( an imperfect
model with no immediate application )
Constructed
here is a fiber bundle over a Base Space comprised of a circle
S1 with Fiber given by a
fractal with
Zn cyclic symmetry group ( n = degree of cyclical symmetry ) and
produced by an
Iterative Function System, IFS. The Total Space E is comprised of the fractal parametrically winding or
twisting about it's local center and following the global orbit line around
a non-local axis until it matches back to the Fractal Fiber's starting position.
The angle of local TwistT can be in
integral multiples of N*(360/n), where {
N = 0, 1, 2 ...} and { n =
degree of cyclic symmetry}. A Fiber Bundle is said to
have a TwistT
which can determine the orientability of the Total Space. The Twist Group T , invariants, soliton, abelian,
non-abelian, trivial, non-trivial, Gauss-Bonnet, symplectic, Gauge Group,
Fiber Bundles are spaces which are locally Products but not
necessarily globally. A discrete fiber
space produces a fibration that is a covering
space, like a helix covering a circle. A vector fiber
space creates a vector bundle and is classified either
algebraically by characteristic classes which are elements of homology
groups OR geometrically by maps to classifying
spaces. Also, a loop space produces the space
of loops in a given space.
These invariants are called
characteristic classes and are viewed formally as elements in the cohomology
groups of the manifold, the duals of the homology groups. Characteristic classes
can also be defined purely topologically by investigating the global motion of
the tangent space as the manifold is traversed. They measure to some extent the
failure of the manifold to possess a globally defined notion of parallelism: for example, a global
parallelism exists on the torus in which Euler-Poincaré characteristic = 0, but not on the two-dimensional sphere in which
Euler-Poincaré characteristic =
2.
(... the Euler-Poincaré characteristic number
is related to the Euler graph number for the graph that is related to the
surface ? ...)
Fibers are objects in some
Category. All fibers must be isomorphic, homeomorphic, or
self-similar. Every Category is built with a notion of
isomorphism. Vector Spaces, Groups, Topological Spaces, Manifolds,
G-sets, and Fractals. Two Fibers are homeomorphic if
one Fiber can be deformed into another by a one-to-one mapping and a continuous
function. Homeomorphism is too weak to preserve the fractal
nature of objects.
For
example, the Koch curve is actually homeomorphic to the plain old non-fractal line
segment of length 1.
People usually talk about self-similar fractals.
A similarity is stronger than a homeomorphism. A one-to-one and onto mapping is
called a similarity if there is a constant c (called the similarity factor) such
that for any two points
x1 and x2 in X their images y1 and y2 in Y have the
property that the distance from y1 to y2 is c times the distance
from x1 to x2.
A fractal is called
self-similar if it it can bew divided into finitely many pieces
:
X = X1
union X2 union ....
such that there is a similarity f1 from X onto X1, a
similarity f2 from X onto X2 and so on.
In that sense the Koch curve is
self-similar because it can be divided into four pieces, each of which is 1/3
the size of the original Koch curve.
A Fiber Bundle can be a Fibration, but not necessarily
everytime.
( ... something
about the parallel fibres .... in the special case of the affine space where the
connection ..., parallelism of Aji
means the constancy of all the components. Properties of the connections
imply certain types of manifolds falling in certain types of characteristic
classes ... parallel transport ...
Connection
A <> Curvature dA <> Torsion A ^ dA <> Hypervolume dA ^ dA
...
Gauge Theory <> Connection
Manifold <> Choice of A - Gauge Freedom)
Each Fiber F in the Total Space E
projects to exactly one point in the Base Space B. The Product of a Fiber F and a local Neighborhood U of the Base Space B creates the Total Space E. The Product can be denoted as
U x F ->E or B x
F->E.
The
inverse image of a point in the Base Space B is the Fiber F
.
A Fiber Bundle with a Fractal as a
Fiber F is a Map f : E -> B where E is called the Total Space of the
fiber bundle and B the
Base Space of the fiber bundle. Showing that for every point in the Base
Space ( b in B ) has a Neighborhood
U so that f1(U)
is Homeomorphic to U x F
by some continuous function and one-to-one
mapping.
Moebius Twisted Ring with
Fractal Cross-Section
Scroll Down for
Image ( sounds effects will be removed in the future ) :
If you cannot see the above flash
object, goto
MacroMedia's homepage
to download a free Shockwave player...or see the below picture ...
Troy Christensen