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The Plastic Fractal Magnet
bedessen writes "An article at NewsFactor summarizes the developments in new plastics that exhibit magnetic fields of fractal dimensions. Whereas a simple bar magnet produces magnetic fields that go from the north pole to the south pole, the fields of the new hybrid plastic sprout like branches of a cactus lined with secondary fields that resemble needles. As these fields become increasingly interlocked, they exhibit a unique kind of order. This intensely ordered structure might one day be key to storing information with a very high density. The researchers behind this are Arthur Epstein, director of the Center for Materials Research at Ohio State University, and Joel Miller, a professor of chemistry at the University of Utah. There's also this PDF overview of the subject, which is quite technical but still readable."
I'm by no means qualified to answer this, but heck i'm a-gonna do it anyway!
Yes, the volume of the water would be 3D. The volume changes from 1/2D to 3D because you are changing the geometry of the object! Honestly, I think the answer *is* as simple as that..
What's your GCNSEQNO?
The snowflake would have a true 3D volume because it is not perfectly thin; it is a physical approximation of a mathematical concept.
The analogy of the snowflake refers to the edge of the snowflake. Imagine that you took a thread and tried to put it along the edge of the snowflake. Assuming that the thread was very thin it would take an infinitely long thread to cover the entire edge, because of the way it is folded. Thus the 'edge' can be said to have a dimension higher than 1 (it does not fit into one dimension). Using mathematical techniques one can also demonstrate that the the infinite thread takes zero space in 2D, thus the dimension is somewhere between 1 and 2; it is a fractal.
Tor
Fractal implies a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry. While there is a finite volume associated with a snowflake, you are right (I think) in attributing the "fractional" dimensionality of the snowflake to its geometry. Obviously the fractal nature is lost as soon as the flake melts. Consider the question: how doe we determine the volume of the snowflake without destroying its fractal nature? That is to say, what linear measurements of length can be used to calculate its volume? Note: weighing the fractal and using density also avoids the question of a geometric calculation of volume and would be cheating.
I'm a jackass, forgot it was html formatted.
The dimension D of an object made of N exact copies of itself, each shrunk by a factor of S is:
log(N)
------
log(1/S)
So, a fractal is an object with a non-integer D.
Your hybrid is not saving the environment. Its purpose is to make you feel good about buying something.
From the pdf link: Perhaps its all about the ordering (which could be due to the geometry of the molecular structure).
Note: the pdf file also states (towards the end):Its interesting that where we are looking at is (I think, perhaps) a non-bulk form of magnetism, and the statement is perhaps overstating a requirement.
Carbon nanotubes are used to transport single electrons. The wavelength of said electrons are such that the dimensions of the conduit result in what can best be modeled as a one dimensional potential well (as taught in senior-level Intro Quantum classes via ODEs, as a way to avoid the math of 3D potential wells and PDEs). So perhaps it could be said that 1D does exist for very very small, bound objects.
No, the volume is changed both qualitatively and quantitatively. Even with classical geometry, the volume isn't conserved. Melting an ice cube changes its volume. Why shouldn't melting a snowflake? As has been mentioned, the alteration of the configuration does indeed affect volume.
The Mongrel Dogs Who Teach
Imagine that you took a thread and tried to put it along the edge of the snowflake. Assuming that the thread was very thin it would take an infinitely long thread to cover the entire edge, because of the way it is folded
Isn't this comparable to the Paradox of Achilles and the turtle [openetwork.com]? Meaning that the thread does not have to be infinitely long? Well this is a valid but unfortunately rather complicated discussion. When you add an infinite number of objects with size zero (or approaching zero), the sum can turn out to be finite or infinite depending on exactly in what way the objects approach zero size (and sometimes, if I remember correctly, it even depends on the order in which you add them).
In the case of this 'paradox', you add an infinite number of objects (stretches of time) that approach zero so quickly that the total is actually finite. This is what some of the Greek thinkers did not realize.
For fractals, on the other hand, when you add the infinite number of small (approaching zero size) objects they end up taking infinite amount of space. This is a necessary condition; if you add them all and the total is finite then it is not a fractal.
Tor
But doesn't the analogy breakdown because the pattern can't truelly repeat scaling down forever? That is, there will have to come a level at which the resolution of the molecules destroy the ever repeating pattern, like grain in a photograph
Yes, this is true for all fractals with a physical manifestation. There is always some lower and upper scale where the fractal properties break down. The lower scale is often, as you suggest, on an atomic level.
A mathematical fractal is an abstraction that has infinite resolution. Such abstractions can be useful to study the properties of physical fractals, even though we know that these are only approximations.
Tor
Everthing you can see or interact with, from snowflakes to magnetic fields, exists in a 3d universe. Such things as electrons, quarks, superstrings etc might not, but I've never seen one.
The snowflake exibits a fractal dimension over a wide range of scales. If you took a microscope you could magnify it many times over and keep finding the same level of detail being revealed. So we say it has a fractal dimension. Without knowing the fractalness of a paticular snowflake, the dimensions of the snowflake wouldn't be enough to tell you how much water was in it with much accuracy.
A coastline has the same property on a human scale. As the size of your measuring stick decreses, the length of the coastline increases.
"A language that doesn't affect the way you think about programming, is not worth knowing" - Alan Perlis
A 2D object's area (or "volume" if you will, since there are only two dimensions) changes as x^2 as you scale the object. A 3D object's volume changes as x^3 as you scale the object. An object with fractal dimension has a volume that scales as some non-integer power as you scale the object.
(additional story link where Epstein confirms this)
Checkout the link in the previous story The Top Ten Physics Highlights of 2002, Highlight #7,Magnets open the gate to nanoscale logic , to see how nano-sized mangetic structures could be used. The hard part is going to be interfacing to this structures. These structures are *small*.Note: this is digital logic without transistors, but with nanoscale ferromagnetic wire.
Is here
Among other results it is shown that Great Britain's coastline has a fractal dimension of 1.24, while that of South Africa is very nearly 1.
Any sufficiently advanced libertarian utopia is indistinguishable from government.
Norway: Fractal dimension 1.52 (here and here, apparently from Feder.)
Google is your friend
I suspect the swedish coast has nearly as high a dimension, with Denmark a bit lower.
Any sufficiently advanced libertarian utopia is indistinguishable from government.
If the disjoint union of n disjoint copies of a fractal F makes a similar (in the geometric sense) one k times as big, then the fractal dimension of F is (log n)/(log k) = log base k of n.
This makes the fractal dimension of a square 2 because it takes four of them to make a square twice as big and log 4 / log 2 = 2. The fractal dimension of the Sierpinski Gasket is log 3 / log 2 because you can assemble 3 copies of it to get one twice as big.
The dimension of the Cantor set (that's the one where you start with the unit interval and remove the middle third of every line, or equivalently the numbers between 0 and 1, inclusive, whose base-3 expansion contains no 1s) is log 2 / log 3 which is less than 1.
The dimension of the rational points in a square is still 2, even though it has fewer points than the Cantor set. So, fractal dimensions are "freaky."
I hereby place the above post in the public domain.
I first learned about chaos theory, from James Gleick's excellent book 'Chaos' about ten years ago. I've been hooked ever since.
The thing that stuck in my head was Fiegenbaum's number 4.669, which BTW is irrational. This ratio is everywhere and most profound of all, is visible in the architecture of our bodies. The main artery from the heart called the Aorta, is like the trunk of a tree, point being is, if you measure the distance between the heart and the first bifurcation, divide that distance by 4.669, it gives you the statistical length of the two branches from the first bifurcation. Now here is the kicker:- it is that ratio, all the way down to the smallest cappillary, to enable a blood supply for every cell in our bodies.
GM technology worries me, not because I'm scared of engineering. But because to my knowledge, we do not yet understand the mathematics of morphogenesis. DNA is a simple four bit code and yet somehow or other, nature manages to store a cellular doubling number in that four bit code.
We all start out as one cell, that doubles in a binary progression. Our body plan is formed by the x,y,z matrics of those doublings. The fractal like architecture of our bodies, gives us a hint to how, the miracle of storing our entire code base, in about four gig might be acomplished.
This new discovery excites me, who knows where it will lead, a new understanding of life maybe? New math? New electronics? The list is endless.
Cutting edge indeed.
Peter
It's called an elephant's trunk whereas it is in fact, an elephant's nose, a nose by any other name would smell as sweet
You use the one-dimensional potential well because you don't need to take angular momentum into account (which would require the 3-D Schrodinger equation). Think about modeling a baseball in freshman physics. You could do that in 3 dimensions--but you could also beat yourself over the head with a club because it's so much fun when you stop. You model the baseball in 2 dimensions, because that is mathematically equivalent. That doesn't mean that ballistic motion is 2-dimensional.
Personally, I think the word dimension should be banned from from all discussions of popular science. But hell, there are enough Ph.D.'s out there that shoot thier mouths off like they are unclear on the concept.