#### Crinkled Canopy:

#### Random Fractals

This research aims to explore the scope of applying the concept of fractal geometry in the field of architecture and construction. There are mainly two different types of fractals – self-similar fractal and random fractal. In this research, both types of fractals are used to design a nature-inspired architectural structure with the strategy of exploring the potency of fractal geometry as a geometric framework that can offer new structural forms. Based on themathematical formulations of self-similar fractal shape and random fractal shape, tree-inspired branching supports and natural terrain inspired unsmooth crinkled roof are modeled using the algorithms of Iterated Function System and Midpoint Displacement (Diamond Square Algorithm) method respectively. Fractal dimensions are calculated to assess the visual complexity of the roof surface and branching supports. Finite element analysis is performed to assess the structural strength of the model with respect to changing of fractal dimensions.

In the proposed canopy structure, the roof form is made crinkled inspired by the morphology of natural terrain, which is an example of nature’s random fractals, sometimes known as the ‘fractal landscape’. The selection of crinkled surface over the flat surface for designing the roof, as mentioned before, is because of the apparent higher stiffness of the crinkled surface than that of a flat surface since folds in the crinkled surface act as self-stiffeners. Crumpling of a piece of paper is a relevant and ubiquitous example of a stress-induced morphological transformation in thin sheets (Blair and Kudrolli, 2005). As shown in ‘Figure 7’, if a flat piece of thin paper is placed at the top center of a bottle, then the two sides of the sheet will bend down. But, if that paper after being crumpled and unfolded is placed at the same position on the bottle, then the crinkled version does not noticeably bend down. It, rather, stays almost non-deformed, because ridges of crinkles, i.e., its random folds after crumpling act as stiffeners. Furthermore, the crinkled paper sheet can carry the load to some extent with a negligible deformation. Each crumpling produces a unique pattern of random folds or crinkles. Therefore, countless crinkle patterns can be obtained by the countless crumpling of a thin piece of paper.

The method DS Algorithm that produces fractal terrain. Red dots are the new vertices while the blue dots represent previous vertices. Top and middle - Plan views; Bottom – Isometric views.

Natural terrain-like crinkled surface generation from a flat surface using DS Algorithm

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Deformations of a roof due to self-weight when its flat surface is transformed into crinkled surface

The displacements of the roof under the gravity and mesh loads (1 KN/m2) with the changing of fractal dimensions, while z-limit is constant (1.5).

Finite element analysis has been performed on this model to assess its structural behavior, thus explored the relation between the factor of irregularity, i.e., the fractal dimension and the structural strength. Besides, it has also analyzed the relation between the fractal dimension of the roof form and its weight. Finite element analysis has been used here as a first-hand assessment of structural testing which is helpful in the early stage of conceptual design development. A brief structural analysis in this study has confirmed the structural feasibility and the enough strength of such an irregular canopy structure. Finite element analysis has shown that, as compared to a flat roof, a crinkled roof exhibits a unique self-stiffened quality which encouraged us to develop this crinkled canopy structure.