Abstract
Geometry and architecture both have a long trajectory in the history of western thought. Geometry offers the possibility to interpret the physical structure of the world and to develop rational thinking, while architecture provides the capacity to transform the physical substance and meaning of our surroundings. Diverse developments in the field of geometric representation have determined the characteristics of architectural space: from the modulated rigour of Classicism and the birth of Euclidean geometry, to contemporary informalism with the incorporation of digital mathematical calculation and intense questioning and reconsideration of traditional Cartesian space. The two-dimensional constant and dynamic projection of a three-dimensional spatial situation has been upheld since the time of simple spatial-temporal allegories of the architectural project up until the new developments with unconventional instrumental resources, generating innovative structural, formal, spatial and technological solutions.
TopSince antiquity, geometry has become established as the most powerful tool with which to conceive and design architecture. Functional, technological, symbolic and cultural aspects are upheld by geometric logic, modified in accordance with changes and ruptures in our ways of thinking, doing and designing, in line with cultural and temporal parameters. The Greeks gave us a completely original way of designing architecture based on a modular system: the classical orders are a synthesis of connected geometric forms and relationships that enable a work of architecture to be described as an interconnected system of measurements. This form of regulated design remained decisive for architecture up until the Industrial Revolution. Although the system of proportions used in antiquity was brought back into use during the Renaissance and then lost force in modern times, the modular concept as a basis for the rationalisation of form prevailed through the centuries. The work of nineteenth century engineers marked the formal and conceptual break with architectural building traditions, rejecting the system of geometrical proportion passed down from antiquity. Instead, they imposed a strong practical line of thought to resolve problems, wherein the new geometric modularity is adapted in response to the results of graphic calculation, scientific analysis and the demands of other construction materials and systems. Descriptive geometry, with its orthogonal projections, provided another mechanism with which to understand and analyse determined forms and spaces, and is still the most frequently used geometric construction today in most architecture and design work. Modern rationalism established the functional principles and incorporated the three-dimensional reticulated structure now possible with reinforced concrete in order to organise the spatial volumetric syntax. All parts of the architectural whole were now subject to the three-dimensional repetition of measurements, in the form of a prismatic cage. This is the fullness of Cartesian space: a homogeneous one-directional continuity within the field of geometrical rigour; the realization of an isomorphism within a measureable and unlimited space. (Chiarella, 2009)
Through the use of computer graphics, the new (non-linear, dynamic and unpredictable) digital processes of mathematical calculation modify existing notions of spatiality by distancing certain geometric qualities that we historically identify with architecture; more precisely, by moving away from the three-dimensional stability and modular rigour inherent in the Cartesian space of predominately Euclidean geometries. Luis Fernandez-Galiano in “Form of the Formless” (Fernandez-Galiano, 1996) describes these trends characterized by the use of complex geometries such as: “Formless architecture is very formal, ..., If architecture is above all about construction and form, stripping it of any of these attributes can only be interpreted as polemic intention and literary provocation.”; “and following the deep currents of a sordid and sublime romanticism, we can imagine a drift that takes us from the fractures of constructivism to the tensions of expressionism, and from there to the tensions of expressionism, and from there to the foreshortened warpings of informal organicism; from folded and geometrical skins to bones, trees and clouds, and from there also to a universe of innards a caves. These are shifts which, within the general framework of a shattered abstraction, lead through the path of the romantic repertory from crystals and the broken geology of mountain peaks to opaque and secret woods, crowned with foliage, and labyrinthine caverns: an entire panoply of expressive resources to endow the formless with form.”
Key Terms in this Chapter
Digital Fabrication: Digital fabrication is a type of manufacturing process where the machine used is controlled by a computer. Are classed according to the processes acting upon the physical and/or chemical properties of the materials used. These are: additive procedures, subtractive procedures and formative procedures (according Brankolevic and Malkawi, 2003 AU57: The in-text citation "Brankolevic and Malkawi, 2003" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ), and joining procedures ( Schodek, 2005 ).
Algorithm: Finite sequence of operations feasible, unambiguous, the execution gives a solution to a problem. Or what is easier to say, an instruction to do something. Featured on the new design processes to systematize computer graphics operations.
Pattern: Set of partial results show systematically behavioral traits associated with a particular situation, entity or object.
Fractal: Geometric pattern repeated at progressively smaller scales, where each iteration is about a reproduction of the image to produce completely irregular shapes and surfaces that can not be represented by classical geometry. Fractals are generally self-similar (each section looks at all) and are not subordinated to a specific scale. They are used especially in the digital modeling of irregular patterns and structures in nature. Defined in 1975 by the mathematician Benoit Mandelbrot.
Fold Space: We talk about pleats, folds as possible deployments and dynamic trajectories. Paths to-scalar between structures and organizations, devices and cities, scenarios and projects related to evolutionary geometries (rhizomatic formations, trimmings and lugs, arracinamientos, trusses and kinks, fluctuations revesas and swerving). Open geometries in which whole and fragment parameters respond to open self-similarity between diagrammatic and logical decision-or evolutionary development designed to generate complex configurations from elementary initial codes.
Complexity: Complexity is a phenomenon that involves a lot of interaction and interference between a very large number of units. It is related to chance, while analysis involves uncertainties and random phenomena. With regard to chance and uncertainty the goal of complexity theory is or constant traffic movement in this direction: order-disorder-organization. The Science of Complexity is a rapidly developing corpus dedicated to the study of dynamic natural systems. A set of theories and sub-theories as theories interrelated Chaos of Disasters, of Fractals, and several others related to the phenomenon of self-organization, created and consolidated some of the key concepts in the characterization of contemporary science: chaos; nonlinearity; unpredictability; random; indeterminism; emergency; self-organization; self-similarity.
Rapid Prototyping: Implementation of sound from digital representations using automated manufacturing techniques, such as cutting, milling or thermal prints, allowing get quick reproductions of designs worked on computer models.
Folded Compositions: Folded compositions can be defined like the transformation of a physical form, to double a flat structure (according to the Merrian-Webster Dictionary), as can be seen from the primary experiences (Buri and Weimand, 2010 AU58: The in-text citation "Buri and Weimand, 2010" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ), to contemporary ( Iwamoto, 2009 , Knippers and Menges, 2011 AU59: The in-text citation "Knippers and Menges, 2011" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ; Peters, 2011 AU60: The in-text citation "Peters, 2011" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ).
Representation: The multiplicity of meanings and uses of the term representation makes this term is almost always ambiguous in three ways: in psychology, in epistemology, and the relationship between epistemology and any psychological elements that are adduced to clarify the nature and forms of knowledge. (Ferrater Mora). “Representation” in creative activity, implies that the image updated by perceptual-cognitive device that operates in the subject, which has prior existence. “Prefiguration”, means that the image anticipates what does not yet exist. In designing the project and images re-present ideas, and material conditions and symbolic contexts and materializes objects associated with these ideas.
Parametric Design: Parametric design introduces geometry from a mathematical-algorithmic viewpoint (Woodbury, 2010 AU61: The in-text citation "Woodbury, 2010" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. ). It proposes the generation of a geometry based on the definition of a family of initial parameters of dimensions and the relationships between them. In the design process, algorithms and advanced computing resources are not simply used to represent forms and control complex geometries but to create dynamic and variable design possibilities. The result is not a single solution but rather a family of possible solutions. The variables and algorithms create a tree or matrix of geometric relationships by calculating the range of possible design solutions according to the variability of the initial parameters and components selected.
Interaction: Property by which a computer system and respond directly to data entered by the user initiating thus a kind of productive “dialogue” commands. Most programs used collectively possess at least a limited degree of interaction that is located somewhere on a spectrum ranging from low to high interaction. Among the applications that make best use of the high interaction are those corresponding to the field of so-called virtual reality supported on highly sophisticated sensors and effects.
NURBS: Acronym for Non-Uniform Rational B-Splines. While it is generally considered to have been developed to build digital versions of the design lines used to draw the sections of the hulls of ships and aircraft bodies, it is in the fifties and in the automotive industry where it appeared the need curves represent free paths (those that do not respond to simple geometric shapes such as circle and ellipse arc). Two engineers from the automotive industry (P. de Casteljau and Bezier P.) developed independently and almost parallel the principles of what is now known generically as the spline curves. These mathematical structures allow numerically describe a curve whose geometric layout free translation enables control curve graphically in all instances. To operate, control and design algorithms that overcome the complexity of traditional equations and do it only from the “graphic” and intuitive handling of geometry (ignoring the tedious own geometric-mathematical structures that support abstractions) is a of the most important to the different design disciplines that operate on the space and complex geometries contributions.
Complex Geometries: In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis. Geometry lies at the core of the architectural design process. It is omnipresent, from the initial form-finding stages to the actual construction. Modern constructive geometry provides a variety of tools for the efficient design, analysis, and manufacture of complex shapes. This results in new challenges for architecture. However, the architectural application also poses new problems to geometry. Architectural geometry is therefore an entire research area, currently emerging at the border between applied geometry and architecture. Complex geometries in architecture include central concepts on freeform curves and surfaces, differential geometry, kinematic geometry, mesh processing, digital reconstruction, and optimization of shapes.