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