QuasiSculpt Four-dimensional modeling, quasicrystal tilings, self-assembling systems and the geometry that connects them.

Vladimir Sedach January 29, 2007

1 Abstract The purpose of this document is to act as a proposal for the author's CPSC 503 project in the Winter 2007 term under the supervision of Professor Marina Gavrilova. The goal of the proposed project is twofold: first, to construct a software system for visualizing four dimensional objects using projective geometry methods, and then to utilize that system to investigate the applications of quasicrystal tilings for creating artificial two-dimensional self-assembling systems.

2 Introduction Visualization of four-dimensional objects has a rich history going back to the turn of the 20th century, when after being explored by mathematicians it was quickly adopted by artists such as Pablo Picasso and became the foundation for cubism [Robbin06],[Henderson83]. Projective geometry was first developed by Kepler and Desargues in the 17th century by extending Euclidian geometry with points at infinity, and has become the foundation of three-dimensional computer

graphics

[SempleKnee60],[Rosenbaum63],[Coxeter74],[PennaPat86],[Herman92].

The discovery of aperiodic crystalline structures that produced diffraction patterns with five-fold rotational symmetry, dubbed quasicrystals, in the 1980s was quickly followed by the finding that such n-dimensional crystalline structures could most generally be produced by projecting regular (hypercubic) lattices from m>n dimensions, and in particular many types of quasicrystal tilings (of the plane) can be generated by projection of hypercubic lattices from 4-space [Janic88],[Janic89],[KramSchlott89],[BaakZeid90],[Senechal96],[Peterson01]. This means that quasicrystal tilings can be generated from regular hypercubic lattices by the same projective

method used to visualize objects in the fourth dimension. The proposed project aims to do exactly that - extend an existing three-dimensional computer graphics system, extend it to show four-dimensional objects by projection, and then use it to generate quasicrystal tilings in two dimensions. These tilings will then be investigated for their applicability to the creation of artificial two-dimensional self-assembling systems. Self-assembly is one of the principal methods of organization of matter in nature [Cramer93]. Artificial self-assembling systems is a recent development to attempt to apply the same principles of organization to produce systems with specific desirable properties (solar cell arrays, etc.) from components greater than molecules in size [WhiteBoncheva02],[Bhalla04].

3 Previous Work With the emergence of computer graphics, there has been a renewed interest in visualizing the fourth dimension, this time using interactive three dimensional graphics software, and several systems have been developed to this end [Noll67],[FeinBesh90],[Banks92],[Robbin92], [Emmer93],[Peterson01],[Robbin06]. Eugenio Durand has developed a software system for generating quasicrystal tilings by the projection method, called QuasiTiler [QuasiTiler]. The necessity of geometric constraints has been recognized as being critical to the construction of self-assembling systems [Bhalla04],[WhiteBoncheva02]. Although crystals are an example of natural self-assembling systems, [Ball99] to the author’s knowledge no one has yet investigated the application of quasicrystal tilings and the geometric constraints they embody to the construction of artificial two-dimensional self-assembling systems.

4 Proposed Work The QuasiSculpt project will consist of two stages. The first stage of the project will consist of building a software system to perform procedural transformations of objects in four dimensions and interactive visualization of four-dimensional objects by projection onto three and two dimensions. This component will be based on an existing Free Software/Open Source 3d modeling system, Wings 3D [Wings3D]. To the author’s knowledge, there is currently no available four-dimensional visualization systems integrated with computer graphics tools intended to be used by artists. The goal of the QuasiSculpt software will be to allow all users to

visualize four-dimensional objects and the effects of various procedural operations on three-dimensional objects in four-space, while allowing more sophisticated users a way to specify those procedural operations and specific parameters for projection to lower-dimensional spaces to achieve a particular effect. The interactive part of the QuasiSculpt software will be used by Professor Gerald Hushlak at the Faculty of Fine Arts to create abstract artwork. The second stage of the project will consist of using the software system developed in the first stage to explore the application of quasicrystal tilings to two-dimensional artificial self-assembling systems. These lattices can be generated procedurally using the QuasiSculpt software developed in stage one. The goal of this part of the project is to explore the resulting quasicrystal patterns for desirable properties the tiles may have as components in artificial self-assembling systems. Part of this research will be to establish some suitable evaluation criteria and set bounds on the parameters of the self-assembling systems being considered. Due to the limited time available, the evaluation will likely take place by considering a single artificial two-dimensional self-assembling system and seeing how a number of different tilings can be used for making components for that self-assembling system. The evaluation will give some preliminary results as to the viability of using quasicrystal tilings for generating the shape of the components for self-assembling systems.

5 Project Timeline

QuasiSculpt is planned to be in a working state (defined as being usable for interactive explorations of 4d objects) at the end of reading week. From that point onwards, the project will focus on exploring the applicability of quasicrystal tilings to self-assembling systems.

6 Conclusion The fourth dimension and quasicrystal tilings are intimately linked via projective geometry. Artificial self-assembling systems is an emerging field of research, where geometry plays a key role. The goal of the proposed project is to construct an interactive software system for visualizing four dimensional objects using projective geometry methods, and then to utilize that system to investigate the applications of quasicrystal tilings for creating artificial two-dimensional self-assembling systems. In addition, QuasiSculpt will provide artists with a way to interactively view and construct four-dimensional objects. Due to the time constraints on the project, both of these parts of the system will not be as complete as they could be. In particular, the investigation of quasicrystal tilings for creating artificial self-assembling systems will be very preliminary. A more in-depth investigation is a possible area for future research. Investigating the space of possibilities for the artistic application of four-dimensional visualization with current graphics software is another interesting idea for future work.

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