shrinking tangents

[AntonioLao]

Starting with a unit circle, two diametrical tangents with endpoints at A and B are constructed. The other endpoints extend to infinity forming pair of parallel lines. By holding point B stationary, point A moves along the circumference toward point B in a CCW direction. At every instant, congruent pairs of tangent intersect at point C. Point C gets closer and closer to point B. The instant the shrinking tangents become orthogonal unities at point K, point B makes a quantum jump to point P and the process kicks start all over again.

Attachment 245

These weblinks discuss tangent bundle for differential manifold http://mathworld.wolfram.com/TangentBundle.html and http://en.wikipedia.org/wiki/Tangent_bundle

[Guille]

Antonio,

Could it be that the circle joining A, B and P is a moebius strip? I say it because if it is possible, then we could explain that chaos theory is a product from space-time itself, and not from the entities of the universe (matter, energy, force...) as it has been believed until now.

[AntonioLao]

Could it be that the circle joining A, B and P is a mandelbrot manifold?

I don't know anything about this. I'll exit and look it up be back as soon as possible.

[AntonioLao]

Could it be that the circle joining A, B and P is a mandelbrot manifold?

Preliminary search indicates that the Mandelbrot set is not a manifold.

http://en.wikipedia.org/wiki/Mandelbrot_set and http://mathworld.wolfram.com/MandelbrotSet.html

M. Shishikura, The boundary of the Mandelbrot set has Hausdorff dimension two, Complex analytic methods in dynamical systems (Rio de Janeiro, 1992), Asterisque 222 (1994) 389.

....AH( 1 (M) has infinitely many components. It seems likely that the boundary of AH( 1 (M) is quite complicated and interesting. In this paper, we only prove that AH( 1 (M) is often not a manifold. In contrast, not only is it known that the Mandelbrot set is not a manifold, but Shishikura [36] proved that its boundary has Hausdorff dimension 2. There is a conjectural parameterization of the Mandelbrot set, known as the abstract Mandelbrot set (see Branner [9] which provides a complete picture of the global topology, while in the study of AH( 1 (M) no such conjectural picture ....

[harmonygirl]

Antonio, I'm not sure I understand the reference, but it certainly doesn't fit with my understanding of Mandlebrot set...

[AntonioLao]

I'm not sure I understand the reference, but it certainly doesn't fit with my understanding of Mandlebrot set

That makes two of us, I don't understand it either. I'm trying to create my own math using only real numbers. I think Mandelbrot set is based on random complex numbers.

[AntonioLao]

Guille mentioned this link to Moebius strip http://en.wikipedia.org/wiki/M%C3%B6bius_strip and I edit one of the images as given below
Attachment 246
If now we cut along the lines indicated with the arrows, two linked circles are created, one being twice the radius of the other.

[AntonioLao]

Furthermore, incorporating directional invariance to the one-sided surface, four distinct topologies emerged as shown below
Attachment 247

[harmonygirl]

I'm not sure that Mandlebrot used random complex numbers. The set is transformative, rather than equal, so x becomes itself squared (and a constant). I haven't really looked at the range of values that can be plugged into x which includes complex numbers. I think it would be pretty fascinating, tho. (If you find one, please let me know).

[AntonioLao]

haven't really looked at the range of values that can be plugged into x which includes complex numbers.

As long as the range contains complex numbers, I'm not interested in doing further research. I've decided to limit all my works within the real domain.