Chapter 2 carries out the first of three consecutive iterations of the methodology outlined in Chapter 1. In doing so, it puts forward a continuous ontological domain, embedding both the spatial categories that consolidate the classical notion of Beauty and those yielding contemporary scenarios that drift away from the classical. To that extent, the argument draws prominently from the work of George Bataille and his commentators, particularly in regards to the notions of the Formless and the Excessive. Closer to the architectural discipline, this chapter also looks into Mark Dorrian and Mark Cousin’s reflections on the aesthetic foundations of ugliness. These insights are mobilised to articulate a three-dimensional space of possibilities that is posed as a framework for architectural trans-Beauty. Its three dimensions pertain to, respectively, the positional, the organisational and the boundary conditions of architectural objects within the space. The reflective argument examines the regions of the domain that operate far from the dimensional combinations that manifest classical Beauty, identifying topological geometry as a suitable operational domain for them. Following this line of thought, the work on topological forms of Scottish biologist and mathematician D’Arcy Wentworth Thompson is foregrounded, and discussed alongside the contemporary notions of topological systems formulated – among others – by Ben Van Berkel and Bernard Cache. Ultimately, the framework of topological diagrams is put forward as a generative framework for the expanded domain of trans-Beauty.