Misconception spreading about Wheeler foam being a fractal

2nd December, 2010.

E-infinity Communication No. 39
Misconception spreading about Wheeler foam being a fractal
This communication is prompted by a fundamental misconception among theoretical physicists not really specialized in nonlinear dynamics and fractals who think that the Wheeler foam is a fractal. Of course there is no mathematical definition for what Wheeler foam is except that it is multi-connected and not simply connected space and in addition that such structure is a consequence of the vacuum fluctuation and that happens near to the Planck length. The mistake or misconception would be harmless if it would not lead to excellent physicists thinking wrongly that fractals exist only at the Planck length and are therefore relevant only to quantum gravity. This is completely wrong. In fact highly sophisticated theoretical physicists who also know about fractals mathematically, such as the singularly clever Gerard ‘t Hooft told Prof. El Naschie on several occasions that fractals are irrelevant for quantum mechanics. When he asked why is it used by some of his collaborators, he answered this is only maybe relevant for quantum gravity at the Planck length. For this reason it is obvious that we need to clarify this point.
First we should say that most of those working outside our group and the group of Prof. Nottale and Prof. Ord mention fractals but do not generally use much of what the powerful theory of fractal sets offers. They may occasionally use a Hausdorff dimension or some kind of fractal dimension to specify an important point here or there, but they do not really use the powerful machinery of transfinite set theory and transfinite theory of dimensions. The exception here is without a trace of a doubt is Prof. Alain Connes. Mathematically speaking Alain Connes’ noncommutative geometry is the most powerful and also the most complex machinery known to our trade. Alain Connes knew that fractals exist long before one reaches the Planck length. Fractals mean zero and empty sets. Any mechanics will somehow incorporate zero and empty sets. Some physics can muddle through without these concepts being said clearly from the outset. Other physical disciplines would be a muddle without a clear conception of the set theoretical basis, namely zero set and empty set. Quantum mechanics is such a physical subject which would be hopelessly complicated to understand without clear understanding of the role of the empty and the zero set.
Let us try to go into the historical reason of how this misconception came about. When Wheeler drew a picture of metric fluctuation due to vacuum fluctuation, it looked like foam. Some people called it Wheeler foam. Somehow it entered into the book of Mandelbrot, ‘The fractal geometry of nature’. Mandelbrot calculated the fractal dimension for a cube made of 27 smaller cubes when we iteratively remove the core cube. In other words, we get a fractal dimension equal to ln26 divided by ln3 which is 2.9656. For comparison the Menger sponge has a dimension equal to ln20 divided by ln3 which equals 2.7… From now on the misconception multiplies. Since it is now called fractal foam it is a fractal and since it is the foam of Wheeler, it is connected to metric fluctuation at the Planck length. Forgive us for saying this is complete nonsense. This misunderstanding crept into the work of some of the best and