Tightening curves and graphs on surfaces

Any continuous deformation of closed curves on a surface can be decomposed into a finite sequence of local changes on the structure of the curves; we refer to such local operations as homotopy moves. Tightening is the process of deforming given curves into their minimum position; that is, those with minimum number of self-intersections. While such operations and the tightening process has been studied extensively, surprisingly little is known about the quantitative bounds on the number of homotopy moves required to tighten an arbitrary curve. An unexpected connection exists between homotopy moves and a set of local operations on graphs called electrical transformations. Electrical transformations have been used to simplify electrical networks since the 19th century; later they have been used for solving various combinatorial problems on graphs, as well as applications in statistical mechanics, robotics, and quantum mechanics. Steinitz, in his study of 3-dimensional polytopes, looked at the electrical transformations through the lens of medial construction, and implicitly established the connection to homotopy moves; later the same observation has been discovered independently in the context of knots. In this thesis, we study the process of tightening curves on surfaces using homotopy moves and their consequences on electrical transformations from a quantitative perspective. To derive upper and lower bounds we utilize tools like curve invariants, surface theory, combinatorial topology, and hyperbolic geometry. We develop several new tools to construct efficient algorithms on tightening curves and graphs, as well as to present examples where no efficient algorithm exists. We then argue that in order to study electrical transformations, intuitively it is most beneficial to work with monotonic homotopy moves instead, where no new crossings are created throughout the process; ideas and proof techniques that work for monotonic homotopy moves should transfer to those for electrical transformations. We present conjectures and partial evidence supporting the argument

The Usefulness of Truth : An Enquiry Concerning Economic Modelling

This thesis attempts to justify a normative role for methodology by sketching a pragmatic way out of the dichotomy between two major strands in economic methodology: empiricism and postmodernism. I discuss several methodological approaches and assess their aptness for theory appraisal in economics. I begin with the most common views on methodology (i.e. empiricism and postmodernism) and argue why they are each ill-suited for giving methodological prescriptions to economics. Then, I consider positions that avoid the errors of empiricism and postmodernism. I specifically examine why the two major strands of methodological criticism fail to give helpful methodological advice to economists and sketch out a pragmatic approach that can do this

Logic and intuition in architectural modelling: philosophy of mathematics for computational design

This dissertation investigates the relationship between the shift in the focus of architectural modelling from object to system and philosophical shifts in the history of mathematics that are relevant to that change. Particularly in the wake of the adoption of digital computation, design model spaces are more complex, multidimensional, arguably more logical, less intuitive spaces to navigate, less accessible to perception and visual comprehension. Such spatial issues were encountered much earlier in mathematics than in architectural modelling, with the growth of analytical geometry, a transition from Classical axiomatic proofs in geometry as the basis of mathematics, to analysis as the underpinning of geometry. Can the computational design modeller learn from the changing modern history, philosophy and psychology of mathematics about the construction and navigation of computational geometrical architectural system model space? The research is conducted through a review of recent architectural project examples and reference to three more detailed architectural modelling case studies. The spatial questions these examples and case studies raise are examined in the context of selected historical writing in the history, philosophy and psychology of mathematics and space. This leads to conclusions about changes in the relationship of architecture and mathematics, and reflections on the opportunities and limitations for architectural system models using computation geometry in the light of this historical survey. This line of questioning was motivated as a response to the experience of constructing digital associative geometry models and encountering the apparent limits of their flexibility as the graph of dependencies grew and the messiness of the digital modelling space increased. The questions were inspired particularly by working on the Narthex model for the Sagrada Família church, which extends to many tens of thousands of relationships and constraints, and which was modelled and repeatedly partially remodelled over a very long period. This experience led to the realisation that the limitations of the model were not necessarily the consequence of poor logical schema definition, but could be inevitable limitations of the geometry as defined, regardless of the means of defining it, the ‘shape’ of the multidimensional space being created. This led to more fundamental questions about the nature of Space, its relationship to geometry and the extent to which the latter can be considered simply as an operational and notational system. This dissertation offers a purely inductive journey, offering evidence through very selective examples in architecture, architectural modelling and in the philosophy of mathematics. The journey starts with some questions about the tendency of the model space to break out and exhibit unpredictable and not always desirable behaviour and the opportunities for geometrical construction to solve these questions is not conclusively answered. Many very productive questions about computational architectural modelling are raised in the process of looking for answers