrich
Dave L. Renfro
[sci.math Sep 29 2002 11:56:41:000AM]
http://mathforum.org/discuss/sci.math/m/444784/444784
wrote
This has to do with the non-continuity of arc length (relative to
"convergence of curves"). Arc length is lower semicontinuous, though.
If curves C_n approach a curve C and limit(n --> oo) of length(C_n)
exists and equals L (which could be oo), then L is greater than
or equal to length(C). For an example where inequality occurs,
besides what you've given, consider the hypotenuse of a right
triangle each of whose legs has length 1. The hypotenuse has
length sqrt(2), but it is easy to find stair-case curves having
length 2 that are arbitrarily close to the diagonal.
Incidentally, the *upper* semicontinuity of the length of timelike
curves plays an important role in general relativity. See Section 6.7
"The existence of geodesics" (pp. 213-214) in "The Large Scale
Structure of Space-Time" by Stephen Hawking and George Ellis. You
can also find this as Proposition 2.2 on page 2064 of Gregory J.
Galloway and Arnaldo Horta, "Regularity of Lorentzian Busemann
functions", Trans. Amer. Math. Soc. 348 (1996), 2063-2084 --->>>
http://www.ams.org/tran/1996-348-05/S0002-9947-96-01587-5/S0002-9947-96-01587-5.pdf
For more about the lower semicontinuity of arc length, see --->>>
sci.math.research ("Lower semicontinuity of length measures?")
http://groups.google.com/groups?th=27c90db27adc97fa
50 posts in which the phrase "lower semicontinuous" appears
http://groups.google.com/groups?as_epq=lower%20semicontinuous
3750 hits google-hits for the phrase "lower semicontinuous"
http://www.google.com/search?as_epq=lower+semicontinuous
975 of the 3750 hits above include the word "length"
http://www.google.com/search?as_q=length&as_epq=lower+semicontinuous
Dave L. Renfro
Dave Rusin
In article <T9Fl9.1305$hT3....@nntpserver.swip.net>,
Daniel Janzon <bert...@hotmail.com> wrote:
>Consider a Koch fractal K(n), where n is the number of iterations:
>
>0 iterations: ______
>
>1 iteration: __ / \ __
>
>Etc. Also, let K be defined as K(n) when n -> Infinity.
>
>Then Length(K(n)) -> Infinity as n -> Infinity.
Today in seminar we were discussing the number N(r) of lattice points in
Z^2 which lie inside the circle of radius r. Obviously the number
is approximately pi r^2. Trickier is the estimate of | N(r) - pi r^2 |;
a bound of the form O(r^(2/3)) is possible, but a simpler estimate
O(r) seems clear because the length of the bounding circle is O(r).
So now consider a Koch snowflake K, scaled to bound a region of area 1.
If N(r) is the number of lattice points inside the scaled figure r K,
what bounds are possible for | N(r) - r^2 | ? Obviously an appeal to
the length of the boundary is no longer possible.
dave
Robert Israel
Dave Rusin <ru...@vesuvius.math.niu.edu> wrote:
>Today in seminar we were discussing the number N(r) of lattice points in
>Z^2 which lie inside the circle of radius r. Obviously the number
>is approximately pi r^2. Trickier is the estimate of | N(r) - pi r^2 |;
>a bound of the form O(r^(2/3)) is possible, but a simpler estimate
>O(r) seems clear because the length of the bounding circle is O(r).
>So now consider a Koch snowflake K, scaled to bound a region of area 1.
>If N(r) is the number of lattice points inside the scaled figure r K,
>what bounds are possible for | N(r) - r^2 | ? Obviously an appeal to
>the length of the boundary is no longer possible.
If the snowflake has finite d-dimensional Hausdorff measure (where
1 <= d < 2), then the region within distance 1 of the boundary of r K
should be contained in the union of O(r^d) unit squares, so you should get
a bound O(r^d).
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2