We present Peirce’s Existential Graphs as systems of geometric logic, where some inversions are proposed between the logical and the topological, the logos and the topos. Imagination, visualization, abduction and pragmatism can thus be comprehended independently of the linguistic, syntactic, deductive or normative aspects of logic. The shift of perspectives allows a plasticity and a dynamism characteristic of Peirce’s entire architectonics. As a consequence of the inversions, a full, extended reason, capable of comparing and gluing the local and the global, can be applied to the complex, non-linear, ramified spaces of our contemporary World. The paper is divided into the following sections: (1) a synthetic account of Peirce’s Existential Graphs; (2) some inversions of the logos and the topos in the Existential Graphs; (3) some wider topological dialectics along Charles S. Peirce’s architectonics; (4) the "visual turn" of our contemporary World: mathematics, art, and a return to Novalis; (5) category-theory and Peirce’s topo/logical insights one century later.
Keywords: Charles S. Peirce, logic, topology, local/global, obstructions/transits.