Not so sacred geometry - paper

[John G]

My point was that homotopy spheres (your S^2 and S^3 manifolds) don't have anything to do with fractal dynamics since S^2 or S^3 manifolds don't support (non-trivial) Hamiltonian flows (which are required to write any Lagrangian for QM)...Also, S^3 manifolds shouldn't be used because of the zero divisor problem (That's why Hamilton invented quaternions!) unless you're very sure you're in a zero torsion environment (eg infinitely thin disk gyro, or Gibbs' vector space EM theory, etc).

Fractals have more uses than just modeling QM, I don't think Ark is saying all his fractals are physically realistic, you kind of want your discrete structure in spacetime to be 4-dim or 3-dim for space. Given that Ark is using conformal transformations, he has torsion I would think and the following may apply:

The compact manifold that represents 4-dim spacetime is RP1 x S3, the Shilov boundary of the bounded complex homogeneous domain that corresponds to Spin(6) / (Spin(4)xU(1)).

 

[ark]

The cantor set is a [0,1] function of n

Cantor set is not a function of n. It is a set, not a function. You can identify it with the characteristic function, that is true. But the characteristic function is a function on the real interval. Not on n. Can you explain this contradiction?

 

[ark]

My point was that homotopy spheres (your S^2 and S^3 manifolds) don't have anything to do with fractal dynamics since S^2 or S^3 manifolds don't support (non-trivial) Hamiltonian flows (which are required to write any Lagrangian for QM)...Also, S^3 manifolds shouldn't be used because of the zero divisor problem (That's why Hamilton invented quaternions!) unless you're very sure you're in a zero torsion environment (eg infinitely thin disk gyro, or Gibbs' vector space EM theory, etc).

Fractals have more uses than just modeling QM, I don't think Ark is saying all his fractals are physically realistic, you kind of want your discrete structure in spacetime to be 4-dim or 3-dim for space. Given that Ark is using conformal transformations, he has torsion I would think and the following may apply:

The compact manifold that represents 4-dim spacetime is RP1 x S3, the Shilov boundary of the bounded complex homogeneous domain that corresponds to Spin(6) / (Spin(4)xU(1)).

First of all quantum mechanics does not need Lagrangian. For instance we have quantum mechanics of pure spin 1/2, no Lagrangian.

No, I do not have any torsion, though I may have one, if I want, but that is completely irrelevant. And my fractals have nothing to do with Spin(6) / (Spin(4)xU(1)), as I have conformal transformations and fractals on S^n for any n.


Now, let me repeat my question to Newton:

Fractals can only be generated with div algs, that is on a symplectic manifold.

Question to Newton: Is it a theorem or is it your hypothesis? If it is a theorem - please provide an explicit proof, including all the necessary definitions (because your definitions may happen to be very personal). If it is a hypothesis - also provide precise definitions, and formulate your hypothesis precisely. Otherwise we are not talking precise science.

 

[Newton]

A fractal is a projection from an orbit on a symplectic (spin) manifold onto a Riemanniam manifold. A symplectic manifold adds a constraint relative to Riemannian space which 'fractionally' lowers its dimension. - My best definition.

Notice that all fractional transforms (which are canonical transforms) are mappings involving symplectic manifolds.

I have never found a counter example for my definition.

The Laplacian is native to symplectic spaces, that is irreduceable 2nd order PDEs have a state-space on a symplectic manifold. The Jacobian is 1 (or -1, there's a double cover of a sphere) from a Riemannian space - this is the symplectic form, which gives that mirror symmetry.

When del-squared = 0, the manifold is Kahler (both Riemannian and symplectic).

So, I know that the Cantor is some discrete sampling on a projection of some Legendre polynomials. It's not quite clear what the exact mapping is.

But ordered sets are the basic definition of functions in Topos theory. You give an input and the get an output, eg C(1) = 101

And regarding QM - Your dealing with stochastic flows rather than Hamiltonian flows (like in string theory). I've found that one can convert Bosonic sting solutions which are deterministic to probabilistic QM solutions with Gaussian transforms.

 

[Newton]

I read you paper. You definitely have a QM view of the world rather than a string theory view. I noticed that Tony Smith provided some input, but having read his papers, virtually all his mathematics - Lie groups/algebra, etc is fundamental string theory! All his manifolds, and hence his wave functions, are all deterministic. His exact EM waves need to be averaged with a Gauss transform to look like QM.

Suppose all that 'randomness' in QM isn't random at all. Instead it's just a Euclidean view of dynamics from a symplectic manifold (Hamiltonian flow) - the same dynamics mathematically as a polynomial random number generator...

 

[ark]

A fractal is a projection from an orbit on a symplectic (spin) manifold onto a Riemanniam manifold.

So I was right that you have a a private definition of a fractal, which is not the same as the one adopted among fractal community!

According to wikipedia (which is not a Bible, yet in mathematics it is often reliable):

The term fractal was coined in 1975 by B. Mandelbrot, from the Latin fractus, meaning "broken" or "fractured." In colloquial usage, a fractal is a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely complex." Mathematicians avoid giving the strict definition and prefer to call 'fractal a geometric object that usually

* has fine structure at each scale and can not be easily described in traditional Euclidean geometry language.
* is self-similar (at least approximatively or stochastically)
* has Hausdorff dimension greater than its topological dimension
* has a simple and recursive definition
* has natural appearance.

Fractals have all or most of these features (see Falconer(1997)). Not all self-similar objects are fractals - for example, the real line (a straight Euclidean line) is formally self-similar and has natural appearance but fails to have other fractal characteristics.

It is difficult to discuss with you, as you are using your private terminology, without defining it first of all. Your "fractals" are not the same animals as my fractals.

 

[Newton]

We need a precise definition! I offered one. What you reference is *not* a definition. We're talking mathematics here, not art!

Your definition of fractal reads more like "some characteristics of" rather than a definition. To that list you should add that fractal statistics are 'cyclostationary'.

If you don't like my definition, give me your mathematical precise definition, in other words "looks natural in appearance" doesn't hack it. Perhaps the more technical term 'pink noise' might do.

And I believe your paper is incorrect - Spin manifolds and fractals are not associated with quadratic curvature in any way. All Spin manifolds are symplectic, which do not have quadratic curvature.