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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
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.
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
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.
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.