Shape grammar theory, as established by George Stiny,4 provides an important counterpoint to this largely pervasive view of design shape as occupancy of point set topology. Collectively they can be seen as positing a view of ‘shape as space’; moreover as connections via mappings among a disparate network of Euclidean, non-Euclidean and more general topologies. Manifolds can be embedded into any topological space where locally continuous measurement by real numbered coordinates holds. While much of the application of this system has been concerned with developing substitution grammars of Euclidean transformations implicitly deployed in the context of a Euclidean spatial medium, the shape grammar system has demonstrated applicability to problems involving non-Euclidean elements and their transformations as well. This containing space is presumed Euclidean: linear, continuous, absolute and singular; there is only one such space in which all shape occurs. In the purely digital realm, both intrinsic and embedding spaces are by necessity Euclidian – real numbered coordinate

Figure 4. The measurements and mappings, historically the realm of craft, are now conducted through increasingly sophisticated machines providing direct and continuous transformation between numerical coordinates and physical location. x= s(u)

u‘ = t’lx)

Figure 3. We can in fact rigorously consider manifolds that bridge from the digital into the ‘worldly’ topologies and transformations of physical space. These geometries occur as R2 ж R3 mappings between two or more distinct topological spaces: an intrinsic two-dimensional parametric space, and the containing three-dimensional space outside the surface. Within this system, Euclidean geometries and spaces take a natural place as the restricted class of linear transformations in the more general class of differentiable mappings. These geometries are no longer seen as monstrous or pathological, devised to challenge the limits of the Euclidean, but rather as generalisations of the classic geometries, formalisms of utility and applicability to architecture, and indeed of everyday experience. At the scale of human experience, we may arbitrarily select a containing Euclidean space worldview – of specific dimensionality, measured by a specific coordinate system, etc, and normalise shape by its embedding into this arbitrarily privileged frame. This work re-prioritises shape over space, and re-establishes an axiomatic system of shape as an algebraic topology of shapes and their parts. largely inaccessible to anyone beyond topologists. Most significant for design is the migration of form’s locus, which emerges not simply as the occupancy of any specific Cartesian space, nor its numerical description, but resides in and as the connection between spatial frames – the intrinsic and the extrinsic, the Euclidean and non-Euclidean, the continuous and the discrete, the digital and the physical. Euclid’s Elements1 establish geometry through assertions on constructions of shapes – the lines and arcs, their measurements, angles and intersections – without directly referencing a spatial medium. As shapes aggregate through their combinatorics, so do their individual connected spaces connect into larger networks. systems within the machine. Gehry Partners, Walt Disney Concert Hall, Los Angeles, 2002

The enclosure detailing connects features from two distinct mappings between 2-D and 3-D space: the ruling lines of the global developable surface, and a patterning of lines in the unfolded surface space, injected back into 3-D space as geodesic curves. This broad class of admissible embedding topologies includes the affine, vector and tensor spaces among many others. What emerges is a view of space and shape that is a radical expansion of the Cartesian system. The existence of such geometries has been supposed over the past three centuries, but prior to digital computation they could be treated only in their most general forms and through their most simplistic examples,

Figure 2. As with the Euclidean elements, shape grammars form a complete system of shape description whose closure is independent of any containment space. However, this manifold structuring applies in a formally rigorous manner when extended to a much wider spectrum of embedding topologies. The nature of space, in which the constructions occur and the axioms hold, has been debated throughout the history of spatial ontology,2 and specifically whether space is absolute, discrete from geometries it contains, or relational, sufficiently defined by relationships between spatial phenomena. Shapes emerge from, within and as a system of spatial networks of heterogeneous dimension and signature, no longer inert but active and dynamic, continuously created, connected and destroyed by design. The most visible examples are the tensor manifolds of non-uniform rational b-spline (NURBS) surfaces. They establish ‘shape as construction’. By extension to alternative dimensions, the two – and three-dimensional Euclidean and non-Euclidean shapes including points, lines, curves, surfaces and volumes are described. The extrinsic space contains the shape as points of occupancy, while the intrinsic space – the space of the surface – is the basis by which its shape is described, measured and traversed, and the perspective from which its continuity emerges. Shape exists in – and as – this network of spatial connection. This evolution of the spatial fabric presupposed by contemporary geometry, of shape as space, and of space as relational, localised and connected, is arguably the central ontological advance of contemporary form-making and associated architectural description. The spaces are not atomic, but in turn disaggregate into subspaces of individual parameters, subshapes and their products. The tangent developable surface

The tangent developable surface is a locally continuous surface that has a global singularity at the edge of regression. The fact that the Euclidian constructions hold when described as coordinates is one of the remarkable achievements of the Cartesian system. This framework extends directly to the parametric, wherein shapes are instances of geometric functions driven from spaces of discrete parametric values.