A Note on Fractals and Banach-Tarski Extraction

28 January 2011


Stephan Banach and Alfred Tarski

Further to my recent posts on fractals and the Banach-Tarski Paradox (A Question for Philosophically Inclined Mathematicians, Fractals and the Banach-Tarski Paradox, A visceral feeling for epsilon zero, and Adventures in Geometrical Intuition), I realized how the permutations of formal methodology can be schematically delineated in regard to the finitude or infinitude of the number of iterations and the methods of iteration.

Many three dimensional fractals have been investigated, but I don't know of any attempts to show an infinite fractal such that each step of the interation involves an infinite process. One reason for this as no such fractal could be generated by a computer even in its first iteration. Such a fractal can only be seen in the mind's eye. Among the factors that led to the popularity of fractals were the beautifully detailed and colored illustrations generated by computers. Mechanized assistance to intuition has its limits.

The Banach-Tarski Paradox involves a finite number of steps, but for the Banach-Tarski paradox to work the sphere in question must be infinitely divisible, and in fact we must treat the sphere like a set of points with the cardinal of the continuum. Each step in Banach-Tarski extraction is infinitely complex because it must account for an infinite set of points, but the number of steps required to complete the extraction are finite. This is schematically the antithesis of a fractal, which latter involves an infinite number of steps, but each step of the construction of the fractal is finite. Thus we can see for ourselves the first few iterations of a fractal, and we can use computers to run fractals through very large (though still finite) numbers of iterations. A fractal only becomes infinitely complex and infinitely precise when it is infinitely iterated; before it reaches its limit, it is finite in every respect. This is one reason fractals have such a strong hold on mathematical intuition.

A sphere decomposed according to the Banach-Tarski method is assumed to be mathematically decomposable into an infinitude of points, and therefore it is infinitely precise at the beginning of the extraction. The Banach-Tarski Paradox begins with the presumption of classical continuity and infinite mathematical precision, as instantiated in the real number system, since the sphere decomposed and reassembled is essentially equivalent to the real number system. There is a sense, then, in which the Banach-Tarski extraction is platonistic and non-constructive, while fractals are constructivistic. This is interesting, but we will not pursue this any further except to note once again that computing is essentially constructivistic, and no computer can function non-constructively, which implies that fractals are exactly what Benoît Mandelbrot said that they were not: an artifact of computing. However, the mathematical purity of fractals can be restored to its honor by an extrapolation of fractals into non-constructive territory, and this is exactly what an infinite fractal is, i.e., a fractal each step of the iteration of which is infinite.

Are fractals a mere artifact of computing technology? Certainly we can say that computers have been crucial to the development of fractals, but fractals need not be limited by the finite parameters of computing.

Once we see the schematic distinction between the finite operation and infinite iteration of fractals in contradistinction to the infinite operation and finite iteration of the Banach-Tarski extraction, two other possibilities defined by the same schematism appear: finite operation with finite iteration, and infinite operation with infinite iteration. The former — finite operation with finite iteration — is all of finite mathematics: finite operations that never proceed beyond finite iterations. All of the mathematics you learned in primary school is like this. Contemporary mathematicians sometimes call this primitive recursive arithmetic (PRA). The latter — infinite operation with infinite iteration — is what I recently suggested in A visceral feeling for epsilon zero: if we extract an infinite number of spheres by the Banach-Tarski method an infinite number of times, we essentially have an infinite fractal in which each step is infinite and the iteration is infinite.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Grand Strategy Annex

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