 # Frac Shell

### Completed, Summer 2014

Over the course of 6 months, students of the Computer Morphogenisis Lab at the Politecnico di Torino in Italy developed structures through the application of mathematics in architectural design. With the assistance of Karamba, they studied and analyzed fractal geometries which were then fabricated at 1:1 scale.

Text and Images from Iasef Md Rian:

Students had selected and applied the concept of Fractal Geometry, a relatively new geometric system developed by Benoit Mandelbrot in the 1970s. Fractal geometry has unique characteristics of self-similar repetitions and the roughness or irregularity at multiple zooming scales. Mathematically, fractal shapes are the shapes that are not integer dimensional, but non-integer dimensional shapes. In this workshop, students developed a geometric model of a surface based on the fractal concept, which was neither a two-dimensional smooth surface nor a three-dimensional solid-like mesh. Their fractal surface was developed by following Takagi-Landsberg’s mathematical function of the fractal surface. This surface is transformed gradually from a smooth paraboloid surface to an unsmooth fractal surface. The shape transition was controlled by the parameter of Fractal Dimension (DH). Students used Takagi-Landsberg’s fractal surface as a base for constructing a grid-shell structure which they named ‘FracShell’. Construction of Fractal Surface based on Takagi-Landsberg method. The midpoint of each side of a triangle is moved up vertically. It creates 4 new triangles. The process of midpoint displacement is repeated, and recursively continued to each newly born triangle. This iterative process produces a fractal surface which is unsmooth at each zooming scale. The roughness of the fractal surface can be controlled by a factor of relative size value, which is the factor of midpoint displacement value. This value is also related to the value of the fractal dimension. When the fractal dimension is 2.0, the surface becomes smooth and paraboloid, but when its fractal dimension value becomes more than 2.0, the surface turns into unsmooth, i.e., fractal. A quick review confirms that the fractal based grid-shell (b) is not a good stand-alone structure unless an additional framework (a) is added.

Before the construction, the team verified the structural capacity with Karamba. After passing the verification under self-weight, students started to construct the grid-shell completely manually without taking any help of robotic support or CNC. Fractal geometry has a unique quality of self-similarity, which means, each module is a scaled copy of its parent module, and that parent module is further scaled copy of its own parent module, and so on. Thus, this unique fractal scheme helped the students to develop a typical module first that was repeated, saving many hours of fabrication time. The digital model having an additional frame attached to it is developed from the selected fractal shape having a fractal dimension of 2.6. The selected model has been structurally analyzed under three different load cases. The structural behavior in terms of maximum displacement under three different loads are checked and compared with the smooth paraboloid grid-shell. The stress behavior under three different loads are checked and compared with the smooth paraboloid grid-shell and the results are mapped in the adjacent graph.  The buckling behavior under three different loads are checked and compared with the smooth paraboloid grid-shell and the results are mapped in the adjacent graph.  The fractal-based grid-shell has self-similar repetitive modules that are scaled copies of their parents. This scheme of fractal shape has allowed students to make singular module first and then repeated it. The most challenging part was the joints and connections. Students created wooden ball manually and then make holes by following the vectors of the wooden members. The major self-similar units were produced by assembling multiple self-similar modules. The central part of the structure was then added to complete the structure.  Host University: Politecnico di Torino
Host Lab: Computer Morphogenesis
Tutors: Iasef Md Rian, Mario Sassone, Shaghayegh Rajabzadeh
Students: Bruno Iorio, Leonardo Ramondetti, Elisa Pitassi, Samuele Marino, Leonardo Ramondetti, Gabriele Fusaro
Funded by: Politecnico di Torino’s Departmental Research Grant