Quasicrystal Structure Inspired Spatial Tessellation in Generative Design

QUASICRYSTALSTRUCTUREINSPIREDSPATIAL TESSELLATIONINGENERATIVEDESIGN

PENGYU ZHANG1 and WEIGUO XU2 1,2School of Architecture, Tsinghua University [email protected] [email protected]

Abstract. Quasicrystal structure is a kind of quasiperiodic spatial tessellation formed by several kinds of tiles. Compared with periodic or other aperiodic tiling, it shows superiorities but also drawbacks when used for generative design. It can generate attractive and irregular novel forms with controllable cost for construction, but its strict rules restrict its variety. To cover the disadvantages of these tessellations without diminishing their advantages, a new kind of spatial tessellation, named as Periodic-to-Aperiodic (P-A) Tiling is proposed in this paper with a series of installation design cases, inspired by the primary principles and architectural applications of quasicrystal structure.

Keywords. Spatial tessellation; Quasicrystal structure; Generative Design.

1. Introduction Natural forms are prototypes for many generative designs, creating various novel forms. Within the diverse natural forms, biological forms are used as prototypes more than other natural existences. Unlike biological forms with the ability to act and evolve for adaptation, inorganic substances, such as stones and crystals, hold a variety of valid structures or high density to resist the various external pressure or infringement. Thus, they also have special properties to be used as prototypes in generative design, such as the quasicrystal structure. In this paper, a new tessellation inspired by the quasicrystal structure is introduced detailly.

1.1. QUASICRYSTAL STRUCTURE AND PREVIOUS CASES Quasicrystal discovered by Shechtman is formed by a quasiperiodic spatial tessellation in microstructure. Shechtman first observed the tenfold electron diffraction patterns of quasicrystal in 1982, implying the existence of a new kind of three-dimensional tessellation in nature (Shechtman and Blech 1985). It is ordered but not periodic. As an analogy, it is neither periodic like crystals nor unordered like stones. It appears to be a transition from stones to crystals, from unordered to ordered, and from aperiodic to periodic. The two main types of quasicrystal imply its characteristics as a transition: one is periodic in one axis and quasiperiodic in planes normal to it; the other is aperiodic in all directions (Yamamoto 2008). It is inorganic but adaptable and responsive to the surroundings with several states like

T. Fukuda, W. Huang, P. Janssen, K. Crolla, S. Alhadidi (eds.), Learning, Adapting and Prototyping, Proceedings of the 23rd International Conference of the Association for Computer-Aided Architectural Design Research in Asia (CAADRIA) 2018, Volume 1, 143-152. © 2018 and published by the Association for Computer-Aided Architectural Design Research in Asia (CAADRIA) in Hong Kong. 144 P. ZHANG AND W. XU the organisms. And this similarity indicates its potential to be used as prototype in generative design, in a similar way of the organisms. Before the discovery of quasicrystal in nature, in mathematics, one similar 2D pattern, specifically Penrose Tiling (Figure 1), is a kind of quasiperiodic tiling and introduced by Roger Penrose in 1974 (Penrose 1974). It is different from periodic tilings which can form a new copy by translation over a fixed distance in a given direction (Grünbaum and Shephard 1987); with symmetrical axes, it is more regular than common kinds of aperiodic tilings, such as Voronoi. Furthermore, De Bruijn introduced the methods to construct Penrose Tiling in 1981 and viewed it as 2D slices of five-dimensional hyper-cubic structures (De Bruijin 1981). This indicates the potential relationship between the 2D tiling and 3D structures.

Figure 1. Penrose Tiling, a kind of quasiperiodic pattern with two kinds of tiles (Penrose 1974, https://en.wikipedia.org/wiki/File:Penrose_Tiling_(Rhombi).svg; https://en.wikipedia.org/wiki/File:Penrose_rhombs_matching_rules.svg).

Quasicrystal Structure was used as the prototype for building skin generation in the design of Battersea Power Station Redevelopment by Cecil Balmond, making a unique and structurally valid form (Balmond and Yoshida 2006; Figure 2, left); and our studio also conducted a structural design based on its performance (Figure 2, right). Furthermore, its form with the unique and simple formal rules was fully explored by our studio in an architectural design case of Port Terminal Design (Figure 3). In this case, based on the spatial structure of Al-Mn Quasicrystal, the architectural form was generated from an icosahedron (any polyhedron having twenty plane faces) with octahedrons (any polyhedron having eight plane faces). QUASICRYSTAL STRUCTURE INSPIRED SPATIAL TESSELLATION 145 IN GENERATIVE DESIGN

Figure 2. Two Previous Design Cases (The left picture is credited to a+u Cecil Balmond).

Figure 3. Form Generation Diagram and Functional Layouts for Port Terminal Design.

1.2. ADVANTAGES AND DISADVANTAGES The previous cases show the advantages and disadvantages of quasicrystal used as prototypes. The quasicrystal structure also has some superiorities when compared with periodic and aperiodic tiling. Periodic tiling, such as crystals, follows strict mathematical rules to form spatial tessellation by one or several kinds of tiles (Mozes 1997). Comparing it, the quasicrystal structure has more variability and interesting. Every part of the quasicrystal structure is similar but different, attracting the viewers to seek its principles. Aperiodic tiling, such as Voronoi, 146 P. ZHANG AND W. XU is a way to generate spatial tessellations composed of diverse tile shapes by computational geometry (Aurenhammer 1991). In comparison with it, the quasicrystal structure has fewer types of tile shapes cutting down the cost directly. And the cost is almost the most significant aspect in most cases nowadays. Nevertheless, quasicrystal structure also has a shortage. It has just a few types due to its strict rules. This means that its application in architectural design is limited to these few types. Thus, under its inspiration, a new spatial tessellation named “P-A Tiling (Periodic-to-Aperiodic Tiling)”, with its advantages is proposed with the approaches and cases in this paper, to account for its disadvantages in architectural design.

2. Approach and Stages The approach of this research is similar to the way to solve technical problems by searching for biological analogies (top-down) (Knippers and Speck 2012). The proposed tessellation is evaluated and promoted as a new one to cover the shortages of the previous prototypes. Considering the advantages and disadvantages of quasicrystal structure and other tessellations, it comes to an improvement of quasicrystal structure. A new tiling is proposed to account for the disadvantages, generating additional various and adaptable forms with fewer kinds of tiles (Figure 4). Although the quasicrystal structure is of a few types, a variety of forms can be generated by P-A Tiling.

Figure 4. Approach Diagram.

The method for P-A Tiling has four main stages: Basic Geometry, Basic Form Generation, Spatial Tessellation, and Adjustment (Figure 5). Python for Grasshopper is used for programming. At the first stage, basic geometries are selected with two parts. One is the QUASICRYSTAL STRUCTURE INSPIRED SPATIAL TESSELLATION 147 IN GENERATIVE DESIGN repeating parts, composed of one or two kinds of shapes, and make up the basic form and the regular part of the final form. The other part is composed of periodic tiling (PT) that can be a cluster of cubes or other polyhedrons. Second, the basic form (BF) are generated by duplicating and connecting the regular shapes in stage one without any valid intersections. The designers can adjust and optimize BF by hand or by setting rules in programs only if no valid intersection happens. The main task for the third stage is to fill the gaps in BF. Wrap BF in PT, and then remove or reshape some shapes of PT, which are inside of or intersect with BF, to make all the shapes in PT connected with BF face to face without extra intersections. At the last stage, the main task is to optimize the whole shape of the form. At the beginning, remove the outer shapes of PT in order to unveil the wrapped BF. Then reshape the remaining part of PT to fit the surroundings, optimize the entity form, and serve structural purposes. Mostly, merging some PT shapes can effectively reduce the quantity of irregular shapes. Afterwards, inspect the whole shape and fill some gaps by program if needed. In the end, a spatial tessellation is obtained.

Figure 5. Diagram for Form Generating. 148 P. ZHANG AND W. XU

3. Applications This method is used for the generation of the third to fifth version of the interactive installations named as Swarm Nest. Swarm Nest is a series of projects conducted by our studio and exploring the forms that indicate swarm behaviours in the nature. In these cases, the process of form generation is similar while the fabrications are quite different.

3.1. GENERATION As for the generation, dodecahedrons and tetrahedrons are used as the main shape, and it was generated by the following steps as it is shown in Figure 6: To start, the dodecahedron (any polyhedron having twelve plane faces) is taken as the repeating shape of the basic geometry, and periodic tiling (PT) formed by two kinds of tetrahedrons (any polyhedron having four plane faces) serves as the irregular part. The polyhedrons in PT are smaller than the dodecahedrons to make it easier to form the spatial tessellation. Generated by the program, a basic form (BF) composed of dodecahedrons is created and adjusted. By comparing the two connections, “vertex to vertex” can better fit in the tessellation with flexible angles and large gaps. Then, the dodecahedrons are wrapped in the periodic tiling of tetrahedrons and octahedrons. To form a spatial tessellation, the PT tetrahedrons that are fully inside the dodecahedrons of BF are removed. For the tetrahedrons that have intersections with dodecahedrons, the vertices inside the dodecahedrons are moved to the closest vertex in BF. In this way, the intersections can be diminished, and the gaps between the shapes can be filled by adding tetrahedrons or reshaping the neighbouring tetrahedrons. Moreover, the shapes of tetrahedrons are approximately aligned to the regular tetrahedron by moving some of their vertices. Afterwards, a spatial tessellation composed of tetrahedrons and dodecahedrons is almost created. Finally, some outer tetrahedrons are randomly removed to expose the wrapped dodecahedrons, while some inner shapes are removed to form cavities. Furthermore, the outer shapes are reshaped by dragging their vertices to modify the whole form into a better appearance. By merging all the shapes into an entity and calculating its outer surface, the gaps inside will be found and can be filled by tetrahedrons if gaps still exist. Then, some tetrahedrons merge into octahedrons or hexahedrons (any polyhedron having six plane faces) to reduce the quantity of all the polyhedrons in the final form. After all, the final form is obtained. QUASICRYSTAL STRUCTURE INSPIRED SPATIAL TESSELLATION 149 IN GENERATIVE DESIGN

Figure 6. Diagram for Swarm Nest by P-A Tiling Method.

3.2. FABRICATION There are three different ways of fabrication based on different materials and structures. The first one is made up of transparent acrylic sheets and hinges. To make the installation, the whole form is deconstructed into numbered faces, and these faces are made of acrylic by laser cutting. A series of numbered circular acrylic sheets are used for the connection, and the incisions of the sheets fix the angles between faces. The acrylic hinges are also used for the connection. Five students finished this 1.8-meter-long model in 15 days (Figure 7).

Figure 7. Details of Swarm Nest 3. 150 P. ZHANG AND W. XU

The second one is made up of white acrylic and 3D printed joints. Similar to the first one, the whole model is deconstructed into numbered faces that are made of white acrylic. The joints are carefully designed and tested with different shapes to make a perfect a form. The joints are also numbered. And the connection without glue or screw is much easier and faster to fabricate than the former one. The whole form is assembled by one person in less than 5 days (Figure 8).

Figure 8. Details of Swarm Nest 4.

The third one is composed of Aluminium sheets. Different from the former cases, each polyhedron of the whole form is unrolled into a surface with extra shapes on it for connections. And then, fold these Aluminium surfaces into a certain shape and connected them together. Because the Aluminium is light and thin, the whole shape is much lighter and bigger. The whole process of assembling takes 10 students for about 4 days (Figure 9).

Figure 9. Details of Swarm Nest 5.

4. Advantages P-A Tiling shows its advantages based on the Swarm Nest cases. QUASICRYSTAL STRUCTURE INSPIRED SPATIAL TESSELLATION 151 IN GENERATIVE DESIGN

First, the principles of the tessellation make it easier to generate various and adaptable forms, which inspiring the architectural design and making it suitable for much more real sites. A variety of forms can be generated by changing the basic geometries of the kinds of polyhedrons or their positional relationships. The changeable final form makes it adaptive to different surroundings by adding interferences. Second, the kinds of components in the final form can be adjusted to reduce the construction cost or enrich its morphological diversity. The more that duplicates of the repeating parts are generated at the first stage, the fewer kinds of polyhedrons will be included in the final form, making it easier to construct; on the contrary, when the kinds of polyhedrons increase, it shows more variability. Finally, the designer can change the form directly and conveniently with the help of computer programs, making it easier for the designers to control the final form without too much randomness. On the one hand, because the principle of this tessellation is unique from others with strict mathematical rules, the designer can set rules for its generation. For example, the designer can select different kinds of polyhedrons as basic geometries and adjust their positional relationship, resulting in various final forms. On the other hand, different from the previous generative design, the final form is generated not only by programs but also by the designer’s hand-made modification assisted by programs. For example, at the fourth stage of the Swarm Nest, the outer polyhedrons are reshaped by dragging the vertices with the help of programs. In this way, the process is easier than remodelling polyhedrons one by one and is done more directly than changing the parameters in the programs to optimize the polyhedrons. However, some problems still exist and require further research. More polyhedrons need to be used in this tessellation to see if some polyhedrons are not suitable for this method to form a spatial tessellation. The program for aligning irregular shapes still needs to be optimized to meet different requirements, such as the least number or kinds of irregular shapes. Although the connections of repeating shapes were considered in this process, more work is required for a new method to organize the repeating shapes.

5. Conclusion and Future Work As a new spatial tessellation inspired by quasicrystal structure, P-A Tiling can inherit advantages of the quasicrystal structure and minimize its drawbacks, making it more potential to be used in architectural design. On the one hand, P-A Tiling inherits its transitional features. The form based on P-A Tiling is generated by transferring the periodic tiling into aperiodic tiling. Part of the form is formed by repeating one or several kinds of tile shapes like periodic tilings, and others are composed of a series of different irregular shapes similar to aperiodic tilings. Thus, the new tessellation can also generate variable forms with fewer kinds of shapes as does a quasicrystal structure, cutting down the construction cost significantly. On the other hand, more kinds of various and adaptable forms can be obtained by the new spatial tessellation than with a quasicrystal structure. More polyhedrons can be used for P-A Tiling, resulting in more various novel forms 152 P. ZHANG AND W. XU and inspiring the architectural design indeed. Additionally, the form is easy to be modified by designers or programs, making it adaptable to different surroundings or requirements. Nevertheless, some work still needs to be done in the future, such as exploring more types of polyhedrons and connections for the tessellation to promote the efficiency of prototype selection, and optimizing the structural capability with mechanical analyses. Furthermore, the forms need to perform a certain function to explore its application in architectural design. This research is supported by National Natural Science Fund of China (NO.51538006).

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