ARCHITECTURE, MATHEMATICS & NUMBERS. Part II
In the last Article, we could see how Architecture is related to mathematics as it is with geometry, with the roots in Pythagorean theories. The value of mathematics and numbers goes beyond and is related with some symbolism and even religious and spiritual meaning as the one conferred by the golden ratio that was used in Egypt, Greece, Rome and the Islamic world.
In this sense, any building in the world embodies a complex set of calculated dimensions, location principles and shapes based on some geometric and mathematics formulas.
MODERN MATHEMATICS AND ARCHITECTURE
At the end of the nineteenth century, Vladimir Shukhov in Russia and Antoni Gaudí in Barcelona pioneered the use of hyperboloid structures; in the Sagrada Familia design, Gaudí also incorporated hyperbolic paraboloids, experimental arches, helicoids and other kind of surfaces with a mathematical base. In the twentieth century, modern architecture and especially linked with Deconstructivism, explored alternative geometries and construction systems as Richard Buckminster Fuller developed in his geodesic domes thin structures.
Geodesic Dome - Buckminster Fuller
The work of Antoni Gaudi is a clear example of designs based on these types of curves, based on a set of mathematical parameters like a parabola. It is thought that Gaudi may not have used this mathematical approach to achieve these shapes, but he did create these forms based on the catenary shape. He could also invert the arcs these arcs to define his designs, and essentially based these shapes as a physical connotation to the mathematical equivalent of these parameters.
Sagrada Familia Cross Section -Source Structural Ice
In the mid 20th century, designers and engineers, such as Frei Otto, would create these curving designs by utilizing mathematical equations as “parameters” to engineer these curves and forms.
MATHEMATICS AND PARAMETRIC ARCHITECTURE
The relationship between mathematics and parametric architecture has its origin 40 years ago. Since then, the development of computers and design software has simplified the design process and made it possible to perform this sinuosity almost for anyone.
TWA Airport Terminal - Eero Saarinen
One of the first tries into these new mathematical methods was done in the TWA Airport Terminal, designed by Eero Saarinen. It is one of the first attempts into Concrete Parametrical Architecture.
Algorithms and Architecture
The term “parameter” is a key term that defines a variable for which the range of possible values generates a collection of distinct cases in a problem.
In Design, this parameter is basically defining a form or a line. This is particularly important in the present, when we are mostly using our computers as the only design instruments. Using these forms of mathematics in computers, these curves can be created in 3D software systems such as CATIA, Rhino/Grasshopper, Nemetshek and some ofthe newer versions of AutoDesk/AutoCAD offerings.
Parametric Plugin - SourcePinterest
This type of “nodal interconnection” of objects, shapes and forms in a parametric format creates alive structures and architecture that evolves as the software performs the parameters. From NURBS curves and surfaces to fractals and Voronoi diagrams, the level of complexity and range of chances is infinite when entering into the parametric world.
This Parametric Architecture allows us to experiment, change, create, replicate and reproduce any shape that could be described by a mathematical formula, giving us the chance to create our formulas and increase the design chances.
Louvre Museum Abu Dhabi - JeanNouvel
Louvre museum in Abu Dhabi is one good case of what can be achieved with parametric architecture. This project is covered by a 180m-diameter perforated dome with a superimposition of layers of the different layers, designed based on a parametrical design.
Beijing National Swimming Center - PTW Architects
A different form of parametric design and translucency is achieved on the blue Watercube by PTW Architects for the Beijing Olympics, through the synthetic skin on all five exposed faces of a simple cubic shape.
Parametric Tools
Parametric Architecture uses different basic elements and tools in whim the replication-reproduction will be based following the mathematical formulas. Starting from basic NURBS lines and Surfaces, algorithms will be able to transform them into 3D grids.
Grasshopper Archicad Surface Grid- Source Horst Sondemann
With fractals, the parametric design has the tool to create never-ending patterns. Fractals are infinitely changeable patterns based on the repetition of a simple basic geometric element.
As Fractals, we can also recreate complex geometries with Voronoi diagrams, a plane division tool based on generation points that define a polygonal basic shape.
Voronoi Parametric Design - Source Designboom
One of the resources of Parametric Design is the Morphodynamic, defined as an organism’s response to external influences to create a pattern and form in nature. There are many forms in the natural world, such as the spiral horns of a Dall sheep, that were shaped not by evolution, but as a direct consequence of the conditions of growth or the forces in the environment.
Kartal-Pendik Masterplan, Istanbul- Zaha Hadid Architects
The reproduction of these external influences applied to urban planning can result in a perfect shaping experiment like the Kartal-Pendik Masterplan, Istanbul by Zaha Hadid Architects. It is an improvisation experiment in which the result is unknown as it was a real evolution process.
Slime mould - Source Parametric Monkey
Like these examples, parametric design in architecture is making it possible to achieve any shape that our creative minds can imagine. The curious fact is that at the same time design is getting more and more complex, parametric design allows designers to reproduce natural shapes, making architecture get closer and closer to its origin in nature.
These shapes and geometrical compositions have always been there but they were almost impossible to be achieved without computer technology.