A Symbolic Model for Graphical Emergence1 2

John S. Gero and Jos�e C. Damski

Key Centre of Design Computing

Department of Architectural and Design Science

The University of Sydney, NSW 2006 Australia

email : fjohn,joseg@arch.su.edu.au

Abstract

This paper presents a shape representation at the symbolic level that extends the

properties of shape in 2D and 3D using the same formalism. This representation is based

on the concept of in�nite maximal lines and planes. Constraints on their properties are

used to de�ne shapes and objects. A process model of shape emergence which relies on

this representation is developed. Data-driven and hypothesis-driven approaches to shape

and object emergence are demonstrated.

1 Introduction

In an earlier paper Gero and Yan (Gero & Yan 1994) presented an approach to shape emergence

using symbolic reasoning. That work was limited to two-dimensional shapes. Those ideas were

extended directly in three-dimensions in Gero and Damski (Gero & Damski 1994). Here we

generalize those ideas to produce a uniform approach.

In this paper we use shapes as a 2D graphic and objects as a 3D graphic. Shapes and

objects play an important role in presenting ideas, concepts and possible physical worlds. They

are the way we begin to understand the visual world our visual sense brings to us (Marr

1982). Shapes and objects play a dominant role in various design domains and particularly in

architectural design where they are used not simply as the representation of an idea but also

as a representation open to reinterpretation. This reinterpretation is the basis of emergence. A

property that is not represented explicitly is said to be an emergent property if it can be made

explicit.

In the conceptual aspects of designing this reinterpretation of what has been drawn appears

to play a critical role (Sch�on 1983, Sch�on & Wiggins 1992). It provides opportunities for

designers to conceptualise what has been drawn di�erently from what was intended when it

was drawn.

Current CAD systems have not been used extensively during conceptual design for a va-

riety of reasons, one of which is that they freeze the object being represented and do not

allow any other interpretations. Most CAD systems use geometric representations of objects

based on line segments and their endpoints. Line segments are grouped together to form a

plane segment. Plane segments are grouped together to represent an object. Graphical emer-

gence is concerned with �nding other shapes and objects derivable from the initial or primary

1Parts of this paper have appeared in (Gero & Damski 1994) and (Gero & Yan 1994)2To appear in the journal Design and Planning: Environment and Planning B

shape/object. Graphical emergence clearly plays an important role in those design domains

which use graphics to represent concepts. Graphical emergence is the basis of a number of

visual design approaches (Grunbaum & Shephard 1987).

Object emergence has been studied for some time (Gottschaldt 1926, Reed 1974, Kanizsa

1979) and is a recognised phenomenon experienced by virtually all humans. Symbolic models

related to shapes and to a lesser extent to object emergence have been presented extensively

by Stiny (Stiny 1980, 1981, 1986, 1990 ), by Krishnamurti (Krishnamurti 1980, 1981) and

by Krishnamurti and Earl (Krishnamurti & Earl 1992). Tan (Tan 1990) presented a limited

approach to shape emergence as have Edmonds and Sou� (Edmonds & Sou� 1992).

In this paper we present a general and uniform symbolic representation which assists in the

discovery of emergent graphics, for both shapes and objects. We will restrict ourselves to closed

shapes and objects of a kind which are particularly useful in architectural design. Shapes in 2D

are based on the concept of in�nite maximal lines, an extension of Stiny concept of maximal

lines, and objects in 3D on in�nite maximal planes.

2 A Symbolic Representation for Graphical Entities

2.1 Introduction

In order to formalize the concepts of objects at a symbolic level the following de�nitions are

used3.

De�nition 1 (Ent) Let ent ei be a symbol to represent a geometrical entity, such as point,

line and plane.

De�nition 2 (Universe of discourse) The universe of discourse U is a non-ordered set of

ei represented as: U = fe1; e2; :::; eng

De�nition 3 (Relation) Let R be a relation between two ents. Given two ents ei and ej, we

de�ne four relations:

� Parallel - represented as ei k ej.

� Skew - represented as ei � ej.

� Orthogonal - represented as ei ? ej .

� Coincident - represented as ei = ej.

De�nition 4 (Expression) Let E be a expression with the following characteristics:

� a relation is a predicate symbol with truth value True or False but not both;

� a expression is a W� - Well formed formula in propositional logic with predicates as

de�ned in Relation.

De�nition 5 (Intersection1) Let the symbol ijk (or the equivalent ikj) denote intersection1

according to the following formulas:

3All terms used are not related to Euclidian geometry, unless stated otherwise.

2

ej � ek ! ijkej ? ek ! ijkej k ek ! :ijkej = ek ! :ijk

with relations:

� Parallel - represented as ijk k ilm

� Skew - represented as ijk � ilm.

� Orthogonal - represented as ijk ? ilm .

� Coincident - represented as ijk = ilm.

ijk and ijl are called co-planar intersections in the ent ej.

De�nition 6 (Intersection2) Let the symbol ijkl (or its equivalents ijlk, ikjl, iklj, iljk and

ilkj) denote intersection2 according to the following formulas:

ijk � ikl ! ijklijk ? ikl ! ijklijk k ikl ! :ijklijk = ikl ! :ijkl

with relations:

� Coincident - represented as ijkl = imnp.

De�nition 7 (Adjacency) Given an intersection2 ixyz it is said to be adjacent to other in-

tersection2 ixyw if they have two indices in common.

From these de�nitions we can derive some general properties of ents, such as:

R1: ej k ek ^ ek k ep ! ej k epR2: ej ? ek ^ ek k ep ! ej ? epR3: ej k ek ^ ek � ep ! ej � epR4: ej ? ek ! ej � ekR5: ej � ek $ ijkR6: ei = ej $ ei k ejR7: ijk = ijl ! ijk = iklR8: iijk = iklm ^ iklm = iopq ! iijk = iopqR9: iij = ikl ^ ikl = iop ! iij = iop

2.2 Representation of Segments, Regions, Shapes and Objects

In this section we de�ne the elements which can be built with the de�nitions in the previous

subsection.

De�nition 8 (Segment) A segment is denoted by two intersection1s iij and iik, such that

they have one ei in common.

3

De�nition 9 (Region) A region is de�ned by two (or more) non-parallel ents and limited by

their intersections.

There are two types of regions: open and closed. A region is said to be closed if it is limited

only by ixy, and open otherwise.

In this article we are concerned only with closed regions. The general representation of

regions for a given set of n non-parallel ents e1; e2; :::en is a sequence of ixy which has x in

common with all elements of the sequence. This sequence is represented by (ixy1 ; ixy2; :::; ixym)

in an m-sided region. The general declaration for a closed region can now be presented as:

� I is a set of coplanar intersections (ixy1; ixy2; :::; ixym).

� An intersection ixyz is obtained for each pair (ixy, ixz) in I.

� An intersection ixyz is said to be adjacent to ixyw

� The connection between two adjacent ixyz is part of ixy.

� A closed region is a circuit4 with an ordered sequence of adjacent ixyz without revisiting any

ixy except the �rst.

We use ents as the representation primitive to construct a symbolic representation of objects

to support object recognition by symbolic reasoning. The de�nition of objects follows.

De�nition 10 (Shape and Object) A shape (2D) and an object (3D) are arrangements of

ents ei in a �nite universe U .

Using ents as representation primitives, the general form of the symbolic representation of

object/shape is:

O = fN ; constraintsg (1)

where N is the cardinality, i.e., the number of ents constituting object/shape O, and where the

constraints specify properties resulting from the ents, which must be present in O.

2.3 Closedness

2.3.1 Closedness of Shapes

There are two types of shapes: open and closed. A closed shape is composed of segments,

such that each ixy of each segment occurs with only one other of the composing segments.

This can be declared as:

� A closed shape is a set of segments where each intersection ixy is shared by exactly two

segments.

An open shape is not a closed shape.

4treating ijkl as nodes and ipq as arcs in a graph

4

2.3.2 Closedness of Objects

Similarly, there are two types of objects: open and closed. A closed object is composed of

closed regions, such that each intersection ixy of each closed region occurs with only one other

of the composing closed regions.

This can be declared as:

� A closed object is a set of closed regions where each intersection ixy is shared by exactly two

closed regions.

An open object is not a closed object.

2.4 Terminology

A primary graphic is a set of closed shapes or objects that is represented explicitly initially

and thus can be input and manipulated by specifying their properties. An emergent graphic

is a shape or an object that exists only implicitly in relation to the primary graphic, and is

never explicitly input and is not represented at input time. Figure 1 shows examples of primary

shapes (a), emerged shape (b), primary objects (c) and an emergent object (d). The process

of recognizing emergent shapes and objects from primary ones is called graphical emergence.

(a) (b)

(c) (d)

Figure 1: (a) Primary shape, (b) emergent shape, (c) Primary objects and (d) emergent object

2.5 Mapping to Geometry - 2D

Shapes is 2D are represented as a set of straight line segments, i.e., by the endpoints of each

line segment. Two colinear line segments can be extended to a maximal line (Stiny 1980).

5

Maximal lines can be extended to extended maximal lines and then to in�nite maximal

line as shown in Figure 2 (Gero & Yan 1994). These extensions allow us to represent the

primary shape as a set of in�nite maximal lines with geometrical and topological contraints.

There are four mappings between our symbolic representation and geometry:

1. ent is mapped as an (geometric) in�nite maximal line.

2. segment is mapped as a geometric line segment.

3. ijk is mapped as a geometric point de�ned by the intersection of two lines.

4. iijk there is no mapping in 2D.

infinite maximal line:extended maximal line: P

line segment: PP

P

P

P

P

P

1

2

3

4

5

6

1 P3 P2 P4 P5

P6

1 P P4 5 P6

1 P6

maximal line: P

Figure 2: Line type de�nitions

Constraints on 2D shapes allow us to derive a new property in addition to R1-9 de�ned above,

as R10 below.

R10: ej ? ek ^ ek ? ep ! ej k ep

2.6 Mapping to Geometry - 3D

A conventional way to represent an object is to use point coordinates as primitives. In this

representation, a plane segment, which is bounded by line segments, can be described by the

coordinates of the line segments' endpoints. An object as a set of plane segments is represented

through a set of point coordinates and its bounding lines.

An ent can be mapped to an in�nite plane in a geometrical sense.

There are four mappings between our symbolic representation and geometry:

1. ent is mapped as a geometric plane (in�nite).

2. region is mapped as geometric plane segment as shown in Figure 3.

3. ijk is mapped as a geometric line de�ned by the intersection of two planes (or segments),

as shown in Figure 4.

4. iijk is mapped as a geometric point de�ned by two lines (each one de�ned by intersection

of two planes), as shown in Figure 4(a).

6

Note: Other geometric points have no mapping in our symbolic system.

Constraints on 3D objects allow us to derive new properties in addition to R1-10 de�ned

above, as R11, R12 and R13 de�ne below.

R11: ijk � ikl $ ijklR12: ej k ek ^ ej � ep ! ijp k ikp

5

R13: ijk = ijl ! ijk k ijl

p

pp

p

j

i

ii ikil

k

l ij

i

Figure 3: A closed plane segment/region, all pi shown.

iik

iji

ijk

ijki

p

p

pi

j

k

k

jki

P

Pj

iij

Pi iik

iij = i

ik = ijk

j

Pi

P

Pk

(a) (b) (c)

Figure 4: Intersections of three planes pi; pj and pk: (a) ijk � iik ! iijk ; (b) iij k iik; iik k ijk ;(c) iij = iik; iik = ijk

2.7 Constraints

There are two classes of constraints on ents of interest here: topological constraints, on intersec-

tions, and geometrical constraints, on their slopes. Representation (1) of an object O becomes

representation (2).

O = fN ; topological constraints, geometrical constraintsg (2)

5Note that the inverse is not always true

7

2.7.1 Topological Constraint

Topological constraints concern the structures within which intersections and regions are orga-

nized. They are represented as groups of ents and their intersections. There are three kinds of

intersection groups: ordinary groups, adjacent groups and enclosed groups, which respectively

specify three kinds of topological structures, organizing intersections and regions in di�erent

ways.

The most general representation is the ordinary group, which is represented by a pair of

round parenthesis: \(,)". The elements of these groups have intersection properties and are not

constrained by any other association such as adjacency or closure. The order of the intersections

in an ordinary group is of no signi�cance.

Intersections ixy can produce new intersections ixyz. Groups of intersections ixyz can be

represented as an adjacent group, represented by a pairs of angle brackets: \<;>". Two

intersections ixyz; ixyw are adjacent if they share two indices xy in common. It is stipulated

here that the �rst and the last intersections in an adjacent group are adjacent to each other.

Obviously, the order of intersections in an adjacent group is signi�cant. The same set of

intersections ixyz with a di�erent adjacent order represents di�erent sets of regions. An enclosed

group, represented by pairs of square brackets: \[,]", is a subcase of an adjacent group which

must satisfy the object closure constraint.

2.7.2 Geometrical Constraints

The geometrical constraints on any two ents and their intersections are concerned with their

properties as de�ned in Section 2.1.

2.8 Classes of Polyline Shapes

From the symbolic representation in representation (2), we can see that a shape is de�ned and

represented by three attributes: cardinality, topological constraints and geometrical constraints.

Therefore, shapes can be distinguished and classi�ed according to their di�erences in these three

attributes.

According to their cardinality, shapes are classi�ed into N-line shapes. According to whether

they satisfy certain topological constraints, N-line shapes can be classi�ed into unbounded

polyline shapes and bounded polyline shapes. Bounded polyline shapes are of the interest in

this paper. A partial taxonomy of shapes according to these constraints is shown in Figure 5.

2.9 Classes of Objects

From the symbolic representation in representation (2), we can see that an object is de�ned and

represented by three attributes: cardinality, topological constraints and geometrical constraints.

Therefore, objects can be distinguished and classi�ed according to their di�erences in these three

attributes.

According to their cardinality, objects are classi�ed into N-plane objects. According to

whether they satisfy certain topological constraints, N-plane objects can be classi�ed into un-

bounded polyplane objects and bounded polyplane objects.

According to the number of planes constituting the topological constraints, objects can be

classi�ed into four-sided objects, �ve-sided objects, ..., M-sided object, as in Figure 6. Some

8

three-line shapes

cardinality = 3

three intersections

triangles

n-line shapes

four intersections

only two oppositesides are parallel

four-line shapes

four-sided shapes

trapezoids

paralellograms

sides are paralleltwo sets of opposite

rectangles

square

two adjacent sides arethe same length

are perpendiculartwo adjacent sides

... cardinality = n

Polyline Shapes

cardinality = 4

Figure 5: One classi�cation of polyline shapes based on the application of constraints

of them can be further classi�ed according to di�erent geometrical constraints. In the �ve-

sided-plane objects in Figure 6, a pyramid can be a triangular prism if two sides are parallel; a

triangular prism can be further classi�ed into right or oblique according as the parallel planes

are perpendicular or oblique to the other planes (geometrical constraints). A parallelepiped

is a prism where each side is parallel to one other. It may be right or oblique according to

geometrical constraints. A cube is a parallelepiped with three pairs of planes parallel to each

other and orthogonal to each adjacent plane, with the geometrical constraint that all edges are

the same length.

The symbolic representation for objects in Figure 6 (assuming closed objects) are:

Four-plane Objects (pyramid with triangular base):

O1 = f4; [e1; e2; e3; e4]g

Five-plane Objects:

O2 = f5; [e1; e2; e3; e4; e5]g

Pyramid with four-sided base:

O3 = f5; [e1; e2; e3; e4; e5]; i123 = i124; i124 = i234g

Pyramid with rectangular base:

O4 = f5; [e1; e2; e3; e4; e5]; i123 = i124; i124 = i234; i25 k i45; i15 k i35g

Triangular Prism:

O5 = f5; [e1; e2; e3; e4; e5]; e1 k e2; i35 k i45gSix-plane Objects (polyhedron with six faces):

9

4 Plane Objects

cardinality = 4

Prism

o1

5 Plane Objects

2 planes are parallel

4 planes intersect at same point

pyramid with rectangular base o

o

o2

3

4

o5

6 Plane Objects

parallelepiped oblique

parallelepiped right

cube

o

o

o

6

7

8

Polyplane Objects

Figure 6: One classi�cation of polyplane closed objects

O6 = f6; [e1; e2; e3; e4; e5; e6]gParallelepiped (oblique):

O7 = f6; [e1; e2; e3; e4; e5; e6]; e1 k e2; e3 k e4; e5 k e6g

Parallelepiped (right):

O8 = f6; [e1; e2; e3; e4; e5; e6]; e1 k e2; e3 k e4; e5 k e6; e1 ? e3; e3 ? e5; e1 ? e5g

3 A Process Model of Graphical Emergence

3.1 Introduction

Psychologists have reported that, when preceiving a picture visually, a person attaches to it a

certain organization (Granovskaya et al. 1987). This organization involves dividing everything

in the visual �eld into a �gure and a background, and grouping elements in the �gure into

structures. Thus, shape recognition involves two steps: (1) isolating a �gure from a background;

and (2) restructuring elements of the �gure.

However, shape recognition involving the above two steps is not directly applicable for emer-

gent shape recognition which commences with a structured �gure rather than an unstructured

background. Emergent shape recognition restructures an already structured �gure. It is more

di�cult to build a new structure from an already structured �gure than from an unstruc-

tured background because the existing structure results in �xation (Purcell & Gero 1991) that

encumbers the establishment of a new structure.

Emergent shape and object recognition involves relaxing or \unstructuring" existing con-

straint structures, deriving new constraints, and restructuring constraints. A process model for

10

2D-shape emergence and to 3D-object emergence is presented in Figure 7.

Unstructuring GraphicsProcess

Graphical DiscoveryProcess

Primary Graphic

Unstructured Graphics

Emerged Graphics

Figure 7: A process model for graphical (shape and object) emergence

3.2 Graphic (Shape and Object) Unstructuring

Graphic (shape and object) unstructuring makes shapes and objects which are explicitly rep-

resented, become implicit. When they are explicitly represented, the behaviour of ents which

de�ne them are constrained by intersection groups. When constraint groups are destroyed or

relaxed, the explicitly represented shape or object becomes implicit. Graphic unstructuring

transfers structured representations of the graphics into unstructured representations of graph-

ics. The aim of graphic unstructuring is to remove or overcome �xation caused by the constraint

structures of primary graphics.

Based on the symbolic representation of graphics we propose, unstructuring is carried out

by a single operation: ungrouping lines (for shapes) or planes (for objects) using ents con-

cepts. This ungrouping operation is carried out by the operator Ug, which relaxes topological

constraints on graphics by making adjacent and enclosed groups become ordinary groups.

As has been described in Section 2.7, di�erent groups of ent intersections represent di�erent

graphics, and an intersection does not exist if the ents come from di�erent groups. The operator

Ug relaxes topological constraints on ents to require only the existence of ents themselves. For

example, if we apply the operator Ug to:

Ug (f10; [ea; eb; ec; ed; ee; ef ]; [ea; eb; ec; ee; eg; eh]; [eb; ed; ee; ef ; ei; ej];

[eb; ee; eg; eh; ei; ej]; :::g)

we have:

f10; (ea; eb; ec; ed; ee; ef ; eg; eh; ei; ej); :::g (3)

This operation is instrumental in enabling us to �nd new ent intersections and in emerging

new graphics.

11

3.2.1 Shape Emergence

Shape emergence is the process of discovering possible shapes that were not explicitly repre-

sented in the primary shape. It consists of two steps: constraint derivation and shape discovery.

Constraint or property derivation is performed using the properties de�ned as rules, Rn, de-

scribed in Section 2.1.

There are two strategies for shape discovery: hypothesis-driven search and data-driven

search. In hypothesis-driven search a shape schema is prede�ned in the representation and the

unstructured shapes are searched to determine matches to the hypothesised schema. In this

mode of shape emergence only prede�ned shapes can be discovered.

In data-driven search a cue (also called a feature) or a combination of cues is used to

traverse the representation of the unstructured shapes until some stopping rule is satis�ed. In

this work, closure is used as the stopping rule. Shapes discovered in this manner do not need

a prede�ned schema. In the way we have set up data-driven search it may considered as form

of meta-hypothesis-driven search. The di�erence between the two is that hypothesis-driven

search commences with a de�nition of the shape or object being searched for whilst in data-

driven search only characteristics of the data in the representation which satisfy some simple

requirements are used to produce shapes or objects without the need to de�ne any shapes or

objects. In data-driven search, shapes can be discovered by applying the following adjacency

reasoning rules RSn to search intersections of ents until an intersection group is formed, which

represents a shape.

RS1: (ixy)!< ixy >

RS2: (ixy; ixz)!< ixy; ixz >

RS3: (ixy; ix1y1) ^ ixy = ix1y1 !< ixy; ix1y1 >

RS4:6 ixy 2 A ^ ix1y1 62 A^ < ixy; ix1y1 >!< A; ix1y1 >

Where RS1 is the starting rule for the data-driven search. Rule RS2 takes two intersections

and transforms them into an adjacent intersections. Rule RS3 examines an exception to rule

RS2, where two intersections are coincident. Finally, rule RS4 adds new intersections to an

existing group of adjacent intersections.

3.2.2 Object Emergence

In a similar manner to shape emergence, object emergence is the process of discovering possible

objects that were not explicitly represented in the primary object. It consists of two steps:

constraint derivation and object discovery. Constraint or property derivation is performed

using the properties de�ned as rules Rn described in Section 2.1.

There are two strategies for object discovery: hypothesis-driven search and data-driven

search. In hypothesis-driven search an object schema is prede�ned in the representation and

the unstructured objects are searched to determine matches to the hypothesised schema. In

this mode of object emergence only prede�ned objects can be discovered.

In data-driven search a cue (also called a feature) or a combination of cues is used to

traverse the representation of the unstructured objects until some stopping rule is satis�ed. In

this work, closure is used as the stopping rule. Objects discovered in this manner do not need

a prede�ned schema. In data-driven search, objects are discovered by applying the following

adjacency reasoning rules ROn to search intersections of ent until an intersection group is

formed, which represents an object.

6In this rule, A denotes < ::: > a set of adjacent ixy.

12

RO1: (ixyz)!< ixyz >

RO2: (ixyz; ixyw)!< ixyz; ixyw >

RO3: (ixyz; ix1y1z1) ^ ixyz = ix1y1z1 !< ixyz; ix1y1z1 >

RO4:7 ixyz 2 A ^ ix1y1z1 62 A^ < ixyz; ix1y1z1 >!< A; ix1y1z1 >

Where RO1 is the starting rule for the data-driven search. Rule RO2 takes two intersections

and transforms them into an adjacent intersections. Rule RO3 examines an exception to rule

RO2, where two intersections are coincident. Finally, rule RO4 adds new intersections to an

existing group of adjacent intersections.

4 Examples

4.1 Shape Emergence

In this section we present an application of this formalism. Figure 8 shows the centre of

Casalecchio town a south-western suburb of Bologna Italy, designed by James Stirling and

Michael Wilford (Anonymous 1994, pp 118).

Figure 8: Casalecchio plan study designed by James Stirling and Michael Wilford

The 2D view of it is shown in Figure 9. The town square is extracted, re-represented using

in�nite maximal lines and is shown in Figure 10.

The primary shapes S1, S2 and S3, as shown in Figure 10, are de�ned as:

S1 = f6; [ea; ec; ed; ee; ef ; eh]; ea k ed; ec k ee; ef k eg; eh k eig

S2 = f6; [ea; eb; ed; eg; ej; el]; ea k ed; eb k ej; ef k eg; ek k elg

S3 = f6; [eb; ec; ee; ei; ej; ek]; ec k ee; eb k ej; ek k el; eh k eig

i.e., the entire primary shape PS is:

PS = f12; [ea; ec; ed; ee; ef ; eh]; [ea; eb; ed; eg; ej; el]; [eb; ec; ee; ei; ej; ek];

ea k ed; ec k ee; ef k eg; eh k ei; eb k ej; ek k elg

7In this rule, A denotes < ::: > a set of adjacent ixyz.

13

Figure 9: Architectural plan

ee

e

c ee

e

ee

e

e

S

S

S

e

a

b

e

d

e

f

g

h

i

j

k l

1

3

2

Figure 10: Primary shapes S1, S2 and S3, and in�nite maximal lines ea to el

Ungrouping S1; S2 and S3, we have:

PS = f12; (ea; eb; ec; ed; ee; ef ; eg; eh; ei; ej; ek; el); eb

ea k ed; ec k ee; ef k eg; eh k ei; eb k ej; ek k elg (4)

Applying rules R1 to R10, we get:

PS = f12; (iab; iac; iae; iaf ; iag; iah; iai; iaj; iak; ial; ibc; ibd; ibe; ibf ; ibg; ibh; ibi; ibk; ibl;

icd; icf ; icg; ich; ici; icj; ick; icl; ide; idf ; idg; idh; idi; idj; idk; idl; ief ; ieg; ieh;

iei; iej; iek; iel; ifh; ifi; ifj; ifk; ifl; igh; igi; igj; igk; igl; ihj; ihk; ihl; iij; iik; iil; ijk; ijl);

ea k ed; ec k ee; ef k eg; eh k ei; eb k ej; ek k elg (5)

Some intersections ixy from representation (5) are shown in Figure 11. From the representa-

tion of primary shape shown in representation (5) we can emerge shapes using the two di�erent

types of search. These two types are shown in the following subsections.

14

iaj

ibd

iijl

ijk

icj

ici

ich

ide

gh

iac

iik

idf

iaf

iagifh

ee

e

c ee

e

ee

e

e

S

S

S

e

a

b

e

d

e

f

g

h

i

j

k l

1

3

2

ibe

Figure 11: Intersections ixy

4.1.1 Shape Emergence Using Hypothesis-Driven Search

Hypothesis-driven search commences with some pre-de�ned class of shape to look for. Suppose

we want to �nd emergent shapes of the class triangle. We search for any combination of three

ixy that satis�es the closedness criteria (see Section 2.3.1). There are many possible triangles

in this example. One of them is shown in Figures 12, as shape:

S4 = f3; [ibd; ide; ibe]; g

S1

ee

e

c ee

e

ee

e

e

S

S

e

a

b

e

d

e

f

g

h

i

j

k l

3

2

4S

Figure 12: The emerged triangle S4

15

4.1.2 Shape Emergence Using Data-Driven Search

Shape emergence as a data-driven search is carried out by applying rules RS1 to RS4 to repre-

sentation (5) as follows.

(ijl)!< ijl > by RS1

(ijl; igl)!< ijl; igl > by RS2

(ijl; igl; iag)!< ijl; igl; iag > by RS4

(ijl; igl; iag; iaf )!< ijl; igl; iag; iaf > by RS4

(ijl; igl; iag; iaf ; ifh)!< ijl; igl; iag; iaf ; ifh > by RS4

(ijl; igl; iag; iaf ; ifh; ich)!< ijl; igl; iag; iaf ; ifh; ich > by RS4

(ijl; igl; iag; iaf ; ifh; ich; ici)!< ijl; igl; iag; iaf ; ifh; ich; ici > by RS4

(ijl; igl; iag; iaf ; ifh; ich; ici; iik)!< ijl; igl; iag; iaf ; ifh; ich; ici; iik > by RS4

(ijl; igl; iag; iaf ; ifh; ich; ici; iik; ijk)!< ijl; igl; iag; iaf ; ifh; ich; ici; iik; ijk > by RS4

Testing for closedness in the adjacent intersections results in the shape:

S5 = f9; [ijl; igl; iag; iaf ; ifh; ich; ici; iik; ijk]; ek k el; ef k eg; eh k eig

shown in Figure 13(a). Following the same principle many other shapes can be found. Some of

which are shown in Figures 13(b), 13(c) and 13(d). This demonstrates the di�erences between

hypothesis-driven search and data-driven search. In the former only speci�ed shapes can be

emerged whilst in the latter any shape whose representation satis�es the closure criteria can be

emerged.

4.2 Object Emergence

In this section, we present an example to illustrate how emergent objects are produced. Using

the same example as in Figure 8, the 3D view of the town square is shown in Figure 14.

The primary objects O1, O2 and O3 are shown in Figure 15, and de�ned as:

O1 = f8; [ea; ec; ed; ee; ef ; eh; em; en]; ea k ed; ec k ee; ef k eg; eh k ei; em k engO2 = f8; [ea; eb; ed; eg; ej; el; em; en]; ea k ed; eb k ej; ef k eg; ek k el; em k engO3 = f8; [eb; ec; ee; ei; ej; ek; em; en]; ec k ee; eb k ej; ek k el; eh k ei; em k eng

i. e., the entire primary object PO is:

PO = f14; [ea; ec; ed; ee; ef ; eh; em; en]; [ea; eb; ed; eg; ej; el; em; en]; [eb; ec; ee; ei; ej; ek; em; en];

ea k ed; ec k ee; ef k eg; eh k ei; eb k ej; ek k el; em k eng

Note that we are using the same labels used for lines in the example in 2D for plans in 3D.

There are two additional labels em and en.

Ungrouping O1; O2 and O3, we have the primary object as:

PO = f14; (ea; eb; ec; ed; ee; ef ; eg; eh; ei; ej; ek; el; em; en);

ea k ed; ec k ee; ef k eg; eh k ei; eb k ej; ek k el; em k eng

Applying rules R1 to R9, R10, R11 and R12 we get:

PO = f14; (iab; iac; iae; iaf ; iag; iah; iai; iaj; iak; ial; iam; ian; ibc; ibd; ibe; ibf ; ibg; ibh; ibi; ibk;

ibl; ibm; ibn; icd; icf ; icg; ich; ici; icj; ick; icl; icm; icn; ide; idf ; idg; idh; idi; idj; idk;

idl; idm; idn; ief ; ieg; ieh; iei; iej; iek; iel; iem; ien; ifh; ifi; ifj; ifk; ifl; ifm; ifn; igh;

igi; igj; igk; igl; igm; ign; ihj; ihk; ihl; ihm; ihn; iij; iik; iil; iim; iin; ijk; ijl; ijm; ijn;

ikm; ikn; ilm; iln); ea k ed; ec k ee; ef k eg; eh k ei; eb k ej; ek k el; em k eng

16

ee

e

c ee

e

ee

e

e

S

S

S

e

a

b

e

d

e

f

g

h

i

j

k l

1

3

2

ee

e

c ee

e

ee

e

e

S

S

S

e

a

b

e

d

e

f

g

h

i

j

k l

1

3

2

(a) (b)

ee

e

c ee

e

ee

e

e

S

S

S

e

a

b

e

d

e

f

g

h

i

j

k l

1

3

2

ee

e

c ee

e

ee

e

e

S

S

S

e

a

b

e

d

e

f

g

h

i

j

k l

1

3

2

(c) (d)

Figure 13: Some emergent shapes found by data-driven search

Some of these intersections are shown in Figure 16.

Now developing the intersections of these intersections we got the expression at intersection2

level, i.e., ixyz8.

Some intersections from this representation are shown in Figure 17 and can be used to

emerge objects utilising the two di�erent types of search. These two types are shown in the

following subsections.

4.2.1 Object Emergence Using Hypothesis-Driven Search

Hypothesis-driven search commences with some pre-de�ned type of object to look for. Suppose

we want to �nd emergent objects of the class triangular prism. We search for a combination

of ixyz that satis�es the closedness criteria for that class of object. One result is shown in

Figure 18 and the object is:

O4 = f5; [eb; ed; ee; em; en]; em k eng

8We do not show the complete development of the representation for brevity and clarity.

17

Figure 14: The town square of Casalecchio as primary objects

heed

eb

ie

e

e

c

j

e

e

f

g

eek

l

en

em

ea

ee

O1

O

O3

2

Figure 15: Representation at plane level of primary objects O1, O2 and O3

Considerably more complex objects can be emerged.

18

cm

em

im cheh

hmem

am

dm

fm

df

af

gm

bmjm

aj

bd

dg

ag

an

dn

bljlbkjk

jn

bn

en

be

cj

ci

ei

i

i

i

i

i

ii

ii

i

i

i

ii

iii

i

i

ii

i

i

i

ii

i

iiii

i

Figure 16: Representation at plane intersection level of primary objects

i

ii

i

ii i

ii

i

i

i

ii

ii

i

ii

ii

ii

i

i ii

iii

i

ii

ebm

cimeim

chm

dem

acm

afm

agmdfm

dgm

bdm

ajm

blm

jlm

acn

den

afn

agn

bdn

iajnjln

bln

dgnidfniehn

chn

cin

ein

bkn

jkncjn

jkm

bkmcjm

ehm

ebn

Figure 17: Representation at intersection ixyz level of primary objects

4.2.2 Object Emergence Using Data-Driven Search

Objects can be emerged using data-driven search approach by the application of rules RO1 to

RO4. For instance:

(ijlm) !< ijlm >

19

Figure 18: Emerged object O4.

(ijlm; iblm)!< ijlm; iblm >

(ijlm; iblm; ibdm)!< ijlm; iblm; ibdm >

(ijlm; iblm; ibdm; idgm)!< ijlm; iblm; ibdm; idgm >

(ijlm; iblm; ibdm; idgm; iagm)!< ijlm; iblm; ibdm; idgm; iagm >

(ijlm; iblm; ibdm; idgm; iagm; iafm)!< ijlm; iblm; ibdm; idgm; iagm; iafm >

: : :

: : : : : :

Testing for closedness in the adjacent intersections results in the object:

O5 = f11; [ijlm; iblm; ibdm; idgm; iagm; iafm; idfm; idem; iehm; ichm; icim; ieim; iebm; ibkm; ijkm; ijkn; ibkn;

iebn; iein; icin; ichn; iehn; iden; idfn; iafn; iagn; idgn; ibdn; ibln; ijln]; ea k ed; ec k ee; ef k eg; eh k ei;

eb k ej; ek k el; em k engshown in Figure 19. Following the same principle many other objects can be found.

5 Conclusion

Objects play an important role in visually-oriented design. They appear to have a special

role in architectural design in particular. One of the characteristics of objects is that they

can be interpreted di�erently than that intended when they were drawn. Current computer-

aided drawing, computer-aided drafting and computer-aided design systems prevent such an

20

Figure 19: Emerged object using data-driven search

interpretation. Inadvertently such systems have enforced �xation so that it is not surprising

that they are not used in the early stages of architectural design.

The ability to discover emergent shapes readily o�ers opportunities to develop design-

oriented graphics systems which may be more amenable to augment designers during the con-

ceptual stage of design. It has rami�cations for collaborative design where two designers share

the same (computational) workspace synchronously (Maher et al. 1993) but one `sees' di�erent

objects to those drawn by the other designer. The two schemas, the original and the emergent,

could co-exist so that both designers have the same image in front of them but `see' di�erent

objects in those images. It becomes possible for designers to have di�erent co-existing functions

for the same design with computational support for each one of them.

The process of object emergence we have presented here is founded on the general notion of

removing �xation produced by a particular representation. It then utilises another, more gen-

eral, representation which allows for the elimination of that �xation to provide the opportunity

for emergence.

Acknowledgements

This research is funded by a grant from the Australian Research Council. Jos�e Carlos Damski

thanks CNPq Brasilia/Brazil for his scholarship. The authors thank Milad Saad for assistance

with AutoCad, and the anonymous referees who provided guidance on improving the paper.

21

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24