Ana-Maria RUSU

GEOMETRY AND COMPLEXITY IN ARCHITECTURE

Abstract: As Constantin Brancuși (1876-1956) said „Simplicity is complexity itself“, simplicity and regularity through the use of basic geometric forms has always played a central role in architectural design, during the 20th century. A diachronic perspective, shows as the use of geometry and mathematics to describe built form provided a common basis for communication between the processes of design, fabrication and stability. Classic ways of representing geometry, based on descriptive methods, favor precise language of bidimensionality easy to represent in a rectangular coordinate system. In recent years, the importance of geometry has been re-emphasized by significant advances in the digital age, where computers are increasingly used in design, fabrication and construction to explore the art of the possible. Contemporary architecture transcend the limitations of Euclidean geometry and create new forms that are emerging through the convergence of complex systems, computational design and robotic fabrication devices, but which can also achieve higher levels of performance. Freeform architectural shapes and structures play an increasingly important role in 21st century architectural design. Through a series of examples, the paper relates to contemporary architectural explorations of complex, curvilinear surfaces in the digital age and discusses how it has required rethinking the mode in which we traditionally operate as architects. The analysis creates the possibility of comparisons between original and current design.

Key words: Geometry; Free-form; Architecture; Design process.

INTRODUCTION The first category of thin covers can in its own turn be divided and classified based on geometrical form; the As it is well-known, descriptive geometry is a branch resulting categories are: of applied mathematics whose goal is to represent objects • Surfaces of revolution; bidimensionally, on one or more planes, allowing for a • Surfaces of translation; graphical solution to many problems that are connected • Ruled surfaces. to tridimensionality. Nowadays, in the digital era, when A second classification can be made based on the computers play an ever increasing role in the projection Gauss curve: and materialization of architectural designs, there have • Singly curved shells. These shells have a zero been profound changes in the evolution of the whole Gaussian curvature; creation process which have led to the question whether • Doubly-curved shells of positive Gaussian geometry has not become solely a means of perception, curvature; of building explanations, understanding, and representing • Doubly curved shells of negative Gaussian space; or does it still allow for the creation of new curvature. [1] shapes? Although geometry has been the language of Classification based on geometrical developability: choice between different design processes, production, • Developable surfaces and materialization of architecture, it receives new • Non-developable surfaces valences in the combination with digital technology. This The free forms cannot be defined analytically, only in new technology transcends the limitations of the a more subtle way, through a network of coordinates, by Euclidean geometry, generating free-form architectural employing functional calculations that are specific to the volumes, a more organic aesthetics with an increased structurally conditioned optimization procedures. level of structural performance and this makes the Up until now, when looking to create curved answer to our previous question become more and more architectural shapes, one had to take into account: confusing and distant. • First of all must be possible to formulate The complex architectural forms, which demonstrate mathematically; a very free shape, have their beginnings set in the 1920’s, • The second, easy to build; when thin concrete shells started to be used as roof • Thirdly, structure should have expressivity. structures. Being able to make these structures in which Thus, thin continuous covers are known for their curved architecture had to possess an adequate structural structural efficiency resulting from geometry, also called strength, and an equally powerful elegance, represented a self-supporting structures, and for their reduced width huge challenge given that knowledge, as well as compared to the openness. Besides being structurally instruments, at that time was extremely limited. The efficient, these structures are also extremely simple and classification of the thin covers presents two distinct elegant. As these architectural forms evolved, the categories of shapes: analytical shapes and free shapes. structural analysis had a long and difficult history. As they were developed and perfected sometime between 1950 and 1960, at a time when architects were using JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS 59 Geometry and complexity in architecture them as a means of artistic expression, long before the computer ever entered the architectural scene a considerable amount of effort was required to check the designs.

2. SHELL STRUCTURES

One of the best known architects who designed and built a great number of thin covers from concrete shells is Felix Candela. Although an architect by formation, Candela remains known as one of the most important builders and structure engineers. Almost all his works, Fig. 2 The hyperbolic paraboloid. created between 1950-1960 around Ciudad de Mexico, are still in great shape, after years of usage and after the devastating consequences of some strong earthquakes in the area. He used the hyperbolic paraboloid for a number of these structures, whose shape allowed for a remarkably small width in relation to their aperture. Candela posited that „of all the shapes we can give to the shell, the easiest and most practical to build is the hyperbolic paraboloid.“ In order to better understand the creation of a thin structure in the shape of a hyperbolic paraboloid, we must first understand the geometrical characteristics of this geometrical surface.

2.1 THE HYPERBOLIC PARABOLOID The hyperbolic paraboloid is a quadric ruled surface Fig. 3 The hyperbolic paraboloid. generated by a straight line that lies on two straight directrices and is at all times parallel to a director plane. A hyperbolic paraboloid can be also defined by It is built by tracing one generator at a time, as a distinct means of a skew quadrilateral ABCD, Fig.3. variation of the general hyperboloid. The third straight A skew quadrilateral determines one hyperbolic directrix opens upward to the infinite and is replaced by a paraboloid and only one. The axis of the hyperbolic director plane parallel to the surface’s generators. paraboloid is the straight line parallel to the intersection The hyperbolic paraboloid is a doubly ruled skew line of the two director planes; it may be determined by surface. It contains two families of mutually skew lines joining the middle sections of the skew quadrilateral’s that can generate the same hyperbolic paraboloid. The diagonals ABCD. The tip of the hyperbolic paraboloid is first generator family is made of generators parallel to the the point on its surface where the tangent plane in that first director plane, Fig.1. The second family is made of particular point is perpendicular on the axis of the generators parallel to the second director plane. hyperbolic paraboloid. The two generators that pass through the tip of the hyperbolic paraboloid are called main generators. The main generators are the diagonals of a parallelogram that can be obtained by joining the middle of the skew quadrilateral’s sides AbCd. [3] The surface of the hyperbolic paraboloid contains two series of straight generators Fig.4.

Fig. 1 The hyperbolic paraboloid.

The second director plane is parallel to the two straight directrices Γ1 Γ2, which support the generators in the first family, Fig.2. [2] Thus the generators in the first family may become directrices for those in the second family and the other way round. The hyperbolic paraboloid is the only ruled surface with two director planes.

Fig. 4 The hyperbolic paraboloid. 60 VOLUME 10  SPECIAL ISSUE ICEGD JUNE 2015 Geometry and complexity in architecture and the Y-axis divides the structure into two symmetrical They allow the delineation of skew quadrilateral parts. To reduce the tension in the concrete shell at the sections of equal measure. Or, in other words: any skew basis of the surface the foundation has been set and two quadrilateral may be adjacent to a section of a surface of pairs of buttresses, in the front and in the back, sustain a hyperbolic paraboloid. If two straight opposite sides are the charge. After almost 60 years since it was built, the equally subdivided and the subdivision points are united chapel is still in exceptional condition. by straight lines, they become generators of the doubly curved surface by the hyperbolic paraboloid. The paraboloid, as a translation surface, may be also generated by a parabola which is moving parallel to its axis, along with another parabola, having parallel axes and pointed in different directions. There are many examples of hyperbolic paraboloid in constructions and architecture where it can be encountered in the manufacturing of roof systems or in other projects which require a large number of surfaces. From a structural point of view, the double curvature of opposite direction deals very well with the changing game of the internal forces of tension and compression, maintaining balance under any strain, if there is a minimum manifestation of forces. The surface may bear in any point or given direction the compression or applied stress, tangential to its curvature. It may be rest on the two lower points. [4] Fig. 6 Chapel Lomas de Cuernavaca For transmitting the self-load which is the most important load of a roof, the suspended parabolas with the curvature downwards are preferred for the tension efforts, and the parabolas with the curvature upwards are preferred for the compression forces, Fig.5. It’s a positive thing that the parabolas coincide perfectly with the pressure lines, and that they are capable of supporting their own weight. The deviation tendencies of the forces from the parabolic curvature of the thin surface are therefore very reduced from the very beginning.

Fig. 5 The hyperbolic paraboloid

2.2 CHAPEL LOMAS DE CUERNAVACA Made by Felix Candela in 1958, Chapel Lomas de Cuernavaca, Fig.6 only needed one hypar to generate its shape. The entry point was obtained by its intersection with an inclined plane and the hyperbolic sides of the church resulted from the intersection of the hypar with a plane parallel to the ground. Fig.7 represents the drawing of the Chapel, made by Candela, with the two types of generators that form the curved surface and Fig.8 offers a glimpse of the ongoing building process, bringing to our attention the generators that give the chapel its saddle-like shape, with a remarkable width of only 4 centimeters. The grid of the generating lines is made of parallel and perpendicular Fig. 7 Chapel Lomas de Cuernavaca lines, refined at an optimum level for creating the shape, JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS 61 Geometry and complexity in architecture

and F2 respectively. An arbitrary end plane P’ generated by end line D’ intersects with the two arcs in points (1, 1’) and (1, 1’). We obtain thus the line (110, 1’10’) as a generator of the arrière-voussure surface. Similarly, we can determine other generators, for example (220, 2’20’), (330, 3’30’) etc. The crooked arrière-voussure surface covers the opening only between the limit generators 1’10’ şi 5’50’. Between the level plane H’ and the end plane P’ the opening is covered by by a general ruler surface, whose generators’ two points of support are two directing lines (the vertical in a, the end line D) and a directing curve (the arc γ2, γ2’).

Fig. 8 Chapel Lomas de Cuernavaca

The calculation of the determined weight on a single ruler surface, more precisely on a single parabolic hyperboloid, was controlled through studies and calculations without the aid of a computer. But this process was not error-free because what we see today is the second version, only 18 metres high; the first version of the building, 24 metres high, had crumbled. This could have been avoided if the builders and designers would have had access to simulation programs that calculate the degree to which a structure is pressed by its own weight. Computer-assisted design developed in the 70’s and it allowed architects to design more graphic, daring and free shapes, bringing immense ease in their design, construction and creation.

3 PARAMETRIC DESIGN

The new visions in architectural design are based on more complex types of geometry, although on a basic level they develop shapes and surfaces that are already Fig. 9 The arrière-voussure surface. known in descriptive geometry; when multiplied, they vary from one side to the other of the structure. Designing these components by using only classical instruments used to represent tridimensionality would take a very long time and solving the structural problems would represent an equally daunting task. An interesting project that suggests this exact type of relation between crooked geometrical surfaces is Norman Foster’s Sage Gateshead Concert Hall. In order to gain a better understanding of the complexity of this structure, it is important that we notice the geometrical characteristics of the arrière-voussure surface.

3.1 THE ARRIÈRE-VOUSSURE SURFACE The arrière-voussure surface is a crooked surface generated by a line that sits on two directing curved lines situated in two parallel planes and on a line that is perpendicular on the planes of the directing curved lines. This directing line is sometimes called the surface axis. A generator of the surface can be obtained by uniting the intersection points between the two directing lines and an arbitrary plane, starting in the right directing line. Thus, we can choose the axis of the arrière-voussure surface Fig. 10 The arrière-voussure surface. fig.9 the end line D (d, d’) and the directing lines can be the arcs (γ1, γ1’) and (γ2, γ2’), situated in the planes F1 62 VOLUME 10  SPECIAL ISSUE ICEGD JUNE 2015 Geometry and complexity in architecture The two directing curves of the crooked arrière- voussure surface can be defined up to the same level plane, so that the entire opening can be covered solely by arrière-voussure surfaces, Fig.10. Thus, starting from the way of generating this crooked surface these notions can be generalized considering that the crooked surfaces are generated by a line that is supported by directing line and any two directing curves Fig.11.

Fig. 13 Sage Gateshead Concert Hall.

The whole surface is made up of multiple arrière- voussure surfaces, which make possible the existence of straight glass panels, Fig.14. At a bigger scale, the tangent circles create a curve that makes a progressive motion along the curve axis in a direction across the building, Fig.15. Through the complexity of the resulting shape, the static calculation would imply a lot of time and effort. Computer generated design becomes an interactive process, where the designer can easily optimized the final result. Static, individual calculation of these oblique arcs implies that, for the usual charges their rupture is avoided

Fig. 11 The arrière-voussure surface. through the maintenance of the compression axis in central band which must be a third of the width of the 3.2 SAGE GATESHEAD CONCERT HALL oblique section. Digital modelling not only extends the design The horizontal components tend to elongate the arc possibilities, but it has also diminished the difficulty and and to push aside the supporting pillars, but the pressure the design and construction costs of these complex in these arcs considered within the whole construction is geometrical surfaces. difficult to assess. One of the methods that help us generate free form The geometry of this free surface is the result of surfaces is the parametric design, which involves the specific, digital performance simulation criteria. During generation of form through inter-related parameters and the design phase, a series of external forces can be facilitates the adaptability of the form when facing the simulated in order to determine the behavior of the alterations required by the initial design phases. Sage structure, which can thus be transformed and optimized. Gateshead Concert Hall, Fig.12 a project completed in 2004, is the result of such a process, where the surface curve lines have been generated by a series of tangent circles, created through parameters and inter-connected by a process that allows for individual changes only in relation to the others, this implicitly modifying the shape of the whole as well – interconnected, in order to be able to adjust to the whole, Fig.13.

Fig. 14 Sage Gateshead Concert Hall.

Fig. 12 Sage Gateshead Concert Hall.

JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS 63 Geometry and complexity in architecture knowledge of descriptive geometry and without identifying the diverse problems that may result from representing space, everything becomes extremely uncertain because, beyond any doubt, digital design seems to have become a massive phenomenon. It offers a possibility of a revised understanding of the relationship between architectural geometry, the building form and structure in the understanding of geometric concepts and approaches. All of these aspects create a database of information and variables, which are characterized by a deep formal complexity and which imposes one of the most important concepts, the process of adapting to and understanding this plurivalence.

Fig. 15 Sage Gateshead Concert Hall. REFERENCES

4. CONCLUSION [1] C., Siegel, (1968), Forme structurale ale arhitecturii moderne. Ed.Tehnică, Bucureşti,; The impact of these new digital technologies on [2] M., Salvadori, (1983), Construcţii. Lupta împotriva design has been great from many perspectives: gravitaţiei, Ed. Albatros, București,. theoretical, methodological, and technological; [3] A., Tănăsescu, (1975), Geometrie descriptivă , moreover, we are witnessing the creation of new design perspectivă , axonometrie , Ed. Didactică şi processes, which seem to challenge the hypotheses of Pedagogică, Bucureşti,. traditional processes of design. [5] The designer manages [4] A., Beles, M., Soare, (1964), Paraboloidul eliptic și a series of data and information, which used to be hiperbolic în construcții Ed. Academiei, București,. gathered by an entire team of specialists – in this way, [5] B., Kolarevic (2003), Architecture in the Digital Age: the vast processes that have been transformed have Design and Manufacturing, Ed. Taylor & Francis created a totally new type of thinking called digital Group, New York, design thinking, which generates new shapes and relations between designer, the geometry of the shape and information, in a much shorter period of time. This Author: new approach facilitates the trans-disciplinary collaboration and makes it possible for designers to Teaching Assistant Ph.D, Ana-Maria RUSU, include many more variables in their creations. The Technical University of Cluj-Napoca, Faculty of designer can integrate simulations of the object’s Architecture and Urban Planning, E-mail: behavior under specific conditions, the built objects’ life [email protected], tel: 0040747102159. cycle, their economic and environmental consequences in the original design of the object. And still, without

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