Extending the Algebras of Design

Algebras of design have previously been investigated for shapes composed of rectilinear geometric elements, such as lines and planes, and the properties of these algebras have been found to be beneficial for formalising designs, as well as the visual processes used by designers as they manipulate shapes in their design explorations. In this paper, an overview is presented of the application of these algebras in formalising design processes, and this is followed by a discussion concerning issues that arise when the algebras are extended to accommodate non-rectilinear designs, represented by shapes composed of curves, surfaces and solids. Consideration of non-rectilinear shapes introduces new problems not previously identified in the established formalism, resulting from the geometries and topologies of the shapes. These give rise to significant questions about the relationships between shapes and the property of embedding, which is fundamental to the construction of algebras of design. 1 – The Boolean Algebra of a Design Pictorial representations, such as sketches and digital models, play an important role in architectural design for supporting creative processes of ideation and form development, as well as analysis and communication of developing design concepts. As a result, aspects of architectural design can be modelled according to shape computations (Stiny 2006), where shapes are abstractions of the pictorial representations that architects use to support their design processes. Shapes are formally defined according to finite arrangements of the geometric elements used to construct representations, such as points, lines, planes and solids, and can include labels or weights to model non-spatial information attributed to a design. For example, (Koning and Eizenberg 1981) presents an analysis of Frank Lloyd Wright’s prairie style, in which designs are represented as shapes decomposed into Froebelean-type building blocks. The analysis resulted in a shape rule schemata that formalises the style according to spatial relations between volumes differentiated as functional zones, as illustrated in Fig. 1 for the design of the Robie house. Fig 1. The Robie house, decomposed into functional zones, represented as Froebelean-type blocks. Image: authors, after (Koning and Eizenberg 1981) Extending the Algebras of Design Nexus Network Journal 2 Identification of the parts of a design imposes a structure on the representative shape. This structure, defined according to the identified set of parts, corresponds to a Boolean algebra for a finite set and its subsets, and is partially ordered according to a part relation, closed over operations of sum and product, with the complete shape as unit, the empty shape as zero, and complements defined accordingly. The algebra enumerates potential decompositions of the shape according to the identified parts, and is equivalent to a complemented distributive lattice, as illustrated in Fig. 2. Here the main floor level of the design of the Robie house is decomposed according to functional zones, with the union of parts given by the supremum (join) and the intersection by the infimum (meet). This lattice illustrates a particular decomposition of the shape, and is indicative of a fixed symbolic representation according to the identified set of (subsets of) parts. But shapes are visual constructs, and the visual richness that is apparent in even the simplest shape cannot be reduced to a set of symbolic elements (Stiny 2006). Accordingly, the lattice illustrated in Fig. 2 gives only an account of a specific decomposition of the main floor level of the design of the Robie house, one defined according to functional zones, and does not represent the complete Boolean algebra represented by the shape. Fig 2. A lattice of the main floor level of the design of the Robie house Shapes have no inherent parts and alternative analyses of the design of the Robie house would result in alternative decompositions. For example, consideration of room layout, instead of functional zones, would identify a different decomposition of the design with parts representing the rooms of the house. Similarly, analyses of the design according to performance factors such as ventilation or lighting, would suggest other decompositions of the shape, with parts identified according to corresponding metrics. Also, other decompositions can result from other processes that involve applying a description to the design, for example for the purposes of representation or communication (Krstic 2004). It is through these processes of enquiry and description that the structure of a shape is defined, Extending the Algebras of Design Nexus Network Journal 3 and as a result shapes can be continuously reinterpreted to enable the different modes of interrogation and enquiry necessary to support a design process (Jowers and Earl 2012). With respect to the algebraic structure of a shape, this plethora of decompositions gives rise to a Boolean algebra which is partially ordered by the subshape relation, is closed over operations of sum and product, with the complete shape as unit and the empty shape as zero, but is infinite in breadth (Stiny 1990). Consequently design representations can be interpreted and structured according to any subshapes that are recognised as embedded parts, and the visual richness that is inherent in shapes can be utilised in computations within formal algebras. This account of shapes is analogous to mereological sums, which are defined according to part-whole relations, and do not decompose uniquely into parts (Casati and Varzi 1999). Indeed, there is a close connection between Boolean algebras and mereology, as explored in (Hovda 2009), although a direct isomorphism is not possible because mereological structures do not contain a zero element (Eberle 1970). In design, the zero is accounted for by the empty shape which is both philosophically and practically an essential element for realistically formalising design representations and the processes applied to them. Philosophically, the empty shape is the untouched medium: a blank sheet of paper, an empty CAD file, etc. Practically, it ensures closure of shape algebras under shape operations. As visual constructs, shapes also have other properties that differentiate them from mereological sums. In particular, the parts of a shape can be both timeless and temporary, and neither timeless nor temporary parts are adequately accounted for by standard mereological conceptions (Fine 1999). A shape’s parts are timeless, because they are persistent components of the shape. The empty shape is the only shape that does not have any proper parts and removal of a part from any non-empty shape will result in that particular shape ceasing to exist. The parts can also be temporary, because at any particular moment a given part may not be recognised within the structure of the shape. For example, individual rooms are always a part of the design of the Robie house, but may not be recognised as such when the design is decomposed according to functional zones. The temporality of the parts of a shape plays an important role in creative design. It introduces a visual ambiguity that allows shapes to be more than externalisations of ideas; they are an active part of the design process. This is most obviously exemplified in sketching processes, where the sketching activity is not only directed by the architect, and her internal thought processes, but also by shapes recognised in the sketches (Schön and Wiggins 1992). There is a reciprocal interaction between the architect and the representation, which results from seeing shapes in new ways and reinterpreting their structure according to alternative parts. This process involves not only decomposition of shapes, but also transformation via manipulation of recognised parts including components and shape elements, which are moved, added, deleted, stretched, reflected, etc., (Prats et al. 2008; Paterson and Earl 2010). These manipulations are formalised according to shape rules, which recognise parts of shapes under Euclidean transformations, and replace them under Boolean operations. Such rules implement computations in algebras which extend beyond those defined by particular shapes and their parts. 2 – The Algebras of Design In the previous section, a design representation was presented as a Boolean algebra defined over the representative shape and its embedded parts. This is a particular (static) Extending the Algebras of Design Nexus Network Journal 4 view of design representation, which supports reinterpretation according to decompositions identified via enquiry and description. But, consideration of manipulations of representations in a design process forces this view to be expanded to include variations of the shape according to added and transformed parts. For example, representing the design of the Robie house according to Froebelean-type blocks gives rise to a Boolean algebra, which includes all possible decompositions of the shape into blocks, partially ordered by the subshape relation, with the complete design as unit and the empty shape as zero. Manipulation of the design of the Robie house by adding more blocks or by transforming recognised parts of the design results in shapes which are not included in this algebra. Therefore, to ensure closure under such manipulations it is necessary to consider not only the shapes that are parts of the Robie house, but all shapes that are of potential interest even shapes of which the Robie house is a proper part. This gives rise to an algebra of design, which includes all possible design representations composed of Froebelean-type blocks, partially ordered by the subshape relation (Stiny 2006). Other algebras of design can similarly be defined, and the simplest examples of these contain shapes composed of a single type of geometric element, (either points, lines, planes or solids) and are denoted by Uij (i ≤ j), where i indicates the dimension of the geometric element and j represents the dimension of the Euclidean embedding space. For example, the Froebelean-type block representation of the design of the Robie house is in the algebra U33, where volumes are arranged in 3D space. Also contained in this algebra are Froebelean-type block representations of all of Wright’s Prairie house designs, as well as all other shapes defined by arrangements of volumes in 3D space. Similarly, designs represented in 2D sketches are in the algebra U12, and 3D wire frame models are in U13. More interesting design representations, composed of combinations of different types of shapes as well as other non-spatial information, are formalised in algebras which are defined by the Cartesian products of these simple algebras, in combination with algebras of labels, Vij, and weights, Wij (Stiny 1991). For example, the representations of the design of the Robie house in Fig. 1 combine lines, planes, and weighted volumes in the algebra U13 × U23 × W33. Unlike the algebra of a particular design representation, these more general algebras of design do not include a unit. The only exception is the algebra U00 which contains a shape composed of a single point, and is isomorphic to the algebra of Boolean logic, with the point acting as unit, and the empty shape as zero. In other algebras Uij (i, j ≠ 0), the empty shape is zero, but the unit would be a universal shape, which would include all other shapes as parts, and would by definition violate the condition that shapes are of finite extent. Instead, the algebras of design Uij (i, j ≠ 0) form generalised Boolean algebras (Stiny 2006). These simple algebras, Uij, have the property that all shapes within the algebras can be defined from a set of shapes, and the sum operation applied under transformation (Stiny 2006). For U0j, all shapes can be defined from a point, repeated under transformed sums; for U1j, all shapes can be defined from a line, repeated under transformed sums; for U2j, all shapes can be defined from a set of triangles, repeated under transformed sums; for U3j, all shapes can be defined from a set of tetrahedra, repeated under transformed sums. As lattices the algebras are closed under union and intersection, but these are not complete because although every intersection is defined, infinite unions are not. Consequently they are relatively complemented distributive lattices, which are equivalent to Boolean rings, closed under sum and product (Birkhoff 1940). They are also closed under continuous spatial transformations, including solid-body and Euclidean transformations. As such the algebras formalise the shapes, shape operations and spatial transformations that architects Extending the Algebras of Design Nexus Network Journal 5 utilise in manipulations of design representations. Aspects of design processes concerned with pictorial representations can therefore be formally defined as shape computations within these algebras (Prats et al. 2008; Paterson and Earl 2010). In practice, a design process is unlikely to be formalised according to any single simple algebra, Uij. It would, instead be formalised by a complicated Cartesian product incorporating many of the types of shape and non-spatial information that are used to inform and support the development of a design. Consideration of the representational enquiries and manipulations that take place throughout such a process gives rise to a subalgebra of the algebras of design. The extent of such a subalgebra cannot be defined prior to completion of a design process, without compromising the outcome. In some instances such a compromise is required, for example, as explored in (Dounas 2008) with respect to algebraic formalisations of building codes. However, in general, the constructivist nature of design processes mean that the subalgebra which formalises a particular process cannot be determined until after that process is complete (Stiny 1991). Rudi Stouffs and Ramesh Krishnamurti (2007) illustrate the potential for this algebraic representation, when applied to data intended to support a design process and all the various actors and agents involved. The examples reported illustrate the advantages of algebras of design over the point-set formalisations of design representations which underlie computer-aided design (CAD). The most prominent of these is that the algebras support both a reductionist and a constructivist approach to design development. The reductionist approach is commonly seen in CAD and fixes the structure of a design representation. This is beneficial for later stages of a design process when decisions have been made and certain aspects of the design have been fixed (Stacey and Eckert 2003). However, it can be detrimental for creative design since it necessitates that architects anticipate all future ways in which the parts of a design representation will be viewed and manipulated as development continues. Conversely, a constructivist approach does not fix the structure, and instead the parts of a design can be freely interpreted throughout the design process (Jowers and Earl 2012). Under a constructivist approach a design representation can accommodate the needs of all actors and agents in a design process, with its structure continuously changing in accordance with processes of enquiry and description. Such reinterpretation is a vital element in the exploration of a design problem, and the development of a design solution, and is believed to be a decisive component of innovative design (Suwa 2003). 3 – Non-rectilinear Embedding Despite the promise of algebras of design for supporting reinterpretation of design representations, there remain important technical issues which have not, to date, been investigated in sufficient detail. In particular, the (Stiny 2006) formulation of the algebras focuses on rectilinear shapes and takes advantages of the peculiarities of rectilinearity, with little consideration of general application to non-rectilinear geometric elements, such as the curves and surfaces used in the representation of the design of the David and Gladys Wright house in Fig. 3. Extending the Algebras of Design Nexus Network Journal 6 Fig. 3. The David & Gladys Wright house, decomposed into functional zones As discussed in (Jowers and Earl 2011), the most prominent of these peculiarities are concerned with the concept of embedding, which is the fundamental principle from which algebras of designs and shape computations are derived. The subshape relation, which applies a partial order over shape decompositions, is defined according to embedding: a shape A is defined to be a subshape of a second shape B, denoted A ≤ B, if all the geometric elements of A can be embedded in all the geometric elements of B. Other shape operations build on this definition of subshape, including shape identity: a shape A is defined to be identical to shape to a shape B, denoted A = B, if both A ≤ B and B ≤ A;