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I am interested in discrete fractal models of nature that involve discrete self-similarity, which is a form of the discrete dilation invariance of conformal geometry. This is a fairly esoteric topic, but perhaps some readers might be interested in the arguments for an infinite discrete fractal cosmos that are given below.

Here is the situation, as I see it.

Einstein's relativity program had 3 parts.

Part I: Relativity of inertial motion - Special Relativity

Part II: Relativity of inertial and accelerated motion - General Relativity

Part III: "Conformal" relativity that applies to both microcosm and macrocosm.

Part III is a work in progress. Einstein intuitively knew that Part III would require relaxing the restriction to absolute lengths and was drawn towards conformal geometry. Einstein, Weyl, Kaluza, Klein, Schrodinger, Pauli and many others tried all sorts of approaches to generalize geometry for Part III, but no one could ever figure out how to make it work. A major problem was that conformal approaches seemed to suggest that rest masses might be variable, and this violated well-observed atomic scale phenomena. Still, Einstein pressed on and in his last scientific writing he was talking about the inherent self-similarity of his unified field equations.

One thing that no one fully explored before was discrete conformal geometry, wherein the conformal invariance was preserved rigorously only for very large and discrete changes of scale.

Because I observed that nature appeared to have a discrete self-similar organization, I tried this discrete scale invariant [aka discrete conformal, aka discrete self-similar, aka discrete fractal] approach, and I think it works.

See: http://www3.amherst.edu/~rloldershaw/menu.html .

Take the Kerr-Newman solution for any fundamental system in nature. Then divide all lengths by 5.2 x 10^17, divide all times by 5.2 x 10^17 and divide all masses by 1.7 x 10^56. What you get is an identical analogue on the next lower Scale of nature's discrete hierarchy of Scales. Now take your original K-N solution and multiply all lengths by 5.2 x 10^17, all times by 5.2 x 10^17 and all masses by 1.7 x 10^56. What you get is an identical analogue on the next higher Scale of nature.

These analogues on the 3 different Scales, say the Atomic, Stellar and Galactic Scales have the same shapes, the same kinematics and the same dynamics. This is my candidate for Einstein's Part III and it is called Discrete Scale Relativity: http://arxiv.org/ftp/physics/papers/0701/0701132.pdf .

A review of, and initial test results on, the definitive observational test of this new paradigm can be found at: http://arxiv.org/ftp/astro-ph/papers/0002/0002363.pdf .

How does Discrete Scale Relativity handle the variable rest mass problem? Simple! Within any Scale rest masses are invariant. But there is a totally equivalent proton on each Scale of nature. Therefore their rest masses differ by integral multiples of 1.7 x 10^56. In this way the rest masses of fundamental objects are allowed to vary, but only by huge discrete values.

It takes a while to get used to thinking in terms of this paradigm, but the inner voice says: "This is the right path to Part III". I had this idea on December 21st, 1976 and have been working doggedly to get physicists to give it a full and fair hearing ever since. I think I sense that my 33-year journey is nearing its destination.

Yours in science,

Knecht

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# Conformal Relativity

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