Cantor's dust as a topostrategic model

N. Lygeros

Translation: Paola Vagioni

In topology but also in fractal analysis, Cantor’s dust or the triadic model, is presented as the equivalent of the set of real numbers. In reality, via the Benoît Mandelbrot approach, it constitutes a topological intermediate between the points and the straight line. As a compact set, i.e. as a closed and bounded one, it has nothing special. It acquires its singularity under the capacity of accumulation. Specifically, every point of Cantor’s dust is an accumulation point. In other words, for any neighbourhood of a point, there is always another point of the set. This property is not only geometrically important, but topologically fundamental. If we examine this topological entity via mental geostrategy, then it gives us an explanation of its robustness as a mathematical structure. When a structure receives a blow and preserves its order, we name it robust. Topologically, what negatively characterizes Cantor’s dust is the lack of cohesion. While topostrategically, this lack is nothing but seeming. A hard structure will not endure an attack over a certain limit. Therefore, even if we have such a structure, we have to know that it is vulnerable. Thus a single piece appears as more powerful, but this does not necessarily mean more robust. Ice is harder than water in a liquid form but less robust when it receives pressure. This differentiation is essential in order to topostrategically understand the robustness of the Aegean Sea. The latter is presented as a finite stage of induction which creates Cantor’s dust. In this way, when we examine the 3000 islands, we can use it as a topostrategic model. Thus in case of a powerful blow, which is equivalent to a mental wave like the tsunami, our islands are more robust. In essence, the idea is the following, where it is presented in our book Mental Strategy, the breaking of iron or wood appears difficult while the breaking of a cloud is unfeasible. The robustness of the Aegean Sea does not come from the compact, but from the lack of cohesion, which has as a paradox repercussion the non-creation of a front. Without front, the problem of breaking does not exist. The fragmented is not afraid of the breaking. Thus the contribution of topostrategy is the transcendence of topology via the creation of a network which is no longer based on traditional facts.

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