Introduction

First of all the reader must have in mind that necessarily this research plan must have a general nature because it deals with emergent fields confined to restrict experts. If the interdisciplinary public has to understand it, it must be written in an intelligible mode to the great majority.

To design, plan and build sustainable cities for the twenty-first century means essentially to find solutions for the problems that will confront humanity such as overpopulation, water, food, energy, pollution, health and medicine, waste, urbanization, violence and terrorism, migration, clash of cultures, sex war especially expressed as homosexualism and lesbianism and single parenthood and so on. I developed an ecodesign model in my master's dissertation to deal with these problems. Hence it blends design and planning. The root of all these problems lies on social segregation and degradation of the environment. For the Scandinavians, social segregation is above all a scientific and technological problem. I mean if it were possible to build qualitatively for all and all would live in harmony, most of these problems would disappear. I developed an ecodesign model in my master's dissertation to deal with these problems. The model of Primary, Secondary and Tertiary Waves (MPSTW) for the design and planning of sustainable cities is an application of catastrophe, semiotics and graph theory (figure 1). It enables one to model the urban ecosystem as a single organism, an autopoietic entity that is distributed in time and space through recursive partitions that are conceived similarly structurally to adjust to the whole. The partitions that the urban ecosystem is recursively divided encompass the concept of a sustainable planet, continent, country, region, cities, boroughs, neighborhoods, industrial areas, leisure areas, transportation areas, buildings, houses and so on [Lour94]. Since the organizational unity of the urban ecosystem is the eco-system with hyphen, the changes in configuration of the ecosystem or the architectonic arrangements generated by the application of the MPSTW may be immediately evaluated as the sum of changes of "isolated" exergies in the eco-systems with hyphen. Exergy is defined as free energy, i.e., disposable energy or free energy of the system relatively to the environment. It expresses energy with a built-in measure of quality. It measures the distance from the "inorganic soup" in energy terms. Hence exergy increases from molecules of clay, clay, bricks, bricks ordered in a cube, a house to a cathedral. An approach of this nature is being unfolded only recently. Its lack has led to the loss of control of urban growth. Megacities grow like a cancerous system, but simultaneously it is well known that the greatest contribution to the GNP comes from them. Evidently the contribution of the environment is not taken into account. The new field of ecological economics has been introducing methods to evaluate the price of the environment such as the EMERGY method as well as new indicators of economy such as the Green GNP [Pill93], [Odum93].

This definition as subsystems of entities or less autonomous entities integrated in their environments is more useful for ecological purposes than the traditional categories of loose architectonic objects in the environment. The representation of architectonic objects as eco-systems endows them with the propriety of defining and being defined by the ecosystem. The scientific treatment of the architectonic object as an intrinsic environmental structure is a must for the necessary conceptual changes to promote the development of the ecology of complex systems.

If the nature of the urban ecosystem is too complex to be known in each detail if the modeling is the primary form of operational abstraction, then the models must reflect a unitary vision of the relationship object-environment or a blending of design and planning, because this is the most realistic vision. To enable this, the model presents a further granularity. The eco-system with hyphen is identified with the architectonic object independently of scale. The subeco-systems with hyphen are the items of the brief, namely, activities, environmental comfort, structural systems, hydraulic installations and so on when the MPSTW is applied. The subeco-systems with hyphen correspond to the figures of the natural language. Such nonsigns are designated figures. The language is such that from a limited number of figures that can ever shape new arrangements one can build an unlimited number of signs. The nature of MPSTW is such that from a limited number of subeco-systems one can build an unlimited number of eco-systems with hyphen. The eco-system is the fundamental unity of the urban ecosystem.

Viewing the architectural language as a system of figures instead of signs allows the blending of design and planning. Hence to tackle with global problems of mankind. Let's try to make clear for the reader what this means through the succinct description of the application of the MPSTW to the core items of the design, namely, activities and structures.

Activities

Primary waves dependent on space and time are described by the following four processes.

A. Homeostasis. An activity needs a space that is limited by measures adjusted to the activities: minimum measures (bathroom), exact (sport fields), maximum (circulation). Indeed a transactional magic between the holistic organism and the environmental project takes place. The developmental psychologists put forward the concept of the identity of the place conceived as a substructure of the self identity, consisting in apprehensions about the physical environment that serve to define what the person is. A natural conclusion follows: the basic task of architecture is to act in favor of man, creating a third mesoenvironment whose central design criteria to judge its performance is the physical, psychical and mental health. This favours a phenomenological approach based on the thingness of things leading to a bidimensionalization of space .

Obviously disciplines such as ergonomy [Lour97d], anthropometry and proxemy enable one to create a sound physical place for sheltering life. Ergonomy deals with the use of physical energy to occupy a physical setting and surpass gravity, fiction and inertia implicit in physiological work. Anthropometry studies spatial patterns that the body describes when it performs work: walk, sit, stand up and so on. It deals with the minimum and maximum measures needed to perform these movements around the furniture. Proxemy studies the behavioural consequences of spatial relationships in interpersonal relationships in all scales and types.

B. Continuity. The activities consist in a sequence of actions coupled to different situations. It is fundamental to emphasize the changing nature of spaces in time. Dissecting the needs of each member of the family along his/her life cycle shows the importance of transforming a space for families with children in a space compatible with activities concerned with people wanting to communicate or how to introduce multiuse spaces in housing groups that cope up with the ecology of the human being along the life cycle.

C. Differentiation. Transformations in lay-out include

a. plans for people with great need of communication

b. plans for people with less need of communication

c. plans for people with certain need for individualization

d. plans for people with great need for individualization

D. Repeatability. Here one is expected to proceed to a general evaluation whose goal is to create environments that can shelter life. Thus mimicking the biodiversity of natural ecosystems with its myriad of ecological niches. Galster [Gals86] foregrounds the externality space. It is defined as the area over which environmental changes initiated by others are perceived as altering the well being (psychological, ecological or financial) a given individual derives from the given location. Its three characteristics are:

a) congruence - the degree to which an individual's externality spaces correspond to predefined geographic boundaries.

b) generality - the degree to which an individual's externality spaces for different types of externalities correspond

c) accordance - the degree to which externality spaces for different individuals in the same area correspond.

For example, a designer may want to know to which degree a neighborhood exists over a given space that shelters communitarian ideals. A concept of neighborhood built upon the notion of an individual's externality space can be easily handled if centered on plans designed by the MPSTW. Galster developed formulas to measure these dimensions. Obviously a sound planning centered on human beings in interaction with other human beings and environment will provide an accurate blueprint for building a sustainable city.

The information gathered in the process repeatability of each element will be expanded to generate another model called The tertiary waves or planning of a sustainable city.

Secondary waves

Applying catastrophe, semiotics, linguistics and graph theory, architectural design may be considered as a language [Lour88], [Lour94] exhibiting the planes of the function and form and the stratas substance of the function, form of the function, substance of the form and form of the form (figure 2 and 3).

Substance of the function for designing a housing group are the characteristic activities of the day of the week.

Form of the function. It is depicted as the sequence of characteristic activities through directed graphs.

Substance of the form. It presents two aspects: conditions and qualities. The conditions refer to the abstract relationships in the architectonic space evidenced through graphs as articulation between environments where the nodes are the rooms and the edges are the adjacencies or doors connecting two rooms. The qualities refer to relationships that are geometric like linear, curved, planar and volumetric. They can be depicted through planar graphs. An ideogram is born out of a planar graph. The graphs are the natural outcome of the inspection of the former stratas.

Form of the form. Lines, planes and volumes are the exact shapes of the specific physical setting. It is regulated in an organic approach by the process homeostasis. After laying out the furniture according to anthropometric measures and other behavioural concerns and guided by the planar graph, one finally arrives at a tiling. A plane tiling is a family of sets called tiles that covers the plane without voids or overlapping. We shall be concerned with the three regular tilings consisting of equilateral triangles, squares and hexagons [Grün87] to define the basic modular grids.

The concept of a prototile which means a tile with a different shape enables us to display lay-outs that treat each room as a whole. We shape meaningful objects called prototiles and characterizing each room from the three regular tilings. It enables the generation of a free plan. The prototile concept clearly enables design as if grown from inwards towards outwards, embodying the ecology of the human being in all stages of his life cycle in each room of the different settings outlined in the previous stratas.

If one maps each pattern of events onto each unit of division of an interior or exterior space characterized as a prototile, a floor plan embedded in its landscape environment is not more than a finite portion of a tiling. If one works out the resulting prototile shapes, trying to match them and to uncover the pattern that allows for their expansion the result is a tiling. It is necessary to uncover its evolving matrix form in order to reach a Gestalt composition.

The next stage to increase the organizational complexity is to introduce a symmetry group theoretic approach through lattice, frieze and crystallographic groups. Frank Lloyd Wright and Le Corbusier, the two greatest form-makers of the twentieth century among other architects since Leonardo da Vinci employed the notions of symmetry and pattern - structure in the architectural design generation.

Structures

The following processes shape the primary waves.

A. Homeostasis. One has to evaluate the site form the viewpoint of excavation, erosion and the bearing capacity of the supporting earth to determine the nature of the soil and foundation requirements at a site. So it would be interesting to display geological charts. The complexity varies from simple recommendations for a general region in developing countries to full-fledged computational modeling to place foundations at the most rigid soil conditions thus building economic foundations and saving materials.

B. Continuity. One has to examine maps about geological substrate, unconsolidated materials and surface waters. The site should be accessed here preferably from a GIS or a terrain data modeler. So homeostasis will call continuity to access the geological charts for example.

C. Differentiation. One has to deal with horizontal subsystems, vertical subsystems, linear components, foundation subsystem. Thus info must be dealt with tridimensionally. Differentiation will send messages to the substance of the form of the static function to detail this process.

D. Repeatability. One has to deal with the effective use of material resources for infrastructure and finish materials (floor, wall and ceiling finish). Hence it would be convenient to utilize the reference materials system, a process flow description of the conversion of renewable and non-renewable resources to finished materials for infrastructure [Hoff94]. Here also the EMERGY method can be utilized to calculate energy use during the entire life cycle of building design [Odum94]. Focusing on the externality space concept from activities and now on evaluating the amount of energy and materials used in the building process helps to protect man and the environment and establish a sound urban basis for ecological economics.

Secondary Waves

The strata substance of the static function are the own forces that act in a section of an element of a building.

Form of the function identifies the mechanism of load transference.

Substance of the form of the static function studies the geometric aspects of the static function:

- linear forms with a one-dimensional static function (columns, cables and arcs)

- composite forms incorporating elements with a one-dimensional linear static function (nets, trusses, space frames, geodesics)

- forms with a two-dimensional surface statical function (plane surface bearing systems include beams, cantilevers, deep girders, frames, whereas curved surface bearing systems include membranes and cells.

- composite forms incorporating element with a surface statical function include box-type (hollow section) beams, multiplanar frames, flat or curved lamella grids, structures of interlocking thin walls that function as deep girders and folded plate structures.

- form with a mass statical function include slabs such as a two-way ribbed waffled slabs, walls acting in compression and bending, grids composed of beams with some thickness

and form of the form encompasses the structural systems.

The fundamental ergonomic, engineering geological, constructive (use of sustainable materials), eMergetic (deal with energy taking into account the price of the environment), geometric, informatic and telematic dimensions pervade the ecodesign model. The faithfulness to them to shape the urban ecosystem is the crucible where solutions to the global problems of mankind will be met.

Object

The object of the Ph.D. thesis is the expansion and implementation of the MPSTW. The former deals with the geometric modeling and the latter with the prototype based object oriented programming language Self.

The geometric modeling

In my master's dissertation [Lour88], I dealt mainly with the primary waves or the processes of interaction of the architectonic object with the environment. The Submodel of the Architectonic Sign or the secondary waves (figures 1, 2, 3, 6.a and 6.b) is concerned with the design processes, namely the geometric modeling. Architecture is above all art. Its aim is to create a true (scientific), good (to care about the well-being of all) and beautiful environment. To ease understanding of the geometric modeling and implementation, I will not be concerned with trying to refer to the integration with the MPSTW as I did in the foregoing section.

At the root of my inspiration for the geometric modeling are the organic forms of the architect Alvar Aalto and the breakthrough in visual thinking unraveled by the graphic artist Maurits C. Escher who wanted to become an architect.

Hence the core of my concerns deals with the theory of tilings and frieze and crystallographic groups. A tiling is a regular division of the plane consisting of a family of closed sets which cover the plane without gaps or overlaps [Grün87]. Grünbaum, Shephard and Schmitt {Schm87} tried to mimic Escher and start introducing tilings whose "tiles" are of a small number of different shapes and called them prototiles. With this introduction, once can consider each room and the landscape as a different prototile and map the concept of tiling onto the free plan. My programmed work entitled The autopoietic nature of the urban morphogenesis developed as part of the requisites for the Ph.D. program in architecture at FAUUSP (80 pages) describes tilings, frieze and crystallographic groups and their interaction with fractals. Previously, in 1992 I developed a much deeper work concerned with tilings and crystallographic groups [Lour92].

Several architects from Leonardo da Vinci to Frank Lloyd Wright and Le Corbusier, the two greatest form makers of the twentieth century applied groups to their projects (figure 4). The three regular tilings are shown in figure 5. These patterns have a certain amount of symmetry. We apply translations, reflections, rotations and glide reflections to them. The symmetry is measured by a group, in this case, the appropriate subgroup of the Euclidean plane whose elements are the isometries of the plane which send a given pattern to itself. We shall classify the groups which can arise in this way as symmetry groups of two dimensional repeating patterns and call them the seventeen crystallographic groups of the plane. An isometry is a very special case of homeomorphism. While an isometry is a transformation that preserves properties such as direction of a line, angles between lines and area, the homeomorphisms do not preserve them. An application of a homeomorphism to a tiling preserve valences of vertices and the number of adjacencies and neighborhood of each tile. Lionel March shows an example of homeomorphism in three projects developed by Wright (figures 6.a and 6.b). Grünbaum and Shephard [Grün87] introduced the notion of combinatorial isomorphism and combinatorial equivalence and showed that the concepts of topological equivalence and combinatorial equivalence coincide. So what matters in this research is this equivalence. Even if I am not going to deal with it explicitly in the Ph.D. thesis. However its unfolding in the future points towards this direction.

Escher depicts the same motif (figure 16) in the square grid (figure 17) and in the circle (figure 7). Escher felt awfully alone with his discoveries. Yet they illustrate my ideas gracefully.

To deal with the seventeen crystallographic groups and the seven frieze groups in detail is a suprahuman task. Therefore my research group believes if just one case is well characterized taking into account three different approaches namely:

a) an application of the theory of isometries (a geometric description) [Mart82], [Lede82];

b) presentation by generators and relations (an algebraic description) [Lynd 86], [Mag66]

c) an element of the Euclidean plane E₂ thought of 'abstractly' as a linear transformation A, coupled with a translation vector a (matrix representation: fixing a basis of R², one can compute the matrix of the linear transformation and the coordinates of the translation vector. Thus A can be thought of as a matrix and a as a column vector of coordinates) [Mack85].

There are five possible types of lattices. Given a lattice L, there are orthogonal transformations that form a group and preserve L. The point group of any wallpaper group which has L as its lattice must be a subgroup of this group. This limitation on the point group is then sufficient to allow us to enumerate the different wallpaper groups with lattice L. The chosen case to be studied is the square lattice. The group of the orthogonal transformations which preserves L is the dihedral group of order 8 generated by a rotation of 90 and a reflection about the x-axis. The point groups C₄ (the cyclic group of four rotations) and D₄ (the dihedral group of the square) are a subgroup of this group. The pictures in the following pages characterize the crystallographic groups p4, p4m and p4g (figures 8.a, 8.b, 8.c, 8.d and 8.e), (figure 9). It is fundamental to understand the action of the point group over the lattice in each of the three possible cases p4 (figures 10, 11, 12.a, 12.b); p4m (figure 13) and p4g (figures 14, 15, 16, 17). Especially important is to understand the axes of glide reflections in the crystallographic group p4g (figure 8.e).

Self

Since my master's dissertation presented in June 1988, I have been searching the adequate language to implement my ecodesign model. Since 1989, the programming paradigm that seemed most adequate to the implementation was the object oriented. Hence the object oriented programming languages C++, Eiffel, Sather, CLOS and object oriented development methods were studied [Rumb91], as well as the artificial intelligence technique called Blackboard [Enge88] to develop a proposal of an object oriented knowledge based system. The object oriented programming language Beta [Mads93] and its Mjoelner environment [Knud94] seemed the best fit to implement my ecodesign model. It was not necessary to implement it through the Blackboard due to the concurrency level. Despite the trials to write a final research project to be implemented through it under the advising of Joergen Lindskov Knudsen from the Department of Computer Science, Aarhus University, Denmark, many problems were met especially dealing with the hermeneutic nature of the MPSTW.In my attempt to solve these inconveniences that pointed toward polymorphic languages, I was put in contact with the Self language in the discussions held in the Beta comp.group through Netscape in January 1996.

Curiously the prototype based object oriented languages may start a new promising interactive artificial intelligence technique in the emergent hermeneutic computer science that is opposed to the mainstream formalist computer science. Object oriented paradigm behaves like a Janus-face between them. The hermeneutic nature of design demands an hermeneutic language to deal with implementation.

Hence the MPSTW falls within an hermeneutic trend This is shown in the Programmed Work entitled The hermeneutic nature of an ecodesign model and Self a prototype based object oriented language [Lour98a], [Lour98b].. Indeed, its elements and processes behave themselves with dynamism and flexibility. They are not variations around a theme. Classes are useful when multiple instances of similar objects pervade the problem space. Hence sharing attributes among the objects as well as programming exploratorily is fundamental. The independence of each element or process suggests an object (figure 18). Each object accepts or delegates tasks to the other. Each element and each process is unique. There is no need for classes No clear taxonomy for tasks is defined, hence little need for inheritance [Grog97]. Moreover, MPSTW is built for cooperative work among designers and citizens covering total synergetic interaction among its members.

Martin Abadi and Luca Cardelli [CA97] insist on that everything can be better represented in terms of objects, even functions and classes. The basic constructions are simpler, flexible and powerful. Hence naturally a prototype based object oriented language tunes with the hermeneutic nature of design enhancing it. Wegner insists on that objects, classes and inheritance are not orthogonal. Classes are defined in terms of objects, inheritance in terms of classes. The essence of a class can be defined independently of the object and the essence of inheritance independently of classes and objects [Wegn87].

Inheritance as a mechanism to share resources defined incrementally internalizing shared resources treat the latter as part of an extended self (identity). Hence the definition of inheritance in terms of a particular mechanism of self-reference enabling the internalization of remotely defined operations as part of the extended identity of the object is called delegation.

The delegation based languages allow the objects to share and internalize operations from ancestral objects called prototypes, that work as instances and templates for the descendants. Hence the prototypical languages are languages based on delegation that carry out delegation by prototypes. And the chosen language to implement MPSTW is Self, a prototype based object oriented language (figure 19).

Ungar and Smith [SU95] emphasize the concreteness of the prototypes because they are examples of objects instead of format descriptions and initializations.

The shared behaviour by a family of objects is hold by a separate object that is the father of all the objects, even of the prototype. This way, the prototype is absolutely equal to any other member of the family. The object that contains the shared behaviour plays the role of a class, except that it only contains the shared behaviour without format information.These parent-objects are called traits-objects.

Self contains graphical objects called morphs that behave exactly like the object. Since MPSTW also has a graphical nature this feature is very relevant.

Methodology

The rise of object-oriented programming may be viewed as the outcome of an effort to allow computers to more directly model physical systems. Yet a Beta program, the best OO programming language, is a collection of abstract conceptual patterns that describe concrete phenomena without themselves partaking of the concreteness of the phenomena [Lour94]. To start programming in Beta one should have a thorough description of the great majority of the elements of the design thus contradicting the hermeneutic nature of MPSTW, where one can start designing by whatever element or process like in real design.

Due to the physicality and thingness of Self, the concreteness of the prototypes are examples of objects instead of format descriptions and initializations. Hence it has the inductive nature of physics. As an outcome, I can deal with the implementation of MPSTW as a case treatment. My research group and I wanted to implement a simple part of the model that would give you a flavour of the whole hermeneutic nature of design. Like in physics, one chooses the most relevant properties of a system that would be enough to represent it or give important information. For example in architecture, sections and plans are enough to give you a glimpse of the three dimensional reality.

The adopted approach consists in generating the architectural plan, structural plan and section of a four-floor building of apartments for singles (figures 20, 21, 22, 23, 24). The figure 20 shows an application of the point group D₄ of the crystallographic group p4m. Of course this is to be thought as being translated over the site dependent on the wish and number of users belonging to this class. The whole approach is concerned with the ecology of the life cycle of the human being. Hence programming and designing from babies to old age. Everybody is expected to live in harmony due to the concept of externality space applied to the sustainable neighborhood. The underlying idea is to integrate people in all dimensions to live in harmony.

In figure 21 one perceives the need for more space to have a workable surface in the kitchen. Then it is necessary to add more 50 cm to the perimeter of the kitchen along the hydraulic wall for example. This will automatically add more 50 cm to the other side of the isosceles triangle. Let's choose to add more space around "the office space". So the office furniture will have to relocated. The small base of the quadrilateral will change as well as the overall dimensions of the circulation space. Although many approaches dealing with the generation of polyhedral chains [Günt88] or arrangements of lines [Bier82], [Bier88], [Edel86], [Stef84], [Bent79], [Alex78], [Chaz87] and [Mehl85] may theoretically help to find a solution to the problem just described, indeed the geometric constraints for building design approach from Kirk Martini [Mart95] seems more adequate to tackle it. Constraints [Knud95] automatically maintain desired spatial relationships, so that a model can be manipulated using a relatively small number of meaningful parameters rather than a large number of coordinates and transformations as is usual in most computer graphics techniques.

Manipulating the geometry the geometry of an assembly such as a building often requires accounting for interaction between constraints governing shape and location. To layout the furniture and the corresponding anthropometric measures or to draw the outline of the resultant architectural plan or to generate structural design is essentially the same problem. This approach will be adopted [Hare88].

The interactive mode of Self wil ease and enhance the general flexibility and malleability of the geometric modeling .

The generation of the lattices in a crystallographic group is the outcome of the action of the point group on the lattice. Then it is enough to choose a basis so that every lattice vector can be expressed as an integral linear combination of t and Rt, where R is a rotation 2/n in the point group G₀. If n > 2, then A, Rt form a lattice basis.

The element of a symmetry group G is written in the form (a,A) where a R² is a translation vector and A is a linear transformation of R² which preserves distance. It is assumed as known the fact that such a linear transformation is either a rotation about the origin or a reflection in a line through the origin. By definition, a symmetry (a,A) sends a point x to (a, A) = a + Ax.

Group multiplication in G is composition of symmetries; thus

(a, A) (b,B) = (a,A) (b + Bx) = a + Ab + Abx

and therefore

(a,A) (b,B) = (a + Ab, AB).

This formula is used to obtain the four invariants of the crystallographic group G:

a) the lattice T;

b) the point group G₀;

c) the action of G_o on T and

d) the shift vectors (responsible for the glide reflections).

Of course this will be applied to characterize the three groups of the case of the square lattice, namely p4, p4m and p4g [Schw74].

Homogeneous coordinates will be applied in order to transform a translation into a linear transformation, easing the operations of multiplication [Fole90].