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Exceeding a Surface, but Evading Volume

The very notion of a ‘mathematics of chaos’ seems to suggest a grotesque domestication, such as the enfeeblement of chaos into a statistically intelligible randomness. For this reason my immediate reaction to the mathematics of chaos was one of visceral suspicion, even though a thread of eighteenth and nineteenth century German philosophy had prepared me for its topic. Nevertheless, it is not easy to imagine a mathematician ceasing to be a Platonist. Nor easy to remain immune to the virological seduction of ‘a geometry of the pitted, pocked, and broken-up, the twisted, tangled, and intertwined’ as Gleick summarizes it in his popularizing book [Ch 94], or to sustain an indifference to topological explorations characterized by mathematical orthodoxy as ‘monstrous, disrespectful to all reasonable intuition about shapes and… pathologically unlike anything to be found in nature’ [Ch 100].

A glance at the purportedly chaomorphic ‘Sierpensky-’ or ‘Menger sponge’ both confirms and undermines such suspicions; it is a shape that is homogenized, saturated with equalities, inanely geometric, yet also irresolvable, paradoxical, unhealthy. A Menger sponge results from the endless recursion of a simple operation. A cube is divided into twenty-seven identical smaller cubes, with the central block and each of the six orthogonally adjacent ones being removed. The resulting frame consists of twenty blocks, which are then all treated in the same way as the initial cube, and so on, recursively. Each transformation increases surface area with a tendency to infinity, and decreases volume with a tendency to zero. However far this process is taken the sponge remains cohesive, and it is possible to trace a line in three dimensions from any point on the surface to any other. In its ideal conception a Menger sponge is thus a model of infinitely complex immanence; a universe of endlessly intricate distances, without inaccessible depths or absolute ruptures. Exceeding a surface, but evading volume, the Menger sponge is a shape of between two and three dimensions, or of a fractional dimension; a fractal to use Mandelbrot’s term. Like the Möbean band of the early Lyotard, or the ‘smooth space’ of Deleuze and Guattatri, it is a libidinal geometry without inaccessible recesses, a topography without transcendent repression.

—Nick Land, The Thirst for Annihilation: Georges Bataille and Virulent Nihilism

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