Analysis of shape grammars: continuity of rules
AAnalysis of shape grammars:continuity of rule applications
SAGE
Alexandros Haridis Abstract
A new method of analyzing rule continuity in computations with shape grammars is presented. Bybuilding on results in topology for shapes in algebras U i , the paper provides: (1) a general formulationof continuous rule applications, based on the continuity of the mappings that describe these ruleapplications, (2) conditions that a mapping must satisfy to be suitable for analyzing rule continuity,and (3) procedures for computing continuous topologies in a given computation. Keywords
Continuity, retrospective analysis, shape grammars, shape topology, design process
Analysis of a computation
A computation with a shape grammar is analyzed after recording all states of transformation of an initialshape. Analysis of rule continuity is a way of showing how the rules applied interact with parts of shapesin the course of a computation, what part-structures emerge from this interaction, and how part-structurescan be reconfigured, so that the rules used in the computation appear to apply in a continuous manner,without inconsistencies between those emerging part-structures. Continuity is a constructed feature ofshape grammar computations. It is meaningful only retrospectively.
Preliminary concepts
Mappings
In computations with shape grammars, rules apply to shapes to generate other shapes ina variety of ways. Mappings describe rule applications, by capturing in a formal notation how parts ofshapes change as a computation unfolds. Department of Architecture, Massachusetts Institute of Technology, USA
Corresponding author:
Alexandros Haridis, MIT Architecture-Computation, Room 10-303, 77 Massachusetts Avenue, Cambridge, MA 02139, UnitedStates.Email: [email protected]
Prepared using sagej.cls [Version: 2017/01/17 v1.20] a r X i v : . [ c s . F L ] D ec Journal Title XX(X)
A one-step rule application S ⇒ S (cid:48) , corresponds to a mapping f : S → S (cid:48) from the parts of S to parts of the new shape S (cid:48) generated by the rule application . A mapping describeshow all of S changes or transforms into the shape S (cid:48) as a consequence of the rule application. Mappingsbetween shapes have their own special properties and come with their own terminology. Here, I coveronly what is useful in the coming sections of this paper. Image and preimage If x denotes a part of the shape S , then f ( x ) is the (unique) image of x determined under f . The image of x is some part y of S (cid:48) , such that f ( x ) = y . When the part x is equal to S itself, we have f ( S ) = S (cid:48) .Recall that a shape, along with the set of all its parts, forms a partially ordered set (poset) whoserelation is defined by x ≤ y just when x + y = y , or equivalently x · y = x , where ≤ is a part relation(embedding) defined in the algebra that the shapes come from (Stiny, 2006). Technically speaking, amapping between two shapes can be understood as a mapping between their respective posets.In this paper, we are interested in mappings which preserve embedding orders between parts. Inparticular, a mapping f : S → S (cid:48) is called order-preserving (monotone) if for all parts x, y of S with x ≤ y , we also have f ( x ) ≤ f ( y ) in S (cid:48) . In this way, we obtain the highly desired property that the imageoperation under f preserves the embedding order of parts. By “ f is a mapping between shapes”, we willalways mean that f is order-preserving .If y is a part of the shape S (cid:48) , denote by f − ( y ) the largest part of the shape S whose image under f is embedded in y . The shape f − ( y ) is called the inverse image or preimage of y (Haridis, 2020). Thenotation f − ( y ) represents a shape and it should not be confused with the action of the inverse of themapping f (i.e. the mapping f − : S (cid:48) → S ). Just like with the image operation, the preimage operationis order-preserving, too.The shape f − ( y ) is taken from a “preimage set”, which is the set of all the parts of S embedded inthe part y under the mapping f . A part y of S (cid:48) has a preimage that is a shape, if and only if, the preimageset of y is nonempty. If, on the other hand, the preimage set is empty, then the preimage of y is said tobe undefined rather than being equal to the empty set (as, for example, in preimages of subsets in settheory). The empty set and the empty shape are categorically two different objects when it comes to thedefinability of preimages. Topology
A mapping between shapes works independently of the topologies assigned to them. Amapping is only a descriptive device, it tells how parts of one shape transform/map into parts of anothershape as a consequence of a rule application. The continuity of a mapping, on the other hand, is atopological issue and it is always studied relative to the topologies assigned to the shapes involved.Indeed, without topologies there is no such issue as continuity (at least not in a mathematical sense).A topology for a shape S is a finite set T of parts of S that satisfies the following three conditions:(1) The empty shape ( ) and S itself are in T .(2) The sum ( + ) of any number of parts in T is also in T .(3) The product ( · ) of any number of parts in T is also in T . Prepared using sagej.cls aridis The parts of the shape S that are members of a topology T for S are called open parts . The smallestpossible topology for a shape is the set { , S } , the indiscrete topology. There exists no largest finitetopology for a shape.A topology for a shape can be “transferred” or relativized to any part of the shape that can be recognizedfiguratively in it (no matter if this part is already open or not). The recognized part inherits a relativizedtopology, subject to the existing open parts in the topology for the original shape. This process ofrelativization is captured by the concept of subshape topology.In particular, let x be any part of the shape S and T a topology for S . The set T x = { x · C | C an open part in T } is a topology for the part x , called the subshape topology . The open parts in the set T x are formed bytaking products between the part x and the open parts of S . If a part x is already open in T , then obviously T x ⊂ T . If x is not open, then the open parts in the subshape topology T x for x , may not necessarily beopen in the original topology T .Any topology for a shape carries a special algebraic structure: a finite distributive lattice of parts, inwhich the top element is the shape itself, the bottom element is the empty shape, and sum and product ofopen parts play the role of join and meet, respectively. When we want to emphasize the algebraic aspectsof a topology for a shape S , we use the notation OS to refer to the lattice of open parts associated withthis topology. The order in the lattice is induced by the part relation associated with the shape S , and itdescribes how open parts are embedded in one another.Most of the time, one focuses on the set-theoretical aspects of topologies. In some cases, however, it ismore insightful to focus on the algebraic ones. Rule continuity in shape grammars is one such cases. Continuity of rule applications
Suppose a rule A → B is applied to a shape S under a transformation t to generate a shape S (cid:48) . Let f : S → S + be a mapping that describes the rule application S ⇒ S (cid:48) , in which f ( S ) = S + and the shape S + is some part of the shape S (cid:48) . Assume also that T and T (cid:48) are, respectively, the topologies for theshapes S and S (cid:48) . Then, the rule application is said to be continuous if the following two conditions aremet:(1) The shape t ( A ) that the rule recognizes in S , is open in T .(2) The mapping f ∗ : OS + → OS , defined as f ∗ ( D ) = f − ( D ) for every open part D in OS + , is alattice homomorphism.Condition (1) is the same as in Stiny (1994), and it is a way of guaranteeing that the left-hand side ofa rule is always a recognized (i.e. open) part. Condition (2) implies that the mapping f is a continuousmapping between the two shapes, S and S + , and it is based on the formulation of continuity given inHaridis (2020).Based on the above definition, the question of rule continuity becomes mainly a question of whetherthe mapping f that describes the rule application is continuous or not. The continuity of f is definedin terms of a homomorphism between two lattices (associated with the topologies for S and S + ) andgoes in the opposite direction of f —that is to say, in the opposite direction of the rule application that Prepared using sagej.cls
Journal Title XX(X) f describes. Besides having a lot of technical (mathematical) advantages that we discuss below, thisformulation recapitulates in a nice way, and in fact emphasizes, the retrospective nature of an analysisof rule continuity in shape grammars—analysis does not interfere with the forward action of a rule, untilonly after the fact (this is one of the merits of the original study on rule continuity in Stiny (1994)).The mapping f ∗ is defined backwards by using the preimage operation f − . The preimage is takenfor every open part in the lattice OS + ; this is the lattice of open parts associated with the subshapetopology for S + . The subshape topology is constructed by relativizing the topology T (cid:48) to the part S + .The appealing feature of this relativization is that it elucidates the structure of S that we wish to preservein a part of the shape S (cid:48) , and in doing so, it allows us to determine the exact relationships among the openparts in the topologies for the two shapes that make continuity possible.As a homomorphism, the mapping f ∗ : OS + → OS preserves all sums and products of open parts in OS + in this way: f ∗ ( D + E ) = f ∗ ( D ) + f ∗ ( E ) and f ∗ ( D · E ) = f ∗ ( D ) · f ∗ ( E ) It follows from (either of) these equations that f ∗ is order-preserving:If D ≤ E in OS + , then this implies f ∗ ( D ) ≤ f ∗ ( E ) in OS .In other words, f ∗ preserves the embedding order of the open parts determined by the structure assignedto the shape S + . Then, “smaller” open parts in OS + which are embedded in “larger” open parts, continue to be related in this way after f ∗ maps them backwards into OS . Open parts that “go together” in OS + continue as open parts that “go together” in OS via f ∗ , so that the open parts (and their relations)recognized in the topology for the shape S (cid:48) are already implied in the topology for the shape S . Thisgives the impression of a seamless structural transformation of the shape S to the shape S (cid:48) , as if the rulewas applied without breaks or inconsistencies amongst the part-structures of the two shapes—only, thiseffect is constructed retrospectively.Some additional properties of the two mappings, f and f ∗ , are briefly mentioned. In general, thepreimage mapping f ∗ is expected to be many-to-one; two or more open parts of the shape S + may havethe same preimage. However, depending on the shapes and the topologies involved, it may also be one-to-one in which case f ∗ is structure embedding (a one-to-one lattice homomorphism). By definition,the forward mapping f maps the shape S into the shape S + , that is, f ( S ) = S + . The preimage f − ( f ( S )) = f − ( S + ) is thus equal to S itself, and it follows that the homomorphism f ∗ preservesthe top element. However, it does not preserve the bottom element— f ∗ (0) is not necessarily equal to0. It is possible to encounter cases where both the top and the bottom elements are preserved, but thisdepends on the shapes, the topologies, and the mappings involved in an analysis. How to choose mappings
The choice of the mapping f : S → S + is key to analyzing rule continuity in a computation. Its exactform can vary depending on the shapes and the rules involved, and it’s very common to find ruleapplications which are continuous with respect to one mapping but not continuous with respect to another. Prepared using sagej.cls aridis I highlighted earlier that f is expected to be order-preserving. This provides an important guide forchoosing its exact form in an analysis. Some additional guides are now given.Let S and S + be any two shapes. Suppose f : S → S + is a mapping from the shape S to the shape S + and assume that OS and OS + are the lattices of open parts corresponding to the topologies for S and S + , respectively. If f ( x ) (cid:54) = 0 for every part x of S , then the mapping f is not continuous. Proof.
Suppose f is continuous. Then, by definition of continuity, the empty shape, which is an open partin OS + , must have a preimage f − (0) open in OS . But, by assumption, there is no part x of S for which f ( x ) ≤ , and so the preimage of the empty shape must be undefined. Thus, f cannot be continuous.This result builds on the observation that the empty shape is a point of discontinuity for any mapping f whose “net result”, or calculated output, is a nonempty shape, no matter which part of the shape S itis given as an input. Such a mapping cannot yield the empty shape, and so the empty shape—which is arequired member of every topology—is left out without a preimage. The following is a corollary of this.A constant mapping f : S → y maps every part of S into the same part y of a shape S + . If the part y is nonempty, then f is not continuous (immediate, from the above result). It follows that f : S → isthe only trivial (constant) mapping between shapes that is continuous.Let S and S + be any two shapes and suppose f : S → S + is a mapping from the shape S to the shape S + . If f (0) (cid:54) = 0 , then f ( x ) (cid:54) = 0 for every part x of S . Proof.
Suppose, for contradiction, that there is a nonempty part x of S for which f ( x ) ≤ . Then, ≤ x implies f (0) ≥ f ( x ) . But this violates the requirement that f is order-preserving. Thus, there is no suchpart x of S .As a corollary, if f (0) (cid:54) = 0 , then f is not continuous.Taking into account the aforementioned results, we now discuss how to choose specific forms for themapping f which are suitable for analyzing rule continuity.In principle, the form of f depends on the shape S + onto which f maps S . As suggested originallyin Stiny (1994), the shape S + is best found in production formulas that fix rule applications in shapegrammars.Production formulas are well-known in the literature of shape grammars. For example, ( S − t ( A )) + t ( B ) is a basic formula used since the inception of the formalism, to describe any one-step rule application S ⇒ S (cid:48) where a rule A → B applies to a given shape S under a transformation t to generate a newshape S (cid:48) (Stiny, 1975). It has universal applicability, in the sense that it can be used to describe any ruleapplication, for any rule, and for any given shape grammar (Gips and Stiny, 1980).The formula ( S − t ( A )) + t ( B ) corresponds to a mapping f : S → S (cid:48) , from a given shape S to a newshape S (cid:48) that the rule application generates, which is defined as f ( x ) = ( x − t ( A )) + t ( B ) Prepared using sagej.cls
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S S + t(B)0 (S – t(A)) + t(B) (S ⊕ t(B)) + t(A)S ⊕ t(A ⊕ B)S – t(A ⊕ B)S – t(A + B) t(A –
B)S � t(A ⊕ B) S � t(B – A) S ⊕ t(B)S ⊕ t(A + B)(S + t(B)) – t(A � B)(S – t(B)) + t(A) t(A)(S – t(B)) (S + t(B)) – t(A) (t(B) – S) + t(A)(t(B) – S) + t(A � B) (t(B) – S) + t(A – B)t(B) – SS – t(A – B) S � t(A + B)S � t(B) S – t(A � B) t(A � B)S – t(A)t(A + B) t(B – A)t(A ⊕ B) t(B)
Figure 1.
Lattice of production formulas characterizing a one-step rule application. Adapted from Krstic(2018). for every part x of S . If the shape t ( B ) is nonempty, then f ( x ) (cid:54) = 0 for all parts x of S , in which caseit follows from a preceding result that f cannot be continuous, no matter the topologies assigned to theshapes S and S (cid:48) .On the other hand, if t ( B ) is empty the mapping f becomes equivalent to f ( x ) = x − t ( A ) , which isindeed a suitable form for f . In particular, we have at least one part x of S for which f ( x ) = 0, namely, x = 0 (note, the suitability of a mapping does not guarantee continuity in every scenario; the continuityof a mapping in a particular scenario is always decided based on the topologies involved in it). Readersfamiliar with the original study on rule continuity in Stiny (1994) will be quick to recognize this mapping,since it was exclusively used in all examples worked out in the paper.There are other formulas, similar to ( S − t ( A )) + t ( B ) , that face similar discontinuity problems. Twoexamples are S + t ( B ) and S + t ( B − A ) (again, assuming that t ( B ) is not empty). The main differencebetween these formulas and a formula such as S − t ( A ) , is that the former ones take account of partsadded to the shape S (a characterization of those parts is in Appendix A), whereas the latter take accountof parts of S that are erased or left untouched.The way to enforce continuity then, that is to say “structure preservation,” is to preserve the structureassigned to S in some part S + of S (cid:48) which is equal to S or “something less” than S (intuitively, whateverit is in S (cid:48) that’s “greater” than S cannot be preserved, since it doesn’t even exist in the first place inadvance of a rule application; you don’t preserve what you don’t know of). And the mapping by whichthis should be achieved, must map at least one part of S to the empty shape. More technically, this leadsus to mappings f : S → S + where S + ≤ S for which there is at least one part x of S such that f ( x ) = 0 .Let’s look into other suitable forms for the mapping f , apart from the one already given, namely, S − t ( A ) .Production formulas that describe rule applications in shape grammars are generally formed bycombining the shapes S , t ( A ) , and t ( B ) —the three shapes that participate in (virtually) any ruleapplication—with the usual operations that determine shape algebras, namely, sum (+), difference (-),product ( · ), and symmetric difference ( ⊕ ). Different “ways of seeing” or selectively monitoring a (one-step) rule application can be devised, depending on how these three shapes are combined in operations. Prepared using sagej.cls aridis Krstic (2018) shows a host of formulas that characterize a one-step rule application—altogether, theyform the thirty-two element lattice shown in Figure 1. These formulas can be used as a practical startingpoint for exploring alternative forms for the mapping f to study rule continuity. However, not all formulaslead to suitable mappings. The suitable ones can be recognized based on the restrictions and guidelinesgiven in this section.In particular, there are three types of mappings that should be excluded as possibilities: order-reversing,constant mappings , and mappings that do not map any part to the empty shape.Examples of order-reversing mappings in Figure 1 are f : S → t ( B ) − S and f : S → ( t ( B ) − S ) + t ( A · B ) and others. Constant mappings are f : S → t ( A ) , f : S → t ( B ) , and f : S → t ( A − B ) .And examples of mappings that do not map any part to the empty shape (besides the ones alreadymentioned) are f : S → ( S − t ( B )) + t ( A ) and f : S → ( S ⊕ t ( B )) + t ( A ) .As a minor technical point, in all of these examples of mappings it is assumed that the participatingterms, e.g. t ( A ) or t ( B ) , are actually nonempty shapes. If any of these terms happens to be empty in aparticular scenario, then continuity may be feasible, but in a scenario like this, the chosen mapping wouldautomatically reduce to an equivalent (albeit simpler) mapping that doesn’t involve the empty terms.Table 1 contains a subset of the formulas in Figure 1, that provide suitable forms of mappings foranalyzing rule continuity. The table excludes the three aforementioned types of mappings. Table 1.
Mappings for analyzing rule continuity. SS − t ( A − B ) S · t ( A + B ) S − t ( A · B ) S − t ( B ) S − t ( A ⊕ B ) S · t ( B ) S · t ( A ⊕ B ) S − t ( A ) S − t ( A + B ) S · t ( B − A ) Continuity of a computation: examples
The continuity of a computation with shapes is easy to define in terms of the rules applied to generateshapes sequentially. Suppose a computation produces the sequence of shapes S , ..., S n , where S isthe starting shape. The computation is continuous for the topologies assigned on the shapes S , ..., S n whenever every rule application in this sequence is continuous.Analysis of rule continuity in a computation can be carried out intermittently as the computationunfolds, or at the end once the computation stops. It is also possible to designate multiple time frames Prepared using sagej.cls
Journal Title XX(X) in a single computation, and analyze rule continuity only in these designated time frames. No matterthe preference, analysis of rule continuity in a computation with shapes is a retrospective process anddoes not interfere with the forward action of rules—it only explains those actions from the standpointof topological continuity, after the fact. The details in this process become more clear in the followingdemonstrations.
Continuity with a single mapping
Let A → B be the following rulethat translates an L-shaped polygon along a horizontal axis (the cross-hairs indicate the originof a standard coordinate system in the Euclidean plane). Consider how the rule is applied undertransformations in this computationwhere these two shapesare matched in the first and second rule application, respectively.Suppose that the three shapes in the computation are denoted by S , S , and S . Analysis of rulecontinuity starts from the last shape, namely S , and moves backwards to the first shape, namely S . Thetopologies in Table 2 are defined when the parts of the shapes matched in the two rule applications, thatis, the shapes t ( A ) , are kept open—this is the first condition for continuity.The topologies in Table 2 are defined minimally. In particular, the set { S, t ( A ) , } is the smallestpossible topology which contains t ( A ) as an open part, for any shape S (the shape t ( A ) functions as a Prepared using sagej.cls aridis Table 2.
Topologies formed when just the parts t ( A ) are open.Shape Topology generator). The last shape in the computation is assigned the indiscrete, two-part topology since no ruleis applied to it.Let the mapping f : S → S − t ( A ) , defined in the usual way as f ( x ) = x − t ( A ) , for every part x of S , describe the action of rules in each step. For simplicity, let the topologies for the three shapes in thecomputation be denoted by T , T , and T , respectively.To decide if the computation is continuous, one must check if the second condition of continuity issatisfied for each rule application. For the two-step computation in Table 2, it works as follows. Thepreimages of all open parts in T must be open in T ; if not, the structure of T must be refined so that itincludes those preimages. Then, the preimages of all open parts in T —including any new parts obtainedafter a refinement in the previous step—must be open in T ; if not, the structure of T must be refined sothat it includes those preimages.In more detail, the topology T of the last shape consists of two open parts whose preimages under f are already open in T —in this case, T needs no updates for rule continuity to hold. The preimagesunder f of the open parts in the second topology T are shown in Figure 2. The rule application is notcontinuous with respect to the existing topology T , because this shapeis not an open part in T . Thus, topology T must be refined in this wayThe refined topology for T , and the given topologies for T and T , make the computation continuouswith respect to the given mapping f . Prepared using sagej.cls Journal Title XX(X)
As this example illustrates, analysis of rule continuity works recursively. Every topology in every step,is adjusted to reflect any updates/refinements made to topologies in previous steps, so that continuityholds throughout. It is possible to carry out the analysis in real time, right after a rule is applied, tosee step-by-step how each subsequent rule application that extends the computation forces a completestructural revision of the computation’s history.Apart from keeping just the part t ( A ) open in each step—the first requirement for continuity—onemay additionally choose to keep the complement of t ( A ) open, namely the shape S − t ( A ) . In this case,the computation becomes continuous with respect to the same mapping, but the topologies that make thishappen are different from those derived in the previous analysis—in particular, see Table 3.Still, the shapes t ( A ) and S − t ( A ) are not the only options available for analysis. One can chooseother shapes to keep open, such as the shape t ( B ) , which is the added piece in every rule application,or its complement S (cid:48) − t ( B ) , as well as other shapes formed in combinations of S , S (cid:48) , t ( A ) , and t ( B ) ,which are readily observed in rule applications according to interest or purpose. The same computationcan become continuous in alternative ways, depending on what parts one chooses to observe in the actionof rules. Analysis of rule continuity is inherently an observational process. Continuity with multiple mappings
Continuity can be studied not only with a single mapping but with multiple mappings, too, which can beused designedly at various stages in a computation. This is based on the mathematical idea that when twoor more mappings are chained, their composition is continuous so long as the component mappings areindividually continuous. In an analogous way, if a rule application corresponds to a mapping f : S → S (cid:48) and the rule application following after corresponds to a mapping g : S (cid:48) → S (cid:48)(cid:48) , then the two-step rule f -1 f -1 = f -1 == Figure 2.
Calculation of preimages for the open parts in the second topology in Table 2. The preimages arecalculated under the mapping f ( x ) = x − t ( A ) . Prepared using sagej.cls aridis Table 3.
Continuous topologies for the shapes in Table 2, formed under the mapping f : S → S − t ( A ) whenthe parts t ( A ) and S − t ( A ) are open.Shape Topology Mapping f : S → S − t ( A ) application is continuous, if and only, if the two rule applications corresponding to mappings f and g are individually continuous. The effects of chaining multiple mappings in a single analysis are telling inlonger computations.Consider how the following two rulesand the starting shapeare used in this computationSuppose in the first two steps, the rule applications correspond to the mapping f : S → S − t ( B ) ,defined as f ( x ) = x − t ( B ) for every part x of S . Then, in the third step the rule application correspondsto the mapping g : S → S − t ( A ⊕ B ) , defined as g ( x ) = x − t ( A ⊕ B ) for every part x of S . Then, inthe fourth step the rule application corresponds to the mapping f , and finally, in the fifth and the rest ofthe steps the rule applications correspond to the mapping g . In all steps, only the mandatory parts t ( A ) Prepared using sagej.cls Journal Title XX(X) are kept open. The topologies that make the complete computation continuous, with respect to both f and g , are given in Table 4.The analysis alternates between the two mappings f and g for two main reasons. First, to illustrate theflexibility and generality of the proposed approach to rule continuity, and the ease with which one candefine and use different mappings to analyze the same computation. Second, to highlight a technical pointin the definition of continuity, according to which if f is some mapping that describes a rule application S ⇒ S (cid:48) , then it must map the shape S into the shape S (cid:48) , that is, f ( S ) = S (cid:48) . In particular, in the third ruleapplication in Figure 3, the mapping f cannot be used because at that step f ( S ) (cid:54) = S (cid:48) , and the mapping g is used instead.As the analysis starts from the last shape in the computation, the structures formed with respect to g in later steps, propagate their open parts backward until the very first shape, intermixing along the waywith the structures formed with respect to f . Obtained retroactively, the structures give the impression ofa seamless forward computation, whereby topologies of shapes in earlier steps look as if they anticipatethe progression of rule applications in subsequent steps and the structures that emerge as a consequenceof their action.As a minor technical point, notice how for each shape in Table 4 a total order is defined amongst itsopen parts; that is, any two of its open parts are comparable (one of the two is embedded in the other).This is a consequence of keeping just the parts t ( A ) open in each step of the analysis. The reader maywant to analyze the same computation when S − t ( A ) , in addition to t ( A ) , is open. The topologies willbe significantly larger, with partial orders defined in all cases (in fact, all topologies will form Booleanalgebras, just as it happens in Table 3). Computability of continuous topologies
Assuming that the shapes generated by a shape grammar have been recorded unanalyzed for a certain(finite) number of steps, analysis of rule continuity amounts to two main computational tasks.(1) The generation and refinement of a shape topology by given parts, and(2) The calculation of preimages of open parts under given mapping(s).The two tasks are carried out for each rule application, starting from the last rule application in acomputation until the first one, working backwards recursively.
Prepared using sagej.cls aridis Table 4.
Continuous topologies formed under the mappings f : S → S − t ( B ) and g : S → S − t ( A ⊕ B ) ,when the shapes t ( A ) are open.Shape Topology Mapping f : S → S − t ( B ) f : S → S − t ( A ⊕ B ) f : S → S − t ( B ) f : S → S − t ( A ⊕ B ) For the first task, since the topologies we deal with are finite, their computability is guaranteed. Aprocedure for generating a topology for a shape, or refining an existing one by given parts, is describedin Haridis (2020). The parts that function as generators of topologies in an analysis, are exactly the partsthat we choose to keep open—say t ( A ) only, or t ( A ) and some other part(s) of interest. In this way,structure is directly tied to observation, the latter driving the creation or refinement of the former.For the second task, the calculation of preimages is mostly straightforward because the mappingsare given in closed form (some extra care is needed when rule applications are irreversible). As thedemonstration in the previous section shows, an existing topology for a shape may need to be refined toaccommodate any newly calculated preimages which are not already open in it. Refinement is nothingmore than a re-evaluation of sum and product between all open parts, old and new ones.A topological concept that is useful for calculating preimages is the concept of a basis . Recall thateach topology for a shape comes with a unique minimal set of basis elements that describes the topology, Prepared using sagej.cls Journal Title XX(X) i.e. generates all its open parts. This minimal set is called a reduced basis (Haridis, 2020). Instead ofcalculating preimages for all open parts in a topology, which may be a time consuming task with largertopologies, one can instead calculate the preimages only for the basis elements. However, this may notyield calculational benefits in every case.For example, for each topology in Table 4, the set of basis elements is equal to the topology itself—there aren’t any open parts that can be formed as sums of other open parts, in any of these topologies.On the other hand, consider a hypothetical scenario in which the two shapes participating in this last ruleapplication from Table 4acquire the topologies shown in Table 5. Suppose S ⇒ S (cid:48) denotes this rule application. To establishcontinuity, the topology for S must be refined in terms of the preimages of the open parts in the topologyfor S (cid:48) (e.g. under the same mapping as in Table 4). Instead of calculating the preimages for all eight openparts of the shape S (cid:48) , one can instead calculate the preimages only for the basis elements—there are fourbasis elements in total, labelled 1 through 4 in the table.More generally, every open part in a topology for a shape is formed as a sum of basis elements andthe preimage operation preserves those sums. When we refine the topology for a shape S in terms ofthe calculated preimages of the basis elements in a topology for shape S (cid:48) , the refinement (i.e. the re-evaluation of sums and products) automatically accounts for the preimages of the rest of the open partsin the topology for S (cid:48) , without calculating them explicitly. Table 5.
Two examples of topologies for the last two shapes in Table 4. The numbers in the second topologyindicate its basis elements.Shape Topology1 2 3 4
Prepared using sagej.cls aridis Discussion: continuity in design
The study of rule continuity in shape grammars, is an important approach toward understanding keyproperties of computational processes independently of the design domain or the area of application.In this sense, rule continuity can be understood as a research problem that contributes to a theory ofcomputation in design.Most computations we encounter in design are inherently continuous. In computations where the rulesdo not apply directly to line elements (or higher dimensional elements, like planes and solids), but applyto point-set based representations of them which are decided in advance and their purpose is to controlshapes indirectly, there is little to no room left for discontinuities and inconsistencies in how shapes aredescribed as a computation unfolds: continuity becomes a built-in feature of such rules. Continuous ruleapplications, however, that only apply to predefined substitutes of shapes (representations), and that donot take advantage of their appearance in the course of a computation—what we can see figuratively inthem—are essentially rules that are “blind” to how shapes look. Such rules are always bereft of noveltyand surprise from the standpoint of calculating.
Acknowledgements
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Notes
1. Mappings can also describe what a rule A → B does, i.e. how A changes to B in the definition of the rule. Suchmappings are best captured by a more general framework known as schemas (Stiny, 2006).2. A mapping between two shapes can be order-reversing, too, depending on the exact form of f . Contrary tonumber algebras, shape algebras do not come with absolute complements. We obtain order-reversing mappingsby using the operation of relative complement. Here, we opt for order-preserving mappings because of theirappealing behavior in general, but also because we find the order-reversing ones as purely pedagogical in nature,having no clear practical role in analyzing computations with shapes.3. The mapping f : S → is technically a valid candidate, but practically vacuous in the context of rule continuity. References
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Environmentand Planning B: Planning and Design 24 (3), 359–384.Krstic D (2018) Grammars for making revisited. In
Design Computing and Cognition ’18 (J. Gero, Eds.), 479–496,Springer.Stiny G (1994) Shape rules: closure, continuity and emergence.
Environment and Planning B: Planning and Design21 (7), s49–s78.Stiny G (1996) Useless rules.
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Prepared using sagej.cls Journal Title XX(X)
Stiny G (2006)
Shape: talking about seeing and doing . Cambridge, MA: The MIT Press.
Supplemental material
A. Parts that a rule adds