Ralph Abraham, former Director of the Visual Mathematics project at UC Santa Cruz and author of Dynamics: The Geometry of Behavior turned to the ancient but only recently rehabilitated topological mathematics of the torus in order to describe the complex behavior of dynamical systems. David Bohm also turned to topological mathematics to reveal the deeper structures of quantum theory acknowledging that the Cartesian grid–even with curvilinear coordinates–was incapable of revealing the deeper structures of quantum theory that had no need for underlying space-time at all, only for the self-resonating, interconnective relations that topological geometry permits.
Like physics, architecture has also remained imprisoned in the conceptual mathematical straightjacket of the grid, whose static limitations have sorely restricted the arena of invention and bred the form-functional contradictions that erupted out of modernism’s emphasis on formal purity as well as the host of reactions to the modernist aesthetic that have emerged over the past two decades, postmodernism and deconstructivism among them. Postmodernism was an attempt to relieve the sterility of modernism by reintroducing a sense of historical meaning and continuity, even though ironic; Deconstructivism was the impulse to symbolically de-center functions that have been repressively center (read controlled) in actual fact by the static, profoundly non-dialectical logos of modern western thought. But both have failed as has modernism, because their field of functioning was too narrowly conceived and because they did not understand the notion of asymmetric, dialogical centering, the open-ended symmetry-breaking and reforming that occurs within a many-leveled, self-organizing system. Nor did they have at hand an alternative geometry on which to model this range of functioning.
As I hope has been made clear, an existentential decentering must take place in the context of the ongoing creating and recreating of a concrete reality–of a full dialectical, material, and social praxis–where the conditions are under the control of, and provide the media for, ongoing communal development.
Buckminster Fuller, too, was vehemently anti-grid and cube and wrote in Synergetics II: “There is no universal space or static space in Universe. There is only omnidirectional, conceptual ‘out’ and the specifically directioned, conceptual ‘in.'” On these topological principles and the four dimensional observer-plus-observed tetra-hedra rather than the cube as his “fundamental minimum structure,” Fuller constructed a geometry of thinking called “Synergetics” to account for the properties of wholeness and regenerativenes that three-dimensional planar geometries left out. In this metaphysically-driven universe, there are only time relationships or “events” which are generated in a “rubber donut jitterbug” within the sphere’s nucleus he call the Vector Equilibrium. Thus Fuller’s “spherics” seem in fact to be toroidal, though he never uses the term.
Gregory Bateson said that our efforts at building a bridge between mind and matter have begun at the wrong end, matter, when we should have begun with that which patterns matter–mind. Beginning afresh with a firm underpinning in epistemology–of “how knowing is done” –the topology of the bridge comes increasingly into focus. We thus escape the grid not by eschewing matter but by mediating it in good dialectical fashion, via the mind-matter vortex structure, the torus. Topological design, mirroring and facilitating nonlinear functioning, provides architecture with a fully-articulated epistemology that permits it to become, as Hegel envisioned, “a second body for the mind” –whose home is not in three dimensional space but in in primordial time and phase space.
In Destiny and Control in Human Systems, mathematician and authority on time Charles Muses constructs a time-and-desire configured chronotopology: a structure of resonant causation that couples cosmic and psychic rhythms where through metaphoric/symbolic insight, one can align oneself with “the waves of time breaking on the beach of occurrence.” “In the nature of time,” Muses explains, “experience or consciousness–both in inner or felt (intensive) nega-space–and in outwardly perceived (extensive) posi-space–moves along the arc of the wave form, dipping both into the future and the past as time.” “The object is to surf on time,” Muses said in an interview–a nonspatial dance in which “we come around each time a little differently, as on a spiral.”