Notes on Fractals and Fiber Bundles In the Hopf fibration, the concentric tori "interpolate" between the round
circle and the one which looks like a vertical line. There is a flow which
moves all the tori towards the vertical line, and hence in some sense any two
of these tori are the same. (What is also true and not so obvious is that
any two circles in the Hopf fibration are the same.)
As for fiber bundles, the most general definition of a fiber bundle has a
fiber F which could be any space, a group G of symmetries of the fiber,
and a base space B. A fiber bundle over B with fiber F and gauge group G,
is a space E with a projection pi:E-->B which is locally a product and
has gluing maps in G. This definition works perfectly well for F a fractal
and since many of the standard fractals are highly symmetric, one should be
able to get some interesting results.
Most classification questions tend to be more tied to the group
G than the fiber F, so those probably won't be much fun. You would certainly
get some weird spaces. (..RS..)
Indeed fiber bundles do play a key role in
string theory...we imagine that a higher dimensions
(that is, dimensional) universe that string theory
requires may indeed be a fiber space over a
four dimensional spacetime, with the fibers being
theextra space dimensions. This is but a small
part of a larger story...my book does not go into
any of this mathematical detail, but you will see
pictures that are locally of the form R^2 x (sphere or torus)
as a lower dimensional analogy of the higher dimensional
version. And I also have pictures of R^2 x (Calabi-Yau manifolds)
where the latter are 6 dimensional spaces required by the theory.
All the best,
Brian Greene
(A similar statement holds for many of the classical fractal curves, for instance, the 'devil's staircase' or
Cantor-Lebesgue curve.)
No, it is not necessary that the fiber is a group. In general
the fiber is a topological space (so that you know what a continuous
function is). In geometry the fiber is a manifold, so that you know what a
differentiable function is.
> In each of the examples above the Fibers have different geometries.
Lines, 1-spheres, etc. I have seen one with that curved triangle.
So I'll try again: What effect does the fiber geometry have
on the fiber bundle? And must they be differentiable?
Well, one studies fiber bundles in different settings (categories, to be
technical). The most general setting is where the fiber, the total space
and the base are topological spaces and the projection is just a
continuous function. However, many of the examples are smooth: all the
spaces are manifolds and the projection is smooth. You can even go further
and look at the case where the spaces (fiber, total and base) are
algebraic varieties and the projection is algebraic.
> If the fibers are groups, what is the signifance of the symmetry
> and other aspects of the group?
In case the fiber is a group and some other condition is satisfied you
get what is called a Principal Fiber Bundle. Then you get for every
representation of the group an Associated Vector Bundle. This is the
situation in gauge theory: the gauge field lives on the principal
fiber bundle and the matter fields are related to the associated bundles.
A triangle is not in an obvious way a group and I have never seen it
used as the fiber in a fiber bundle. Why would you want to use a triangle?
A triangle is not a smooth manifold, so it will be difficult to talk about
differentiable functions.
QED is a gauge theory.
Dear Troy,
A fibre bundle is a geometric construction which represents a symmetry group
of the fibre. Thus the Mobius band is a geometic way of looking at the
transformation x --> -x on the fibre which is a real line. This has
applications to physics because it allows us to express symmetries in terms
of topological and geometric concepts.
The point of a fiberspace M is that it looks LOCALLY like a product; i.e.
small neighboorhoods U of points in the base B can be pulled back to UxF in
M, where F is the fiber, and the fiber map projects UxF onto U. In the
example of a moebius band M , U is an intervall and the fiber is an
interval I , and U pulls back to UxI in M . If you assume that the
universe is a 3-dim. space then universe-time space could conceivably be a
4-dim. fiberspace: the base space is the universe and the fibers
(corresponding to time) could be infinite lines or circles.
Fiber spaces and bundles:
The relation between the total space of a bundle, its base space and
the fiber is indeed somewhat subtle. (In your Moebius strip example,
the total space is the entire Moebius band, the base space is the
central circle, and the fiber is a line). The most straightforward
case is the one in which the fibers of the bundle are lines, or planes,
or more generally vector spaces of any dimension. Such bundles are
called vector bundles. The point is that in these, the fibers are
topologically trivial. The topology of the total space thus comes from 2
sources: (a) the topology of the base space,(b) the
`twisting' in the bundle.Think of the Mobius band and the cylinder. Both
are bundles over a circle, and both have a 1-dimensional line as
fiber. The difference in their topologies comes from the fact that one
is a `twisted' bundle and one is not. There is a sophisticated
mathematical theory designed to understand - and measure - the precise
way that the topology of the total space is determined. It involves
contraptions which go by the name of ``characteristic classes'', and
a theorem known as the Leray-Hirsch Theorem. One reasonable
reference for this would be the book ``Differential Forms in Algebraic
Topology'' by R. Bott and L. Tu.Contributions from the twisting
in a bundle to the topology of the total space do indeed play a role
in physics. They show up, for example, in quantum field theories as an
explanation for so-called anomolies. But you should ask a physicist
about that...(..SB..)
There is a very large body of work that all starts from the
paper
Hutchinson, John E.
Fractals and self-similarity.
Indiana
Univ. Math. J. 30 (1981), no. 5, 713--747.
In it Hutchinson shows that
if A is a collection of affine(?)
shrinking self maps of the plane (or some
Euclidean space?) then the
images of the actions represented by infinite
words in the alphabet
A converge to a unique limiting set X(A). This set is
of interest.
It is self similar and often fractal and can be made to be any
of
the commonly known examples. It is also a minimal set that
is
invariant under the actions of A in that is it the union of its
images
under the actions of A.
The phrase "can be made to be any of the known
common examples" is
the contribution of Barnsely. He made the observation
that if a set
is approximately the union of its images under the actions of
A,
then it is approximately the limit X(A). Thus if you can take a set
X
and approximately cover it by a set A of contractions of itself,
then X is
approximately X(A). I don't recall the reference for
this. It might
be:
Barnsley, M. F.(1-GAIT); Ervin, V.(1-GAIT); Hardin,
D.(1-GAIT);
Lancaster, J.(1-GAIT)
Solution of an inverse problem for
fractals and other sets.
Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 7,
1975--1977.
It is a very simple observation (as opposed to
Hutchinson's
which is quite deep), but it has had an enormous sequence
of
applications and consequences.
There are at least a couple of
books to start with.
Barnsley, Michael(1-GAIT)
Fractals everywhere.
Academic Press, Inc., Boston, MA, 1988. xii+396 pp. $39.95.
ISBN
0-12-079062-9
Falconer, Kenneth(4-BRST)
Fractal geometry.
Mathematical foundations and applications.
John Wiley & Sons, Ltd.,
Chichester, 1990. xxii+288 pp. $33.95. ISBN
0-471-92287-0
Falconer,
K. J.(4-BRST)
The geometry of fractal sets.
Cambridge Tracts in
Mathematics, 85.
Cambridge University Press, Cambridge-New York, 1986.
xiv+162
pp. $32.50; \$16.95 paperbound. ISBN 0-521-25694-1;
0-521-33705-4
> I am wondering if you could tell me if there is a "natural"
process
of
> paperfolding for the Koch Curve-Fractal
> as there is for the
Dragon Curve-Fractal?
This is a really good question that I'd have to
admit never occurred to me.
The paperfolding process produces the pattern of
"up" versus "down" folds
that we see in the dragon. (An up fold is one that
points up). If we put 0
for down and 1 for up, the successive generations
goes like
0
0*0*1
001*0*011
0010011*0*0011011
etc.
I put *'s
to mark where the middle fold. The second half after the middle
fold is the
same as the first half in reverse order with 1's and 0's
interchanged.
For
the Koch curve, the first generation has fold pattern
010
2nd gen.
010*0*010*1*010*0*010
etc.
The pattern is that between each of the
previous folds, you stick another
copy of the 010 pattern.
You can achieve
this by folding paper, but I don't know of any simple way.
>
>
Also, I am wondering if you could tell me a bit about the
instances
where
> differential equations can be derived from
> the fractals
involving continuous-time systems. Are there any
such systems
>
demostrating the Koch Fractal's
> underlying
algorithm?
Historically, discrete dynamical systems were studied as an
approximation to
continuous systems specified by differential equations. The
most famous
differential system leading to a fractal is the one that gives
the Lorenz
attractor. It's a nonlinear first-order system for three functions
x(t),
y(t), z(t) of time t.
The attractor is a strange butterfly-like
object that all solutions
converges as time goes to infinity.
There's a
ton of stuff on the Lorenz attractor on the web, including
Java
realizations.
Attractors of differential equations tend not to have
sharp corners, so I
doubt the Koch fractal arises as an attractor of one, but
I really don't
know.
> What are some of the fractals that are
associated
> with continuous-time systems which allow
> for the
expression of them in terms of differential equations?
> Further,
from
> my understanding a Group can give way
> to a matrix which
yields differential equations. Is it possible to
> back-track and find a
group representation for the symmetry
> or other property of a
fractal?
Fractals are more often associated to semigroups. A semigroup
allows
multiplication of any two elements, but not inversion.
The
semigroup for a fractal consists of the transformations that map the
whole
fractal onto its parts that look the same.
For the Koch fractal the semigroup
is generated by
four transformations
f1(x,y) = ( x/3, y/3) f2(x,y) = (
(sqrt(3)*x-y+2)/6 ,
(x+sqrt(3)*y)/6 ) f3(x,y) = ( (sqrt(3)*x+y+3)/6 ,
(x-sqrt(3)*y+sqrt(3))/6 )
f4(x,y) = ( (x+2)/3, y/3 )
Each of these gives
one of the four parts of the Koch curve.
That's why the Koch curve is an
attractor of an Iterated Function System
which I talk about later in my web
book.
A lot of the properties of fractals are studied in terms of the
IFS.
I don't know what you mean by the algebraic group
representation of the Koch fractal. The shape is
generated by similarity transformations so each of these
can be expressed as a matrix and a column vector (for
the translation). But by a group representation I
understand a function from an abstract group to a
matrix group. I don't see how to accomplish this for the
Koch curve, because I don't see a group structure on the
Koch curve. Could you give me a little more detail about
your question? In particular, you say you know the
group is compact. What group do you know is compact?
The boundary of the Koch fractal is not a continuously differentiable
image of an interval since this boundary does not have length. In fact
no portion of the boundary has length.
Applications of the Koch Fractal:
Good question. One obvious use is as a means of generating pure fractals
for evaluating fractal dimension and image analysis algorithms. There
are many examples in the literature of using the Koch curves for this
purpose.
I think there may be a deeper answer. You could have asked the same thing
about a sine wave before Fourier analysis was developed. My recent IEEE
paper developed a basis of fractals for decomposing a signal into
fractal components. This was done using fractal functions, but there
is no reason why you couldn't do exactly the same thing using Koch
curves. I believe that scale space decomposition may prove to be an
important application.
On the Koch: it can be used for data massaging (interpolation of data points) - for
this see the book:
The Science of Fractal Images
it can be used for one obvious use is as a means of generating pure
fractals for evaluating fractal dimension and image analysis algorithms. There
are many examples in the literature of using the Koch curves for this
purpose.
possibly for resolving a signal into koch fractal dimensional components
No connected portion of the boundary (which is more than one point)
has length. Just being 'indenumerably infinite' doesn't mean it
has no length. A circle could be described in the same way. The
boundary of the Koch fractal simply turns out to have no length.
The Koch curve, the limiting shape after the construction process has
been carried out infinitely many times, is nowhere differentiable.
Helge von Koch's goal in defining this curve was to find a continuous
curve that is nowhere differentaible. In a sense, his construction is
a natural generalization of the function x -> |x|, continuous
everywhere but not differentaible at x = 0. Von Koch found an
elegant way to put little absolute value functions arbirtarily
close to every point of the curve, so there can be no derivative
anywhere.
As for its uses in mathematics, mostly the Koch curve is an
example of a simple fractal construction. The curve itself is
thoroughly understood so no new mathematical results about it
are likely to occur. However, it is used as the boundary for
the wave equation ("fractal drums") mathematically by
Michel Lapidus, and in physical experiments by Bernard
Sapoval.
I'm unaware of any connection between the Koch curve and groups,
except for the obvious symmetry group of the Koch snowflake.
To which duality theorem were you referring? There are many
of them.
this continuous function can
be obtained: it is simply the uniform limit of the sequence of continuous functions parametrizing the
polygonal curves approximating the Koch snowflake curve. This does not give you an 'explicit formula'--and, in
fact, no such elementary formula exists. But it does provide you with a way of approximating as closely as you
wish a rather complicated geometric object by a much simpler one.
>So I am concerned with the fiber. In the Moebius strip it is
a line, in the 3-sphere it is a 1-sphere; and I guess with a
torus it is a 2-sphere or a circle. The fiber, F, is a group, right?
What do you mean: "I have a torus..... its fibers";
to talk about fibers etc you need a total space T
a base space B and a projection p:T\to B. So what is your torus supposed
to be?
Email : troy@trinicom.com