The Unexpected Fractal Signatures in Fibonacci Chains
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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. -
The Fractal Dimension of Islamic and Persian Four-Folding Gardens
Edinburgh Research Explorer The fractal dimension of Islamic and Persian four-folding gardens Citation for published version: Patuano, A & Lima, MF 2021, 'The fractal dimension of Islamic and Persian four-folding gardens', Humanities and Social Sciences Communications, vol. 8, 86. https://doi.org/10.1057/s41599-021-00766-1 Digital Object Identifier (DOI): 10.1057/s41599-021-00766-1 Link: Link to publication record in Edinburgh Research Explorer Document Version: Publisher's PDF, also known as Version of record Published In: Humanities and Social Sciences Communications General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 06. Oct. 2021 ARTICLE https://doi.org/10.1057/s41599-021-00766-1 OPEN The fractal dimension of Islamic and Persian four- folding gardens ✉ Agnès Patuano 1 & M. Francisca Lima 2 Since Benoit Mandelbrot (1924–2010) coined the term “fractal” in 1975, mathematical the- ories of fractal geometry have deeply influenced the fields of landscape perception, archi- tecture, and technology. Indeed, their ability to describe complex forms nested within each fi 1234567890():,; other, and repeated towards in nity, has allowed the modeling of chaotic phenomena such as weather patterns or plant growth. -
The Implications of Fractal Fluency for Biophilic Architecture
JBU Manuscript TEMPLATE The Implications of Fractal Fluency for Biophilic Architecture a b b a c b R.P. Taylor, A.W. Juliani, A. J. Bies, C. Boydston, B. Spehar and M.E. Sereno aDepartment of Physics, University of Oregon, Eugene, OR 97403, USA bDepartment of Psychology, University of Oregon, Eugene, OR 97403, USA cSchool of Psychology, UNSW Australia, Sydney, NSW, 2052, Australia Abstract Fractals are prevalent throughout natural scenery. Examples include trees, clouds and coastlines. Their repetition of patterns at different size scales generates a rich visual complexity. Fractals with mid-range complexity are particularly prevalent. Consequently, the ‘fractal fluency’ model of the human visual system states that it has adapted to these mid-range fractals through exposure and can process their visual characteristics with relative ease. We first review examples of fractal art and architecture. Then we review fractal fluency and its optimization of observers’ capabilities, focusing on our recent experiments which have important practical consequences for architectural design. We describe how people can navigate easily through environments featuring mid-range fractals. Viewing these patterns also generates an aesthetic experience accompanied by a reduction in the observer’s physiological stress-levels. These two favorable responses to fractals can be exploited by incorporating these natural patterns into buildings, representing a highly practical example of biophilic design Keywords: Fractals, biophilia, architecture, stress-reduction, -
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. -
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. -
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. -
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. -
Harmonic and Fractal Image Analysis (2001), Pp. 3 - 5 3
O. Zmeskal et al. / HarFA - Harmonic and Fractal Image Analysis (2001), pp. 3 - 5 3 Fractal Analysis of Image Structures Oldřich Zmeškal, Michal Veselý, Martin Nežádal, Miroslav Buchníček Institute of Physical and Applied Chemistry, Brno University of Technology Purkynova 118, 612 00 Brno, Czech Republic [email protected] Introduction of terms Fractals − Fractals are of rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced copy of the whole. − They are crinkly objects that defy conventional measures, such as length and are most often characterised by their fractal dimension − They are mathematical sets with a high degree of geometrical complexity that can model many natural phenomena. Almost all natural objects can be observed as fractals (coastlines, trees, mountains, and clouds). − Their fractal dimension strictly exceeds topological dimension Fractal dimension − The number, very often non-integer, often the only one measure of fractals − It measures the degree of fractal boundary fragmentation or irregularity over multiple scales − It determines how fractal differs from Euclidean objects (point, line, plane, circle etc.) Monofractals / Multifractals − Just a small group of fractals have one certain fractal dimension, which is scale invariant. These fractals are monofractals − The most of natural fractals have different fractal dimensions depending on the scale. They are composed of many fractals with the different fractal dimension. They are called „multifractals“ − To characterise set of multifractals (e.g. set of the different coastlines) we do not have to establish all their fractal dimensions, it is enough to evaluate their fractal dimension at the same scale Self-similarity/ Semi-self similarity − Fractal is strictly self-similar if it can be expressed as a union of sets, each of which is an exactly reduced copy (is geometrically similar to) of the full set (Sierpinski triangle, Koch flake). -
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. -
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: -
Analyse D'un Modèle De Représentations En N Dimensions De
Analyse d’un modèle de représentations en n dimensions de l’ensemble de Mandelbrot, exploration de formes fractales particulières en architecture. Jean-Baptiste Hardy 1 Département d’architecture de l’INSA Strasbourg ont contribués à la réalisation de ce mémoire Pierre Pellegrino Directeur de mémoire Emmanuelle P. Jeanneret Docteur en architecture Francis Le Guen Fractaliste, journaliste, explorateur Serge Salat Directeur du laboratoire des morphologies urbaines Novembre 2013 2 Sommaire Avant-propos .................................................................................................. 4 Introduction .................................................................................................... 5 1 Introduction aux fractales : .................................................... 6 1-1 Présentation des fractales �������������������������������������������������������������������� 6 1-1-1 Les fractales en architecture ������������������������������������������������������������� 7 1-1-2 Les fractales déterministes ................................................................ 8 1-2 Benoit Mandelbrot .................................................................................. 8 1-2-1 L’ensemble de Mandelbrot ................................................................ 8 1-2-2 Mandelbrot en architecture .............................................................. 9 Illustrations ................................................................................................... 10 2 Un modèle en n dimensions : -
Fractal Analysis of the Vascular Tree in the Human Retina
3 Jun 2004 22:39 AR AR220-BE06-17.tex AR220-BE06-17.sgm LaTeX2e(2002/01/18) P1: IKH 10.1146/annurev.bioeng.6.040803.140100 Annu. Rev. Biomed. Eng. 2004. 6:427–52 doi: 10.1146/annurev.bioeng.6.040803.140100 Copyright c 2004 by Annual Reviews. All rights reserved First published online as a Review in Advance on April 13, 2004 FRACTAL ANALYSIS OF THE VASCULAR TREE IN THE HUMAN RETINA BarryR.Masters Formerly, Gast Professor, Department of Ophthalmology, University of Bern, 3010 Bern, Switzerland; email: [email protected] Key Words fractals, fractal structures, eye, bronchial tree, retinal circulation, retinal blood vessel patterns, Murray Principle, optimal vascular tree, human lung, human bronchial tree I Abstract The retinal circulation of the normal human retinal vasculature is statis- tically self-similar and fractal. Studies from several groups present strong evidence that the fractal dimension of the blood vessels in the normal human retina is approximately 1.7. This is the same fractal dimension that is found for a diffusion-limited growth process, and it may have implications for the embryological development of the retinal vascular system. The methods of determining the fractal dimension for branching trees are reviewed together with proposed models for the optimal formation (Murray Princi- ple) of the branching vascular tree in the human retina and the branching pattern of the human bronchial tree. The limitations of fractal analysis of branching biological struc- tures are evaluated. Understanding the design principles of branching vascular systems and the human bronchial tree may find applications in tissue and organ engineering, i.e., bioartificial organs for both liver and kidney.