Comments on: The Koch Curve and Visual Resolution http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/ Tue, 22 Jun 2010 06:21:15 +0000 hourly 1 http://wordpress.com/ By: Arunn http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6542 Arunn Mon, 26 Feb 2007 02:49:11 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6542 bharath: thanks for the informative comments. bharath: thanks for the informative comments.

]]> By: bharath http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6541 bharath Sat, 24 Feb 2007 23:47:38 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6541 nice interesting topic. To show area remains bounded it is sufficient to show the object at no point spills over the initial square drawn around it. This can be shown using 2 observations: 1. Each edge at any step of the way is aligned in one of the 3 directions. 2. Replacing the middle by an equilateral triangle does not spill beyond the square for which the initial edge is the diagonal. try it. the proof will need no computation. And it is *not* necessary to have fractals to get curves of infinite length enclosing finite area. If you take a square of length 1 on each side. pick the top side and introduce simple nested Cauchy intervals. From right end each interval, drop down by 0.5, go horizontally below the left end of the next interval and jump up 0.5 to the left of the next interval. Continue this to form a closed curve. The area will be bounded by 1 at all times and length will increase by 0.5(2)^k every step. which is faster increase in perimeter than Koch curve. nice interesting topic.

To show area remains bounded it is sufficient to show the object at no point spills over the initial square drawn around it.

This can be shown using 2 observations:
1. Each edge at any step of the way is aligned in one of the 3 directions.
2. Replacing the middle by an equilateral triangle does not spill beyond the square for which the initial edge is the diagonal.

try it. the proof will need no computation.

And it is *not* necessary to have fractals to get curves of infinite length enclosing finite area. If you take a square of length 1 on each side. pick the top side and introduce simple nested Cauchy intervals. From right end each interval, drop down by 0.5, go horizontally below the left end of the next interval and jump up 0.5 to the left of the next interval. Continue this to form a closed curve.

The area will be bounded by 1 at all times and length will increase by 0.5(2)^k every step. which is faster increase in perimeter than Koch curve.

]]> By: Math Carnival at Nonoscience http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6543 Math Carnival at Nonoscience Sat, 24 Feb 2007 04:55:32 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6543 [...] of mathematics has been published by Mark at Good Math Bad Math. A good collection for the weekend. The Koch Curve and Visual Resolution from this blog is also featured in the carnival.[...] [...] of mathematics has been published by Mark at Good Math Bad Math. A good collection for the weekend. The Koch Curve and Visual Resolution from this blog is also featured in the carnival.[...]

]]> By: Arunn http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6540 Arunn Sun, 19 Nov 2006 21:17:37 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6540 Kiranjeet: Good to know the post was helpful to you. Keep visiting and discussing. Kiranjeet: Good to know the post was helpful to you. Keep visiting and discussing.

]]> By: Kiranjeet http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6539 Kiranjeet Sun, 19 Nov 2006 21:13:09 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6539 Thank you, this is by far the most helpful website I have found to help me understand the Koch snowflake. Thank you, this is by far the most helpful website I have found to help me understand the Koch snowflake.

]]> By: Arunn http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6545 Arunn Wed, 30 Aug 2006 04:42:03 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6545 Scan Man: My work is as enjoyable as these squiggles and associated thoughts... eganesan: thanks for the link. Swarup: Thanks. Very encouraging to have you around commenting at my blog with your expertise. You are welcome to contribute more at this blog, from your rich experience and knowledge. I am not a trained mathematician like you, but certainly would love to know more new things in that subject, from willing and generous sources like you. Selvakumar: Yes, the nowhere differentiable tag is valid for the Koch curve only when n tends to infinity. Actually, only when n tends to infinity, it is a Koch curve and behaves as a fractal - exhibiting self similar structure. To answer your other question, we need to know a concept called the representative elemental volume (REV) for a porous medium. We will do it in a future post. Venkat: Thanks for your thoughts. Swarup has shared his thoughts on yours. Scan Man: My work is as enjoyable as these squiggles and associated thoughts…
eganesan: thanks for the link.
Swarup: Thanks. Very encouraging to have you around commenting at my blog with your expertise. You are welcome to contribute more at this blog, from your rich experience and knowledge. I am not a trained mathematician like you, but certainly would love to know more new things in that subject, from willing and generous sources like you.
Selvakumar: Yes, the nowhere differentiable tag is valid for the Koch curve only when n tends to infinity. Actually, only when n tends to infinity, it is a Koch curve and behaves as a fractal – exhibiting self similar structure.
To answer your other question, we need to know a concept called the representative elemental volume (REV) for a porous medium. We will do it in a future post.
Venkat: Thanks for your thoughts. Swarup has shared his thoughts on yours.

]]> By: scan man http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6548 scan man Tue, 29 Aug 2006 17:40:56 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6548 'simple algebra' indeed!!! All I can say is that the last figure looks beautiful :) Can't say the same about the squiggles and the numbers in the equation though :( If you do this in your spare time, I can't imagine what your work is like!! ‘simple algebra’ indeed!!!

All I can say is that the last figure looks beautiful :)

Can’t say the same about the squiggles and the numbers in the equation though :(

If you do this in your spare time, I can’t imagine what your work is like!!

]]> By: eganesan http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6547 eganesan Mon, 28 Aug 2006 12:29:52 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6547 Added a link to Nonoscience. Added a link to Nonoscience.

]]> By: Selvakumar.A http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6546 Selvakumar.A Mon, 28 Aug 2006 07:39:56 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6546 Does it mean that while doing an analysis, We must take a resolution level corresponding to an area 1.6 times the actual value within the specified limits, (only) then the results will be comparable with experimental values ? Does it mean that while doing an analysis, We must take a resolution level corresponding to an area 1.6 times the actual value within the specified limits, (only) then the results will be comparable with experimental values ?

]]> By: gaddeswarup http://unrulednotebook.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6550 gaddeswarup Mon, 28 Aug 2006 07:33:55 +0000 http://unrulednotebook.wordpress.com/2006/08/28/the-koch-curve-and-visual-resolution/#comment-6550 A simple closed curve is just the homeomorphic image ( in this case continuous, one-to-one) image of a circle. It need not be rectifiable. Jordan-Brouwer Theorem is valid with that definition. The proofs are still difficult and the modern ones use some algebraic invariants like fundamental group or homology ( I do not know Brouwer's proof) and are based on Alexander's arguments. See Munkres's Topology or Armstrong's "Basic Topology" both available in Indian editions. A simple closed curve is just the homeomorphic image ( in this case continuous, one-to-one) image of a circle. It need not be rectifiable. Jordan-Brouwer Theorem is valid with that definition.
The proofs are still difficult and the modern ones use some algebraic invariants like fundamental group or homology ( I do not know Brouwer’s proof) and are based on Alexander’s arguments. See Munkres’s Topology or Armstrong’s “Basic Topology” both available in Indian editions.

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