DOCSLIB.ORG

Quasicrystal Structure Inspired Spatial Tessellation in Generative Design

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quasicrystal Structure Inspired Spatial Tessellation in Generative Design QUASICRYSTAL STRUCTURE INSPIRED SPATIAL TESSELLATION IN GENERATIVE DESIGN 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
Load more

Recommended publications
  • WHAT IS...A Quasicrystal?, Volume 53, Number 8
    ?WHAT IS... a Quasicrystal? Marjorie Senechal The long answer is: no one is sure. But the short an- diagrams? The set of vertices of a Penrose tiling does— swer is straightforward: a quasicrystal is a crystal that was known before Shechtman’s discovery. But with forbidden symmetry. Forbidden, that is, by “The what other objects do, and how can we tell? The ques- Crystallographic Restriction”, a theorem that confines tion was wide open at that time, and I thought it un- the rotational symmetries of translation lattices in two- wise to replace one inadequate definition (the lattice) and three-dimensional Euclidean space to orders 2, 3, with another. That the commission still retains this 4, and 6. This bedrock of theoretical solid-state sci- definition today suggests the difficulty of the ques- ence—the impossibility of five-fold symmetry in crys- tion we deliberately but implicitly posed. By now a tals can be traced, in the mineralogical literature, back great many kinds of aperiodic crystals have been to 1801—crumbled in 1984 when Dany Shechtman, a grown in laboratories around the world; most of them materials scientist working at what is now the National are metals, alloys of two or three kinds of atoms—bi- Institute of Standards and Technology, synthesized nary or ternary metallic phases. None of their struc- aluminium-manganese crystals with icosahedral sym- tures has been “solved”. (For a survey of current re- metry. The term “quasicrystal”, hastily coined to label search on real aperiodic crystals see, for example, the such theretofore unthinkable objects, suggests the website of the international conference ICQ9, confusions that Shechtman’s discovery sowed.
    [Show full text]
  • Quasicrystals a New Kind of Symmetry Sandra Nair First, Definitions
    Quasicrystals A new kind of symmetry Sandra Nair First, definitions ● A lattice is a poset in which every element has a unique infimum and supremum. For example, the set of natural numbers with the notion of ordering by magnitude (1<2). For our purposes, we can think of an array of atoms/molecules with a clear sense of assignment. ● A Bravais lattice is a discrete infinite array of points generated by linear integer combinations of 3 independent primitive vectors: {n1a1 + n2a2 + n3a3 | n1, n2, n3 ∈ Z}. ● Crystal structures = info of lattice points + info of the basis (primitive) vectors. ● Upto isomorphism of point groups (group of isometries leaving at least 1 fixed point), 14 different Bravais lattice structures possible in 3D. Now, crystals... ● Loosely speaking, crystals are molecular arrangements built out of multiple unit cells of one (or more) Bravais lattice structures. ● Crystallographic restriction theorem: The rotational symmetries of a discrete lattice are limited to 2-, 3-, 4-, and 6-fold. ● This leads us to propose a “functional” definition: A crystal is a material that has a discrete diffraction pattern, displaying rotational symmetries of orders 2, 3, 4 and 6. ● Note: Order 5 is a strictly forbidden symmetry → important for us. Tessellations aka tilings Now that we have diffraction patterns to work with, we consider the question of whether a lattice structure tiles or tessellates the plane. This is where the order of the symmetry plays a role. The crystals are special, as they display translational symmetries. As such, the tiling of their lattice structures (which we could see thanks to diffraction patterns) are periodic- they repeat at regular intervals.
    [Show full text]
  • Simple Rules for Incorporating Design Art Into Penrose and Fractal Tiles
    Bridges 2012: Mathematics, Music, Art, Architecture, Culture Simple Rules for Incorporating Design Art into Penrose and Fractal Tiles San Le SLFFEA.com [email protected] Abstract Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable M.C. Escher-like image that works esthetically as well as functionally requires resolving incongruencies at a tile’s edge while constrained by its shape. Escher was the most well known practitioner in this style of mathematical visualization, but there are significant mathematical objects to which he never applied his artistry including Penrose Tilings and fractals. In this paper, we show that the rules of creating a traditional tile extend to these objects as well. To illustrate the versatility of tiling art, images were created with multiple figures and negative space leading to patterns distinct from the work of others. 1 1 Introduction M.C. Escher was the most prominent artist working with tessellations and space filling. Forty years after his death, his creations are still foremost in people’s minds in the field of tiling art. One of the reasons Escher continues to hold such a monopoly in this specialty are the unique challenges that come with creating Escher type designs inside a tessellation[15]. When an image is drawn into a tile and extends to the tile’s edge, it introduces incongruencies which are resolved by continuously aligning and refining the image. This is particularly true when the image consists of the lizards, fish, angels, etc. which populated Escher’s tilings because they do not have the 4-fold rotational symmetry that would make it possible to arbitrarily rotate the image ± 90, 180 degrees and have all the pieces fit[9].
    [Show full text]
  • A Review of Transmission Electron Microscopy of Quasicrystals—How Are Atoms Arranged?
    crystals Review A Review of Transmission Electron Microscopy of Quasicrystals—How Are Atoms Arranged? Ruitao Li 1, Zhong Li 1, Zhili Dong 2,* and Khiam Aik Khor 1,* 1 School of Mechanical & Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore; [email protected] (R.L.); [email protected] (Z.L.) 2 School of Materials Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore * Correspondence: [email protected] (Z.D.); [email protected] (K.A.K.); Tel.: +65-6790-6727 (Z.D.); +65-6592-1816 (K.A.K.) Academic Editor: Enrique Maciá Barber Received: 30 June 2016; Accepted: 15 August 2016; Published: 26 August 2016 Abstract: Quasicrystals (QCs) possess rotational symmetries forbidden in the conventional crystallography and lack translational symmetries. Their atoms are arranged in an ordered but non-periodic way. Transmission electron microscopy (TEM) was the right tool to discover such exotic materials and has always been a main technique in their studies since then. It provides the morphological and crystallographic information and images of real atomic arrangements of QCs. In this review, we summarized the achievements of the study of QCs using TEM, providing intriguing structural details of QCs unveiled by TEM analyses. The main findings on the symmetry, local atomic arrangement and chemical order of QCs are illustrated. Keywords: quasicrystal; transmission electron microscopy; symmetry; atomic arrangement 1. Introduction The revolutionary discovery of an Al–Mn compound with 5-fold symmetry by Shechtman et al. [1] in 1982 unveiled a new family of materials—quasicrystals (QCs), which exhibit the forbidden rotational symmetries in the conventional crystallography.
    [Show full text]
  • Some Considerations Relating to the Dynamics of Quasicrystals
    Some considerations relating to the dynamics of quasicrystals M. Pitk¨anen Email: [email protected]. http://tgdtheory.com/public_html/. November 12, 2012 Contents 1 The dynamics of quasicrystals, the dynamics of K¨ahleraction, symbolic dynamics, and the dynamics of self-organization 1 1.1 The non-determinism for the dynamics of quasicrystals contra non-determinism of K¨ahleraction . .1 1.2 The dynamics of quasicrystals as a model for fundamental dynamics or high level sym- bolic dynamics? . .2 1.3 Could ordered water layers around biomolecules be modelled as quasicrystal like structure?3 2 What could be the variational principle behind self-organization? 4 2.1 Why Negentropy Maximization Principle should favor quasicrystals? . .5 2.2 Maximal capacity to represent information with minimal metabolic energy costs as a basic variational principle? . .5 2.3 A possible realization for 4-D dynamics favoring quasicrystal like structures . .6 2.4 Summary . .7 1 The dynamics of quasicrystals, the dynamics of K¨ahlerac- tion, symbolic dynamics, and the dynamics of self-organization The dynamics of quasicrystals looks to me very interesting because it shares several features of the dynamics of K¨ahleraction defining the basic variational principle of classical TGD and defining the dynamics of space-time surfaces. In the following I will compare the basic features of the dynamics of quasicrystals to the dynamics of preferred extremals of K¨ahleraction [K1]. Magnetic body carrying dark matter is the fundamental intentional agent in TGD inspired quantum biology and the cautious proposal is that magnetic flux sheets could define the grid of 3-planes (or more general 3-surfaces) defining quasi-periodic background fields favoring 4-D quasicrystals or more general structures in TGD Universe.
    [Show full text]
  • Quasicrystals
    Volume 106, Number 6, November–December 2001 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 106, 975–982 (2001)] Quasicrystals Volume 106 Number 6 November–December 2001 John W. Cahn The discretely diffracting aperiodic crystals Key words: aperiodic crystals; new termed quasicrystals, discovered at NBS branch of crystallography; quasicrystals. National Institute of Standards and in the early 1980s, have led to much inter- Technology, disciplinary activity involving mainly Gaithersburg, MD 20899-8555 materials science, physics, mathematics, and crystallography. It led to a new un- Accepted: August 22, 2001 derstanding of how atoms can arrange [email protected] themselves, the role of periodicity in na- ture, and has created a new branch of crys- tallography. Available online: http://www.nist.gov/jres 1. Introduction The discovery of quasicrystals at NBS in the early Crystal periodicity has been an enormously important 1980s was a surprise [1]. By rapid solidification we had concept in the development of crystallography. Hau¨y’s made a solid that was discretely diffracting like a peri- hypothesis that crystals were periodic structures led to odic crystal, but with icosahedral symmetry. It had long great advances in mathematical and experimental crys- been known that icosahedral symmetry is not allowed tallography in the 19th century. The foundation of crys- for a periodic object [2]. tallography in the early nineteenth century was based on Periodic solids give discrete diffraction, but we did the restrictions that periodicity imposes. Periodic struc- not know then that certain kinds of aperiodic objects can tures in two or three dimensions can only have 1,2,3,4, also give discrete diffraction; these objects conform to a and 6 fold symmetry axes.
    [Show full text]
  • Conformal Quasicrystals and Holography
    Conformal Quasicrystals and Holography Latham Boyle1, Madeline Dickens2 and Felix Flicker2;3 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada, N2L 2Y5 2Department of Physics, University of California, Berkeley, California 94720, USA 3Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Department of Physics, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on a novel structure, a conformal quasicrystal, that pro- vides a discrete model of conformal geometry. We introduce and construct a class of one-dimensional conformal quasicrystals, and discuss a higher-dimensional example (related to the Penrose tiling). Our construction permits discretizations of conformal field theories that preserve an infinite discrete subgroup of the global conformal group at the cost of lattice periodicity. I. INTRODUCTION dom [12{24]. Meanwhile, quantum information theory provides a unifying language for these studies in terms of entanglement, quantum circuits, and quantum error A central topic in theoretical physics over the past two correction [25]. decades has been holography: the idea that a quantum These investigations have gradually clarified our un- theory in a bulk space may be precisely dual to another derstanding of the discrete geometry in the bulk. There living on the boundary of that space. The most concrete has been a common expectation, based on an analogy and widely-studied realization of this idea has been the with AdS/CFT [1{3], that TNs living on discretizations AdS/CFT correspondence [1{3], in which a gravitational of a hyperbolic space define a lattice state of a critical theory living in a (d + 1)-dimensional negatively-curved system on the boundary and vice-versa.
    [Show full text]
  • 2007 136Th Annual Meeting & Exhibition Linking Science and Technology for Global Solutions
    2007 136th Annual Meeting & Exhibition Linking Science and Technology for Global Solutions Technical Program Program-at-a-Glance ..............................................................2 Session Listing .......................................................................8 Monday AM ...........................................................................16 Monday PM ............................................................................63 Tuesday AM .........................................................................117 Tuesday PM .........................................................................164 Wednesday AM ...................................................................223 Wednesday PM ...................................................................274 Thursday AM .......................................................................327 General Posters ..................................................................359 Index ....................................................................................363 2007 136th Annual Meeting & Exhibition Monday Tuesday Wednesday Thursday ROOM AM PM AM PM AM PM AM Materials Materials Intellectual Intellectual Materials Materials Materials Processing under Processing under Property in Property in Processing under Processing under Processing under the Influence of the Influence of Materials Materials the Influence of the Influence of the Influence of External Fields: External Fields: Science: Patents, Science: Patents, External Fields: External Fields:
    [Show full text]
  • Paper Pentasia: an Aperiodic Surface in Modular Origami
    Paper Pentasia: An Aperiodic Surface in Modular Origami a b Robert J. Lang ∗ and Barry Hayes † aLangorigami.com, Alamo, California, USA, bStanford University, Stanford, CA 2013-05-26 Origami, the Japanese art of paper-folding, has numerous connections to mathematics, but some of the most direct appear in the genre of modular origami. In modular origami, one folds many sheets into identical units (or a few types of unit), and then fits the units together into larger constructions, most often, some polyhedral form. Modular origami is a diverse and dynamic field, with many practitioners (see, e.g., [12, 3]). While most modular origami is created primarily for its artistic or decorative value, it can be used effectively in mathematics education to provide physical models of geometric forms ranging from the Platonic solids to 900-unit pentagon-hexagon-heptagon torii [5]. As mathematicians have expanded their catalog of interesting solids and surfaces, origami designers have followed not far behind, rendering mathematical forms via folding, a notable recent example being a level-3 Menger Sponge folded from 66,048 business cards by Jeannine Mosely and co-workers [10]. In some cases, the origami explorations themselves can lead to new mathematical structures and/or insights. Mosely’s developments of business-card modulars led to the discovery of a new fractal polyhedron with a novel connection to the famous Snowflake curve [11]. One of the most popular geometric mathematical objects has been the sets of aperiodic tilings developed by Roger Penrose [14, 15], which acquired new significance with the dis- covery of quasi-crystals, their three-dimensional analogs in the physical world, in 1982 by Daniel Schechtman, who was awarded the 2011 Nobel Prize in Chemistry for his discovery.
    [Show full text]
  • Eindhoven University of Technology MASTER Lateral Stiffness Of
    Eindhoven University of Technology MASTER Lateral stiffness of hexagrid structures de Meijer, J.H.M. Award date: 2012 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain ‘Lateral Stiffness of Hexagrid Structures’ - Master’s thesis – - Main report – - A 2012.03 – - O 2012.03 – J.H.M. de Meijer 0590897 July, 2012 Graduation committee: Prof. ir. H.H. Snijder (supervisor) ir. A.P.H.W. Habraken dr.ir. H. Hofmeyer Eindhoven University of Technology Department of the Built Environment Structural Design Preface This research forms the main part of my graduation thesis on the lateral stiffness of hexagrids. It explores the opportunities of a structural stability system that has been researched insufficiently.
    [Show full text]
  • Nanotechnology and Quasicrystals: from Self Assembly to Photonic Applications
    Nanotechnology and Quasicrystals: From self assembly to photonic applications. Ron Lifshitz Raymond and Beverly Sackler School of Physics & Astronomy Tel Aviv University, 69978 Tel Aviv, Israel. After providing a concise overview on quasicrystals and their discovery more than a quarter of a century ago, I consider the unexpected interplay between nanotechnology and quasiperiodic crystals. Of particular relevance are efforts to fabricate artificial functional micro- or nanostructures, as well as efforts to control the self-assembly of nanostructures, where current knowledge about the possibility of having long-range order without periodicity can provide significant advantages. I discuss examples of systems ranging from artificial metamaterials for photonic applications, through self-assembled soft matter, to surface waves and optically-induced nonlinear photonic quasicrystals. 1 Nanotechnology and quasicrystals? When organizers of the NATO Advanced Research Workshop on nanotechnology, held in St. Petersburg in June 2008, asked me to deliver a keynote lecture on quasicrystals I was certain that they had made a mistake. I have been studying quasicrystals for over 15 years and investigating nanomechanical systems for just about a decade, and although one always finds connections between different scientific fields, I had never expected such an invitation. Nevertheless, the organizers insisted and explained that they wanted to learn about the possibility of exploiting nontrivial symmetries—perhaps imitating what viruses do—and in general, learn the lesson of a scientific community that was forced by nature to keep an open mind and “think outside of the box”. This chapter is motivated by my presentation on quasicrystals at the NATO ARW on nanotechnology in St.
    [Show full text]
  • Eureka Issue 61
    Eureka 61 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Anja Komatar © The Archimedeans (see page 94 for details) Do not copy or reprint any parts without permission. October 2011 Editorial Eureka Reinvented… efore reading any part of this issue of Eureka, you will have noticed The Team two big changes we have made: Eureka is now published in full col- our, and printed on a larger paper size than usual. We felt that, with Philipp Legner Design and Bthe internet being an increasingly large resource for mathematical articles of Illustrations all kinds, it was necessary to offer something new and exciting to keep Eu- reka as successful as it has been in the past. We moved away from the classic Anja Komatar Submissions LATEX-look, which is so common in the scientific community, to a modern, more engaging, and more entertaining design, while being conscious not to Sean Moss lose any of the mathematical clarity and rigour. Corporate Ben Millwood To make full use of the new design possibilities, many of this issue’s articles Publicity are based around mathematical images: from fractal modelling in financial Lu Zou markets (page 14) to computer rendered pictures (page 38) and mathemati- Subscriptions cal origami (page 20). The Showroom (page 46) uncovers the fundamental role pictures have in mathematics, including patterns, graphs, functions and fractals. This issue includes a wide variety of mathematical articles, problems and puzzles, diagrams, movie and book reviews. Some are more entertaining, such as Bayesian Bets (page 10), some are more technical, such as Impossible Integrals (page 80), or more philosophical, such as How to teach Physics to Mathematicians (page 42).
    [Show full text]