The Metamorphic Point. Châtelet, Klee, and the Dynamic Ontology of the Point

Abstract

This article presents a radical deconstruction of the static ontology of the point inherited from the Euclidean tradition, through a comparative analysis of the thought of mathematician Gilles Châtelet and visual artist Paul Klee. Against the notion of the point as an inert, partless entity, both thinkers conceive of it as a dynamic agent and virtual germ: for Klee, it is an “Agens” imbued with an internal Spannungswille (will to tension), while for Châtelet, it is an “operator” resulting from processes of contraction or dilation. The article identifies three fundamental parallels between them: the point as a site of internal tension and mutual attraction, the line as an oriented arrow actualizing its virtual direction, and the point as a germinal centre of irradiation. The synthesis culminates in Châtelet’s later technical elaborations, where the point becomes, algebraically, a quotient of polynomials (a “germinal matrix”) and, topologically, a becoming-hole. Through gestures of extraction and circumvolution, the point transforms into a relational zone. The diagram thus emerges as the operative field where the point’s virtuality, gesturality, and metamorphic nature are enacted, offering a fertile alternative ontology for both mathematical and artistic practice.