Continuous Maps and Their Epistemic Limits

Parametric shapes may be both continuous in state space and their extrinsic instantiation, but may exhibit singularities in either, and no longer maintain any topological similarity between intrinsic and extrinsic views. What we need are richer descriptions of topology that embed also implicit logics of construction and concurrent local discretisation that emerge organically from the global topology itself. This split is fundamental because continuous mathematical functions, algebraically expressed, often obscure rather than reveal spatial or topological facts. Amiens Cathedral, Amiens, France, 13th century
left and opposite top: As systems of one logic are sequentially propagated along surfaces of a divergent logic, ruptures inevitably occur. It follows from the axiomatisation of mathematics in the 19th century. The procedural shapes of subdivision surfaces may be extrinsically continuous but arise out of discrete intrinsic operations. To answer, we must consider why the functional definition of surface geometry has become so distinct from the topological one. The non-Euclidean shapes are intrinsically continuous, but can demonstrate folds and singularities in the embedding space. During this time the mathematician Felix Klein was preoccupied with the question of unifying a multiplicity of theoretical geometries. How can we bridge the gap between the mathematics of continuous discretisation and the syntax of architectural spaces? Such ruptures are common in Gothic design, for example here in the vaults of Amiens Cathedral. Of course the tools we have are new, but this synthetic ambition for deductive relationships of local parts to global whole is a fundamental tension within the project of design itself (Figure 4). The impulse to equate continuous maps to complete definitions of architectural elements is compelling because it has proven so germane to problems of constructability, rationalisation and parametric control (Figure 2). By distancing geometry from visualisation, Klein’s Erlangen programme lay the seeds of the divorce between geometry and design. Klein’s proposal for a unified classification of surfaces through their nested invariant functional meta-behaviour is known as the Erlangen programme. The new geometries are uneasily classified as either continuous or discrete, a dichotomy whose simple and axiomatic distinction in the classical view – between the real and the integer – no longer so simply holds. Klein’s ambition was to classify the varieties of surfaces through the sets of maps or functions that left these surfaces invariant.5 Continuous maps themselves form a space which can be transformed, and these second-order sets of transformations can themselves also be transformed, and so on, creating an infinitely nested sequence of continuous function sets that indirectly describe the properties of the first set of surfaces. Fascinating as issues of surface differential geometry are, the more fundamental formal issues play out at the scale of the global surface – that is, how surfaces enclose and partition spaces, how one circulates among them, and resolution of spatial connectedness and separation.

Updated: 28.10.2014 — 13:31