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 abelian
Joined: 1/12/2008
Msg: 26
How long is the coastline of Britain?Page 2 of 2    (1, 2)

Nearly 70 people have viewed this thread so far, but only 4 have posted. What's going on? Is everybody expecting a movie or something?

Well, since I have a copy of ``The Fractal Geometry of Nature,'' Mandelbrot, Benoit, as well as textbooks which treat chaotic systems, and textbooks on information theory, so I already know something about the subject and don't feel like debating the validity of comments such as:


at any rate it might occur to you to wonder if fractals mimic nature or if nature is based on fractals (there's a topic for a rousing debate and your opinion is as valid as any mathematician's; they fight over it all the time!)

I've already spent time wondering about that and I'm quite certain that your opinion (or mine) is not as valid as any mathematician's. Would you say that the opinion of some random person on the internet about brain surgery is as valid as any brain surgeon's if you were going to have brain surgery? Would you be more inclined to do so if brain surgeons fought over various techniqes to perform the operation? fI can't see much coming out of thread based on such a premise.
 JustDukky
Joined: 7/8/2004
Msg: 27
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How long is the coastline of Britain?
Posted: 4/11/2009 9:08:58 AM
I think I said I wasn't going to post in this thread anymore, so I guess this makes me kind of a liar, but I felt compelled to respond to abelian's post.


I'm quite certain that your opinion (or mine) is not as valid as any mathematician's.


Comments like that are designed to discourage people. Where would we be if (for example) Srinivasa Ramanujan heard it, believed, it and never contacted Godfrey Hardy?

There is an ongoing philosophical debate in the mathematical community between the "Platonists" and the "Formalists" (further complicated by the "Constructivists". How is a "certified" mathematician (whose field might easily be applied mathematics) any more qualified to have an opinion on the philosophy of the discipline? How is "taking a side" with one's opinion {easily backed up using argument by authority (yes I know it's a fallacy) with the opinions of many highly revered mathematicians} an invalid stance so long as one can argue the case and perhaps learn in the doing? If one hasn't written a dissertation on the existence of non-constructable sets, how does that disqualify him from discussion? Is a physicist's opinion not valid because he's not a mathematician? Tell that to Ed Witten; I'm sure he'd be amused.

Quit trying to turn mathematicians into some kind of high priests of an obscure religion that nobody else could understand. I would not discount someone's opinion because he didn't write a dissertation on it. On a good day, I might punch a hole or two in his argument, but that's what it's all about anyway.

You may not realize it, but your stance is the logical equivalent of saying that only party members are competent to vote in elections, or that only high priests should talk about religion, because the rank & file don't have anything to contribute to either politics or religion and their opinions don't matter. I would say that such an attitude is not only elitist, but downright insulting to the common man.


Would you say that the opinion of some random person on the internet about brain surgery is as valid as any brain surgeon's if you were going to have brain surgery?


If he took the same stance as many neurosurgeons, yes. I would look into his arguments.


I can't see much coming out of thread based on such a premise.


The premise of the thread is that fractals are beautiful. Does one have to be a mathematician to appreciate beauty?
 onlines
Joined: 4/6/2009
Msg: 28
The beauty of math known as fractals
Posted: 4/11/2009 6:59:25 PM
Benoît Mandelbrot is one of my favorite mathematicians. His theory, though insanely simple and by definition -- truly infinite, can be applied to anything you see in the world of pattern today. A writer, Nassim Nicholas Taleb who is an understudy of Mandelbrot, recently wrote a book titled, "The Black Swan" which discusses the chaos theory and fractal formation in the financial system, and our every day governing life.

There is a video on video.google.com which goes in great lengths about these images, and was the basis for one of Arthur C Clarke's books. I can't remember the title of it, but i'm sure you can search it up.

Also, go to www.ted.com and search up "fractals" whereby tribes in Africa with absolutely no mathematical background, were able to design their village based on fractals.
 abelian
Joined: 1/12/2008
Msg: 29
The beauty of math known as fractals
Posted: 4/13/2009 1:59:05 PM

Often wonder how many physics folks watch the show.

I'm a ``physics'' guy and I used to watch when it first came on, but haven't lately. When I was watching, I was rather impressed with the fact that they weren't just pulling jargon out of their a$$. I could easily recognize a lot of the math and the mathematics they referenced really was used to do the kinds of things they were purporting to do (but obviously they accomplished more and did it faster than one could do in the real world.) Needless to say, one could not write and debug programs to do what is done on the show in several months, even without the fancy graphics.
 abelian
Joined: 1/12/2008
Msg: 30
How long is the coastline of Britain?
Posted: 4/13/2009 2:02:31 PM

Does one have to be a mathematician to appreciate beauty?

That depends on how deeply you want to appreciate it. There is more to the beauty of an equation than the pretty pictures you can generate. The pretty pictures only scratch the surface.
 OTTO BONN
Joined: 4/20/2006
Msg: 31
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The beauty of math known as fractals
Posted: 4/13/2009 3:49:29 PM
I have been using Fractint for years now to generate a wide variety of fractal images. Fractint is free and is a DOS program.
 13571113
Joined: 5/3/2009
Msg: 32
The beauty of math known as fractals
Posted: 5/22/2009 5:04:58 PM
Hi everyone, this is called necroing a thread.

Yep....its back on the first page woot!
 StrawberryBurns
Joined: 12/13/2010
Msg: 33
How long is the coastline of Britain?
Posted: 1/1/2011 10:13:16 PM
After years of hearing people say that my work reminded them of fractals I finally decided to study up on them. I started with the Arthur C. Clark videos on YouTube to give myself a basic understanding of it all. I've been messing with Apophysis and other fractal software. I can kinda see how my symmetry work would remind people of fractals but my images start with a photo and not a formula.
 Proventus
Joined: 12/26/2010
Msg: 34
The beauty of math known as fractals
Posted: 1/2/2011 9:20:37 AM
While there definitely is beauty in fractal geometry, at this point, we really don't know what it's for. Sure, it has some emerging uses in IT, and probably a few more fields (don't know), but that's where it ends..... For now. We don't know what it is and where it fits in the grand scheme of things.

I used to belong to the MIT club of Ottawa and we'd get speakers in once a month or so. One of the most interesting speakers for me was Alan Guth (Inflationary Universe). He presented a "map" of the universe with galaxies randomly dispersed. Pure chaos. But when you zoomed out to display everything that was mapped, the symmetry emerged. Think of sprinkling 500 grains of sand on a table. Chaos. Now cover the table with sand. It's perfect. Order in chaos.

But my father with no formal education knew it already. One of his favorite expressions was "can't see the forest for the trees." lol.

Someone asking what practical use fractal geometry has reminds me of this quote:
"But what ... is it good for?" An engineer at the Advanced Computing Systems Division of IBM commenting on the microchip.
 RockinDaze
Joined: 7/3/2010
Msg: 35
The beauty of math known as fractals
Posted: 1/2/2011 9:50:36 AM
Okay, I am no math major for certain, but everyone knows that it all relates to problem solving, working equations, etc. As for me, I am more of science fanatic I suppose. Aligning the moons and stars and planets/ Disease, pathology, treatments.... same premise. Kinda. Okay, okay, I'm outta here...
 IgorFrankensteen
Joined: 6/29/2009
Msg: 36
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The beauty of math known as fractals
Posted: 1/2/2011 10:16:14 AM
I wonder what mathematical magic was involved with muddling together TWO old threads of differing titles like this?
I only WISH I understood enough math to understand fractal geometry. I can appreciate that someone who DOES would see beauty in it, just as I as a writer can become excited about a particular juxtaposition of words, in a particular context.
One minor thing I would like to point out about such discussions as have been mentioned in this thread, is that it can be VERY tricky to try to relate a mathematical concept to the rest of reality. I find it is important to recognize that mathematics is an artificial HUMAN CONSTRUCT. We use it, and often use it with great success, to MODEL the real world, in ways that help us to better understand and deal with it, but a mathematical model is JUST a model, and not reality itself. People who become TOO convinced that a given model matches reality completely, are destined to careen off of REAL cliffs while "seeing" a "real" and safe roadway in front of them all the way down.
 RockinDaze
Joined: 7/3/2010
Msg: 37
The beauty of math known as fractals
Posted: 1/2/2011 10:36:41 AM
Does not all beautiful things begin with an idea? It this idea has been proven to work how can one dispel the "myth"? Reality stands in all great architecture, even leaning towers, and beauty will always be in the eye of the beholder. 'Tis a wonderful thing when mind and heart meet. It's complicated. (lol)
 swamp_dude
Joined: 7/23/2007
Msg: 38
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The beauty of math known as fractals
Posted: 1/10/2011 6:00:54 PM
The program I saw on fractels and their use .... said they had discovered that the fractel relationship of branches on the trees in a forest lot seemed to match the relationship of the trees to the forest .... so this allowed them to come to much more accurate estimates on the amount of lumber in a forest.

they implied it was a very unexpected discovery.

it was a few years ago that I saw the show but I believe they also used fractals to work out the evolution of a Teridactyle's [sp?] wing ... and it worked.
 bilby886
Joined: 7/19/2007
Msg: 39
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The beauty of math known as fractals
Posted: 3/6/2012 7:10:40 AM
Learned Fellows,Scholars, Amateur Scientists,Ladies and trolls, can somebody please give me 4 practical outcomes from fractal research?

My understanding is before the advent of easily accessible computers with graphic displays, we had not much Idea of this mathematical pattern based phenomenon.

Except for
1. utilizing the recursive abilities of structured stack oriented programs (like C, Pascal)
2. lighting fast processors busy doing more of nothing
3 creating demand for FPUs in home PCs
4 creating demand for more memory (stack space) goto 1. If not able to then
5 "blowing the stack - exception errors - boundary errors blah blah

what has all this pattern stuff done for us?
Vaccines? A better space shuttle? time travel? World peace? cure schizophrenia? invention of a new type on mongrel btch?
Why ? and what can you use it for?
If you say predicting weather then I can say my method of divining the cracks occurring on fired tortoise shells has the same success rate!

to me it seems the high math version of crayoning the walls of our academies!
I could be wrong and please enlighten me?
 lyingcheat
Joined: 9/13/2009
Msg: 40
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The beauty of math known as fractals
Posted: 3/6/2012 3:59:24 PM

Learned Fellows,Scholars, Amateur Scientists,Ladies and trolls, can somebody please give me 4 practical outcomes from fractal research?

You're right, at its simplest it's about discovering patterns. Do you have something against patterns?

This is a pattern - It's Boyle's law.
pV = k
where;
p denotes the pressure of the system.
V denotes the volume of the gas.
k is a constant value representative of the pressure and volume of the system.

So is this - It's Newtons Law of Universal Gravitation
F = G {m1 m2}/{r^2}
where;
F is the force between the masses,
G is the gravitational constant,
m1 is the first mass,
m2 is the second mass, and
r is the distance between the centers of the masses.

They ^^^ tell us things about the relationships between, in the first instance, pressure and volume, (and often, incidentally, temperature). The second tells us about relationships between distance, mass, and the invisible bonds that connect them. ie; Patterns.


The way sodium and chlorine atoms join reveals the structure of salt crystals (cubic).
The way silicon and oxygen atoms connect to each other is mirrored in the resulting quartz crystals.

The golden ratio -
Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.
http://en.wikipedia.org/wiki/Golden_ratio

How about a pretty basic simple pattern that repeatedly appears in nature...
The Fibonacci sequence - By definition, the first two numbers in the Fibonacci are 0 and 1, and each subsequent number is the sum of the previous two.
0 1 1 2 3 5 8 13 21 34 55 89 144...
http://en.wikipedia.org/wiki/Fibonacci_number


There are (only) five geometric shapes that have certain, quite remarkable, properties.

The Platonic solids.
Tetrahedron - Cube - Octahedron - Dodecahedron - Icosahedron
Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and angles.
The Platonic solids have been known since antiquity. Ornamented models of them can be found among the carved stone balls created by the late neolithic people of Scotland at least 1000 years before Plato. Dice go back to the dawn of civilization with shapes that augured formal charting of Platonic solids.
It is a classical result that there can be no more than five Platonic solids. That all five actually exist is a separate question—one that can be answered by an explicit construction.
http://en.wikipedia.org/wiki/Platonic_solid

The Octave.
In music, an octave or perfect octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". It may be derived from the harmonic series as the interval between the first and second harmonics.


So OK, I haven't given you "4 practical outcomes from fractal research", but why bother. If you don't grasp the value recognising patterns and the mathematical rules that underpin them then 4 examples won't convince you.

http://en.wikipedia.org/wiki/Computer-generated_imagery

____________________________________________________________________________________________________


I find it is important to recognize that mathematics is an artificial HUMAN CONSTRUCT. We use it, and often use it with great success, to MODEL the real world, in ways that help us to better understand and deal with it, but a mathematical model is JUST a model, and not reality itself.

This ^^^ is not really true. It's often the other way around. See above.

Also, check out the these short synopsis notes from a recent program by the BBC...


1. Marcus du Sautoy reveals a hidden numerical code that underpins all nature. A code that has the power to explain everything, from the numbers and shapes we see all around us to the rules that govern our own lives. In this first episode, Marcus reveals how significant numbers appear throughout the natural world. They're part of a hidden mathematical world that contains the rules that govern everything on our planet and beyond.
http://www.bbc.co.uk/programmes/b012xppj

2. Marcus du Sautoy uncovers the patterns that explain the shape of the world around us. Starting at the hexagonal columns of Northern Ireland's Giant's Causeway, he discovers the code underpinning the extraordinary order found in nature - from rock formations to honeycomb and from salt crystals to soap bubbles.

Marcus also reveals the mysterious code that governs the apparent randomness of mountains, clouds and trees and explores how this not only could be the key to Jackson Pollock's success, but has also helped breathe life into hugely successful movie animations.
http://www.bbc.co.uk/programmes/b01320wn

3. Marcus du Sautoy continues his exploration of the hidden numerical code that underpins all nature. This time it's the strange world of what happens next. Professor du Sautoy's odyssey starts with the lunar eclipse - once thought supernatural, now routinely predicted through the power of the code. But more intriguing is what the code can say about our future.

Along the path to enlightenment, Marcus overturns the lemming's suicidal reputation, avoids being crushed to death, reveals how to catch a serial killer and discovers that the answer to life the universe and everything isn't 42 after all - it's 1.15.
http://www.bbc.co.uk/programmes/b0137xfr
(my emphasis throughout)

There's nothing 'mysterious' or 'supernatural' about any of this ^^^. The use of the words "secret" and "code" is actually a subtle joke.

http://en.wikipedia.org/wiki/Giant's_Causeway
 chrono1985
Joined: 11/20/2004
Msg: 41
The beauty of math known as fractals
Posted: 3/7/2012 5:27:05 PM
I've recently learned to work with a type of fractal known as Hilbert Curves, which have some pretty amazing spatial properties. Coordinates on a Hilbert Curve are represented as two integer values: a distance along the curve, and the order of the curve. From these two values one can construct size of the space that contains the curve, full length of the curve, and cartessian (xyz) coordinates. Furthermore when examined in binary form a distinct pattern arises, where removing two bits from the left or right (depending on encoding) yields the distance to the point along a Hilbert Curve of 1 less order, so not only does the coordinates describe it's own location in it's own space, but also it's own location in all lesser order spaces.

When applied to the concept of a Quad Tree, which is a structure where a square space is divided into 4 equally sized sectors, then those divided, until the desired amount of division is reached. Storing the lower tiers of the tree become unnecessary, because the deepest level of the tree describes all levels shallower than it. Revolutionary for spatial indexing and partitioning in computer simulations that must be light in memory usage and efficient on CPU utilization, like in a game where this kind of stuff can be the difference between a slide show and real time play speed.

Additionally when applied to the concept of an antenna, the space filling nature of a Hilbert Curve allows for more surface area in a smaller space, and thus a much more efficient antenna that can sense much more of the EM-spectrum. It's likely that your cell phone (if you own one) has a flat antenna patch inside of it somewhere which is structured from a high order Hilbert Curve or one of it's variants.
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