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Science Forum Index » Fractals Science Forum » Question on fractals
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| Guest |
 Posted: Tue Dec 07, 2004 6:22 am |
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We've been going over fractals in one of my bio-engineering classes,
and i was wondering if someone might have the time to clear a few
things up for me. What exactly is the value of a fractal dimension
without defining some degree of self-similarity within the species of
interest? I can't personally see any. I've run across a few different
defintions of varying depth, but it appears to me as though they relate
the nature of patterns when the scale changes between orders of
magnitude. But all i can see them doing is deriving a decimal
dimension of said pattern between 2 integral embedding dimensions. How
does this infer information about the nature of the pattern? I've
heard of fractals used to describe the branching of trees as well as
turbulent fluid flow, but i don't see how fractals relate any
information about the degree of similarity change with the scale (i.e.
how similar smaller parts of the tree are to the big tree, versus
smaller parts of the fluid to the big fluid). If it can't relate
similarity change, what exactly is it doing? What use is information
about a reoccurring pattern, if you can't state exactly how similar the
pattern is to itself? For all you know you may not even be dealing
with a self similar pattern to begin with. So yeah, what exactly is a
fractal dimension supposed to be relating to me?
Thanks,
Jeremy Q |
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| John Bailey |
| Posted: Tue Dec 07, 2004 7:48 am |
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On 7 Dec 2004 03:22:30 -0800, TommyBojangles@gmail.com wrote:
Quote: What exactly is the value of a fractal dimension
without defining some degree of self-similarity within the species of
interest?
When we state physical quantities, the weight of a man for instance,
we always state the units in which the quantity is given and do not
remark that "it can be anything." We are never suprised when it is a
different value for pounds than for kilograms. Explicitly adding a
qualifier stating the fractal dimensions aas well as units of a
quantity to a number that represents a physical measurement could be a
very natural and useful step. It provides a convenient reminder that
the length (or whatever) needs to be adjusted if the measurement scale
changes.
Consider Mandelbrot's example: the coastline of England. The
specification of a perimeter as a numerical value with the added
specified dimension value, D, provides an unambiguous specification of
the best fitting fractal dimension line. If you want to re-specify,
using say kilometers, you need only follow usual engineering practice
for transforming dimensions and you will get the correct new value.
Starting with the usual assumption that L(r) = k*r^e, generally
plotted as a straight line in log-log coordinates, do some simple
dimensional analysis. The dimensions of k are miles^(1-e) assuming r
is expressed as miles. The dimensions of r^e is miles^e. The plus and
minus exponents of the two e's on the right side of the equation
cancel so that the whole right side of the equation has the dimension:
miles matching conveniently the dimensions of the left. Notice that
1-e = D, the fractal dimension so k is in base units to the D power
(e.g. miles^ 1.3) Furthermore, the specification of a perimeter as a
numerical value with the added specified dimension value, D, provides
an unambiguous specification of the best fitting fractal dimension
line.
The places where boundary measures are misstated abound. For example,
The CIA FactBook, a widely consulted reference on countries and
geography states lengths for every country in km. This is of course,
nonsense. With a simple reorientation, that could be changed.
For more, see:
http://home.rochester.rr.com/jbxroads/interests/sci.fractals/dim_list.html
John Bailey
http://home.rochester.rr.com/jbxroads/mailto.html |
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| Roger Bagula |
| Posted: Tue Dec 07, 2004 1:23 pm |
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Dear Jeremy Q,
John Bailey covered dimension pretty good.
How does it relate to you?
Economic and population changes are
,now, predicted using chaotic functions.
Dr. Mandelbrot's favorite is stock markets,
but any mass movement of humanity usually
involves " roughness" with is with "turbulance"
one way that fractal dimensions come about.
Fractals and chaos kind of take up the stuff
that regular mathematics has let fall between the cracks
because it was to hard in many cases.
Actually fractals are just beginning to be understood.
We have laws for only the most simple cases, in fact.
And the beginning of multifractal laws that Dr. Mandelbrot
loves to study with his Mandelbrot cartoons and Mandelbrot functions.
Anyone who tells you it is all sowed up,
doesn't understand fractals or their dimensions, ha, ha...
TommyBojangles@gmail.com wrote:
Quote: We've been going over fractals in one of my bio-engineering classes,
and i was wondering if someone might have the time to clear a few
things up for me. What exactly is the value of a fractal dimension
without defining some degree of self-similarity within the species of
interest? I can't personally see any. I've run across a few different
defintions of varying depth, but it appears to me as though they relate
the nature of patterns when the scale changes between orders of
magnitude. But all i can see them doing is deriving a decimal
dimension of said pattern between 2 integral embedding dimensions. How
does this infer information about the nature of the pattern? I've
heard of fractals used to describe the branching of trees as well as
turbulent fluid flow, but i don't see how fractals relate any
information about the degree of similarity change with the scale (i.e.
how similar smaller parts of the tree are to the big tree, versus
smaller parts of the fluid to the big fluid). If it can't relate
similarity change, what exactly is it doing? What use is information
about a reoccurring pattern, if you can't state exactly how similar the
pattern is to itself? For all you know you may not even be dealing
with a self similar pattern to begin with. So yeah, what exactly is a
fractal dimension supposed to be relating to me?
Thanks,
Jeremy Q
--
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn |
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| Guest |
| Posted: Wed Dec 08, 2004 7:35 am |
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Is it customary for a state school to be teaching this type of material
at the sophmore level? Do most engineers make use of this type of
stuff? |
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| Roger Bagula |
| Posted: Wed Dec 08, 2004 11:27 am |
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Actually they have been aiming to introduce fractals at an introductory
level
in senor advanced mathematics in high school as the methods mostly
calculus free at the begiining levels.
But it isn't supprising that a biology type course would include
them as the modeling of plants as shapes is, now, almost universally
done using fractal L-systems since
the 80's. Leaves, trees , brush etc. are made using fractal turtle/
memory type processes
It is thought that DNA uses an L-system or similar fractal coding system
to store shape data
for development of all kind.
TommyBojangles@gmail.com wrote:
Quote: Is it customary for a state school to be teaching this type of material
at the sophmore level? Do most engineers make use of this type of
stuff?
--
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn |
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