‘FracShell’: A Grid Shell Design based on Fractal Geometry


‘FracShell’ is an output of a computational design workshop held in Politecnico di Torino, Turin in Italy in 2014. This workshop was based on a theme which was to find a novel form for designing a shell structure using unique geometric shape, and ‘fractal geometry’ was the best choice we had for representing radically a new type of form system and exhibiting its potency to produce an innovative structural system.

Archimedes’ famous ‘midpoint displacement method’ was used in our project to automatically design the basic form of the shell structure. Usually, Archimedes’ method is one of the classic mathematical tools to produce a parabola in the two dimensional space and a paraboloid in the three dimensional space. In Archimedes’s method, the value of height factor ‘w’ (midpoint displacement value) is 1/4 which is the key responsible for producing a parabola and a paraboloid from a straight line and a flat polygon respectively.

Later, in 1910, Teiji Takagi modified Archimedes’ method and replaced the ‘w’ value by 1/2, and found a remarkable change. The smooth curve of the parabola or the smooth surface of the paraboloid turns into unsmooth and rough with statistical self-similarity, which was later defined by Mandelbrot (1983) as a ‘fractal’. This new fractal curve is known as ‘Takagi curve’ and its three dimensional counterpart is known as ‘Takagi mountain’. (Mandelbrot 1987) Much later, George Landsberg further modified the Archimedes’ and Takagi’s method, and generalized the ‘w’ value to a changeable floating number that ranges from 0.25 to 1.0. Parametrically, the changing of ‘w’ value changes the texture of the surface of a paraboloid. In grasshopper (parametric) software embedded in Rhinoceros3D (CAD software), it can be shown animated way that ‘w’ value exploits the dimension of the paraboloid’s smooth surface, thus changes the fractal dimension ( which is fractional and non-integer) of the surface from 2 to 3, especially when ‘w’ ranges from 0.5 to 1.0.

midpoint displacement


fractal shell 1

Based on this geometric venture of exploiting a regular geometric shape into a fractal shape, we intended to apply this shape morphology for structural form design as a gridshell structure, and see how it mechanically reacts and changes structural behavior with the changing of ‘w’ value.

However, after finite element analyses process, as a structural feasibility check, we had decided to make its real scale physical prototype by taking ‘w’ value as 0.5. ‘FracShell’ was a design team which devoted their times to fabricate the structure by facing some critical challenges. However, in the end, we were able to complete the job, and see the physical realization of a mathematical entity as an innovative structural form in architecture.

FracShell construction 3





  1. Mandelbrot, Benoit B. The fractal geometry of nature. Macmillan, 1983.
  2. Mandelbrot, Benoit B. “Fractal landscapes without creases and with rivers.”The science of fractal images. Springer-Verlag New York, Inc., 1988.


Special thanks to , Bruno Iorio, Elisa Pitassi, Gabriele Bonnet, Gabriele Fusaro and Samuele Marino


One Comment to “‘FracShell’: A Grid Shell Design based on Fractal Geometry”

  1. I am highly thankful to Leonardo Ramondetti, Bruno Iorio, Elisa Pitassi, Gabriele Bonnet, Gabriele Fusaro, Christian Allera and Samuele Marino. You all are genius 🙂

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