- Fractal Fiber Bundles -

Fractal Field Theory ( for grins ).

What happens when strings get frayed?

The idea here is create a novel fibration space based on using a fractal as the fiber in fiber bundles :

So for fun

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.

Let F, the fiber be a fractal. Pull it up into the total space or fiberspace, M.
Let it twist or wind or solitonically slide around back into itself, slightly off.
As an example let fiber, F, be Koch curve and the twist be integral units of n*360/6.

Each fiber in the bundle B over the topological manifold M is a closed subspace F of a
(fixed?) metric space (X,d), with some some continuity condition
(homotopic deformation in X?) as you move between nearby fibers, and where
the fiber maps over p in M are certain -lipeomorphisms- of X.

Fractal Twisted Tori, would there be any interesting gauge symmetries locally?

The Koch Curve has 6 positions of which to turn it's self back into it's self. Fractals, right? Are there fractal groups? There are fractal point sets:

Hutchinson, John E.
Fractals and self-similarity.
Indiana Univ. Math. J. 30 (1981), no. 5, 713--747.

Fiber Bundles

Base Space - Total Space ...

The Koch Curve is governed by a set of equations that fall under the semi-group, at least that could transform the initial fiber as it twists its way about some axis:

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 )

Perhaps, "fragments of infinity"?

Or Origami on the Fabric Folds of SpaceTime?

The drawings on the side are the result of my doodling and thoughts while on break at
Mission Control ...

Email : troy@trinicom.com