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Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets
Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets Indubala I Satija Department of Physics, George Mason University , Fairfax, VA 22030, USA (Dated: April 28, 2021) Abstract We describe various kaleidoscopic and self-similar aspects of the integral Apollonian gaskets - fractals consisting of close packing of circles with integer curvatures. Self-similar recursive structure of the whole gasket is shown to be encoded in transformations that forms the modular group SL(2;Z). The asymptotic scalings of curvatures of the circles are given by a special set of quadratic irrationals with continued fraction [n + 1 : 1; n] - that is a set of irrationals with period-2 continued fraction consisting of 1 and another integer n. Belonging to the class n = 2, there exists a nested set of self-similar kaleidoscopic patterns that exhibit three-fold symmetry. Furthermore, the even n hierarchy is found to mimic the recursive structure of the tree that generates all Pythagorean triplets arXiv:2104.13198v1 [math.GM] 21 Apr 2021 1 Integral Apollonian gaskets(IAG)[1] such as those shown in figure (1) consist of close packing of circles of integer curvatures (reciprocal of the radii), where every circle is tangent to three others. These are fractals where the whole gasket is like a kaleidoscope reflected again and again through an infinite collection of curved mirrors that encodes fascinating geometrical and number theoretical concepts[2]. The central themes of this paper are the kaleidoscopic and self-similar recursive properties described within the framework of Mobius¨ transformations that maps circles to circles[3]. FIG. 1: Integral Apollonian gaskets. -
Farey Error Correcting Coding Text, Data and Images
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2207-2211 © Research India Publications. http://www.ripublication.com Farey Error Correcting Coding Text, Data and Images Poornima M1 , S. Jagannathan2, R Chandrasekhar3, S.T. Kumara4 1 Mathematics Department , SJB Institute of Technology, Bangalore, Karnataka, India. 2 Computer Science & Engineering Department, Nagarjuna College of Engineering & Technology, Bangalore, Karnataka, India. 3 MCA Department, PESIT Research, Bangalore, Karnataka, India. 4 A P S College of Engineering, Bangalore, 560082, Karnataka, India. Abstract: 2. PRINCIPLES AND AXIOMS OF THE FAREY SEQUENCE A new scheme for encryption and decryption of plain text, data and the coding of medical images through Farey Farey Sequences for Farey1...Farey8 which are irreducible or Sequence is brought in the paper. It proposes the reduced simple decimal division quantities in [0, 1] for a given n is arithmetic point representations and more of integer and given below. whole number point implementations. The test case and Farey1 = [ 0/1, 1/1 ] outcomes are explained in the section 4 and 5. Farey2 = [ 0/1, ½, 1/1 ] Keywords: Error correcting codes, image coding, reduced arithmetic floating point, fixed point, digital signal Farey3 = [ 0/1, 1/3, ½, 2/3, 1/1 ] processing and farey sequences Farey4 = [0/1, ¼, 1/3, ½, 2/3, ¾, 1/1] Farey5 = [ 0/1,1/5 ,1/4, 1/3, 2/5, 1/2 ,3/5, 2/3, 4/5, 1/1] 1. INTRODUCTION The efficacy of Farey sequence applications for error correcting codes and image processing are brings out in the Farey6 = [ 0/1, 1/6,/5, ¼, 1/3, 2/5, ½, 3/5, 2/3, ¾, 4/5, paper. -
Multidimensional Farey Partitions by Arnaldo Nogueira a and Bruno
Indag. Mathem., N.S., 17 (3), 437-456 September 25, 2006 Multidimensional Farey partitions by Arnaldo Nogueira a and Bruno Sevennec b a Institut de Math~matiques de Luminy, 163, avenue de Luminy, 13288 Marseille, Cedex 9, France b Unit~ de MathOmatiques Pures etAppliquOes, Ecole Normale Sup&ieure de Lyon, 46, allOe d'Italie, 69364 Lyon, Cedex 7, France Communicated by Prof. R. Tijdeman at the meeting of October 31, 2005 ABSTRACT The linear action of SLfn, Z+) induces lattice partitions on the (n - 1)-dimensional simplex An-b The notion of Farev partition raises naturally from a matricial interpretation of the arithmetical Farey sequence of order r. Such sequence is unique and, consequently, the Fareypartition of order r on Al is unique. In higher dimension no generalized Farey partition is unique. Nevertheless in dimension 3 the number of triangles in the various generalized Farey partitions is always the same which fails to be true in dimension n > 3. Concerning Diophantine approximations, it turns out that the vertices of an n-dimensional Farey partition of order r are the radial projections of the lattice points in Zn~ N [0, r] n whose coordinates are relatively prime. Moreover, we obtain sequences of multidimensional Farey partitions which converge pointwisely. 1. INTRODUCTION The Farey or intermediate fractions are our motivation. We introduce the notion of matricial Farey tree, where matrices of the unimodular group replace the fractions. The matricial approach generalizes to higher dimension the so-called Farey sequence of order r (Hardy and Wright [6, p. 23]). For r 6 N = {1, 2 ... -
A Scaling Property of Farey Fractions
European Journal of Mathematics (2016) 2:383–417 DOI 10.1007/s40879-016-0098-0 RESEARCH ARTICLE A scaling property of Farey fractions Matthias Kunik1 Received: 18 September 2015 / Revised: 26 January 2016 / Accepted: 28 January 2016 / Published online: 25 February 2016 © Springer International Publishing AG 2016 Abstract The Farey sequence of order n consists of all reduced fractions a/b between 0 and 1 with positive denominator b less than or equal to n. The sums of the inverse denominators 1/b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions fn.Forn →∞we also obtain a certain limit function, which describes an asymptotic scaling property of functions fn in the vicinity of any fixed fraction a/b and which is independent of a/b. The result can be obtained by using only elementary methods. We also study this limit function and especially its decay behaviour by using the Mellin transform and analytical properties of the Riemann zeta function. Keywords Farey sequences · Riemann zeta function · Mellin transform · Hardy spaces Mathematics Subject Classification 11B57 · 11M06 · 44A20 · 42B30 1 Introduction The Farey sequence Fn of order n consists of all reduced fractions a/b with natural denominator b n in the unit interval [0, 1], arranged in order of increasing size. Dropping just the restriction 0 a/b 1 gives the infinite extended Farey sequence Fext n of order n. We use these notations from Definitions 2.1 and 2.3 and present an overview of new results in each section. -
The Arithmetic of the Spheres
The Arithmetic of the Spheres Je↵ Lagarias, University of Michigan Ann Arbor, MI, USA MAA Mathfest (Washington, D. C.) August 6, 2015 Topics Covered Part 1. The Harmony of the Spheres • Part 2. Lester Ford and Ford Circles • Part 3. The Farey Tree and Minkowski ?-Function • Part 4. Farey Fractions • Part 5. Products of Farey Fractions • 1 Part I. The Harmony of the Spheres Pythagoras (c. 570–c. 495 BCE) To Pythagoras and followers is attributed: pitch of note of • vibrating string related to length and tension of string producing the tone. Small integer ratios give pleasing harmonics. Pythagoras or his mentor Thales had the idea to explain • phenomena by mathematical relationships. “All is number.” A fly in the ointment: Irrational numbers, for example p2. • 2 Harmony of the Spheres-2 Q. “Why did the Gods create us?” • A. “To study the heavens.”. Celestial Sphere: The universe is spherical: Celestial • spheres. There are concentric spheres of objects in the sky; some move, some do not. Harmony of the Spheres. Each planet emits its own unique • (musical) tone based on the period of its orbital revolution. Also: These tones, imperceptible to our hearing, a↵ect the quality of life on earth. 3 Democritus (c. 460–c. 370 BCE) Democritus was a pre-Socratic philosopher, some say a disciple of Leucippus. Born in Abdera, Thrace. Everything consists of moving atoms. These are geometrically• indivisible and indestructible. Between lies empty space: the void. • Evidence for the void: Irreversible decay of things over a long time,• things get mixed up. (But other processes purify things!) “By convention hot, by convention cold, but in reality atoms and• void, and also in reality we know nothing, since the truth is at bottom.” Summary: everything is a dynamical system! • 4 Democritus-2 The earth is round (spherical). -
Some Applications of the Spectral Theory of Automorphic Forms
Academic year 2009-2010 Department of Mathematics Some applications of the spectral theory of automorphic forms Research in Mathematics M.Phil Thesis Francois Crucifix Supervisor : Doctor Yiannis Petridis 2 I, Francois Nicolas Bernard Crucifix, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. SIGNED CONTENTS 3 Contents Introduction 4 1 Brief overview of general theory 5 1.1 Hyperbolic geometry and M¨obius transformations . ............. 5 1.2 Laplaceoperatorandautomorphicforms. ......... 7 1.3 Thespectraltheorem.............................. .... 8 2 Farey sequence 14 2.1 Fareysetsandgrowthofsize . ..... 14 2.2 Distribution.................................... 16 2.3 CorrelationsofFareyfractions . ........ 18 2.4 Good’sresult .................................... 22 3 Multiplier systems 26 3.1 Definitionsandproperties . ...... 26 3.2 Automorphicformsofnonintegralweights . ......... 27 3.3 Constructionofanewseries. ...... 30 4 Modular knots and linking numbers 37 4.1 Modularknots .................................... 37 4.2 Linkingnumbers .................................. 39 4.3 TheRademacherfunction . 40 4.4 Ghys’result..................................... 41 4.5 SarnakandMozzochi’swork. ..... 43 Conclusion 46 References 47 4 Introduction The aim of this M.Phil thesis is to present my research throughout the past academic year. My topics of interest ranged over a fairly wide variety of subjects. I started with the study of automorphic forms from an analytic point of view by applying spectral methods to the Laplace operator on Riemann hyperbolic surfaces. I finished with a focus on modular knots and their linking numbers and how the latter are related to the theory of well-known analytic functions. My research took many more directions, and I would rather avoid stretching the extensive list of applications, papers and books that attracted my attention. -
From Poincaré to Whittaker to Ford
From Poincar´eto Whittaker to Ford John Stillwell University of San Francisco May 22, 2012 1 / 34 Ford circles Here is a picture, generated from two equal tangential circles and a tangent line, by repeatedly inserting a maximal circle in the space between two tangential circles and the line. 2 / 34 0 1 1 1 3 / 34 1 2 3 / 34 1 2 3 3 3 / 34 1 3 4 4 3 / 34 1 2 3 4 5 5 5 5 3 / 34 1 5 6 6 3 / 34 1 2 3 4 5 6 7 7 7 7 7 7 3 / 34 1 3 5 7 8 8 8 8 3 / 34 1 2 4 5 7 8 9 9 9 9 9 9 3 / 34 1 3 7 9 10 10 10 10 3 / 34 1 2 3 4 5 6 7 8 9 10 11 11 11 11 11 11 11 11 11 11 3 / 34 The Farey sequence has a long history, going back to a question in the Ladies Diary of 1747. How did Ford come to discover its geometric interpretation? The Ford circles and fractions Thus the Ford circles, when generated in order of size, generate all reduced fractions, in order of their denominators. The first n stages of the Ford circle construction give the so-called Farey sequence of order n|all reduced fractions between 0 and 1 with denominator ≤ n. 4 / 34 How did Ford come to discover its geometric interpretation? The Ford circles and fractions Thus the Ford circles, when generated in order of size, generate all reduced fractions, in order of their denominators. -
The Hyperbolic, the Arithmetic and the Quantum Phase Michel Planat, Haret Rosu
The hyperbolic, the arithmetic and the quantum phase Michel Planat, Haret Rosu To cite this version: Michel Planat, Haret Rosu. The hyperbolic, the arithmetic and the quantum phase. Journal of Optics B, 2004, 6, pp.S583. hal-00000579v2 HAL Id: hal-00000579 https://hal.archives-ouvertes.fr/hal-00000579v2 Submitted on 15 Dec 2003 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The hyperbolic, the arithmetic and the quantum phase Michel Planat† and Haret Rosu‡ § Laboratoire de Physique et M´etrologie des Oscillateurs du CNRS, † 32 Avenue de l’observatoire, 25044 Besan¸con Cedex, France Potosinian Institute of Scientific and Technological Research ‡ Apdo Postal 3-74, Tangamanga, San Luis Potosi, SLP, Mexico Abstract. We develop a new approach of the quantum phase in an Hilbert space of finite dimension which is based on the relation between the physical concept of phase locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As a result, phase variability looks quite similar to its classical counterpart, having peaks at dimensions equal to a power of a prime number. Squeezing of the phase noise is allowed for specific quantum states. -
Hofstadter Butterfly and a Hidden Apollonian Gasket
Hofstadter Butterfly and a Hidden Apollonian Gasket Indubala I Satija School of Physics and Astronomy and Computational Sciences, George Mason University, Fairfax, VA 22030 (Dated: April 2, 2015) The Hofstadter butterfly, a quantum fractal made up of integers describing quantum Hall states, is shown to be related to an integral Apollonian gasket with D3 symmetry. This mapping unfolds as the self-similar butterfly landscape is found to describe a close packing of (Ford) circles that represent rational flux values and is characterized in terms of an old ((300BC) problem of mutually tangentp circles. The topological and flux scaling of the butterfly hierarchy asymptotes to an irrational number 2 + 3, that also underlies the scaling relation for the Apollonian. This reveals a hidden symmetry of the butterfly as the energy and the magnetic flux intervals shrink to zero. With a periodic drive, the fine structure of the butterfly is shown to be amplified making states with large topological invariants accessible experimentally. PACS numbers: 03.75.Ss,03.75.Mn,42.50.Lc,73.43.Nq Hofstadter butterfly[1] is a fascinating two-dimensional Model system we study here consists of (spinless) fermions spectral landscape, a quantum fractal where energy gaps en- in a square lattice. Each site is labeled by a vector r = nx^ + code topological quantum numbers associated with the Hall my^, where n, m are integers, x^ (y^) is the unit vector in the x conductivity[2]. This intricate mix of order and complexity (y) direction, and a is the lattice spacing. The tight binding is due to frustration, induced by two competing periodicities Hamiltonian has the form as electrons in a crystalline lattice are subjected to a magnetic X X i2πnφ field. -
What Do Bloch Electrons in a Magnetic Field Have to Do with Apollonian Packing of Circles?
Journal of Physics A: Mathematical and Theoretical PAPER What do Bloch electrons in a magnetic field have to do with Apollonian packing of circles? To cite this article: Indubala I Satija 2021 J. Phys. A: Math. Theor. 54 025701 View the article online for updates and enhancements. This content was downloaded from IP address 129.6.222.101 on 16/12/2020 at 18:00 Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 54 (2021) 025701 (32pp) https://doi.org/10.1088/1751-8121/abc65c What do Bloch electrons in a magnetic field have to do with Apollonian packing of circles? Indubala I Satija∗ Department of Physics, George Mason University, Fairfax, Virginia, United States of America E-mail: [email protected] Received 16 June 2020, revised 26 October 2020 Accepted for publication 30 October 2020 Published 16 December 2020 Abstract Integral Apollonian packing, the packing of circles with integer curvatures, where every circle is tangent to three other mutually tangent circles, is shown to encode the fractal structure of the energy spectrum of two-dimensional Bloch electrons in a magnetic field, known as the ‘Hofstadter butterfly’. In this Apol- lonian–butterfly-connection, the integer curvatures of the circles contain ina convoluted form, the topological quantum numbers of the butterfly graph—the quanta of the Hall conductivity. Nesting properties of these two fractals are described in terms of the Apollonian group and the conformal transformations. In this mapping, Farey tree hierarchy plays the central role, revealing how the geometry and the number theory are intertwined in the quantum mechanics of Bloch electrons in a magnetic field. -
Sequences, Their Application and Use in the Field of Mathematics
SEQUENCES, THEIR APPLICATION AND USE IN THE FIELD OF MATHEMATICS Gabe Stevens 5c1 STEVENS GABE – 5CL1 – 2019/2020 Stevens Gabe 5CL1 2019/2020 Table of Contents Table of Contents ............................................................................................................. 1 Introduction ..................................................................................................................... 3 What is a sequence ........................................................................................................... 4 The difference between a set and a sequence ................................................................... 4 Notation ........................................................................................................................... 5 Indexing (Rule) ......................................................................................................................... 6 Recursion ................................................................................................................................. 7 Geometric and arithmetic sequences ................................................................................ 8 Geometric sequences ............................................................................................................... 8 Properties....................................................................................................................................................... 8 Arithmetic sequence ............................................................................................................... -
Arxiv:1710.02996V5 [Math.CO] 21 May 2018 Solutions of Problem II Such That A1 + A2 + ··· + an = 3N − 6, and Triangulations of N-Gons
PARTITIONS OF UNITY IN SL(2; Z), NEGATIVE CONTINUED FRACTIONS, AND DISSECTIONS OF POLYGONS VALENTIN OVSIENKO Abstract. We characterize sequences of positive integers (a1; a2; : : : ; an) for which the 2 × 2 matrix ! ! ! an −1 an−1 −1 a1 −1 ··· is either the identity matrix Id, its negative −Id, or 1 0 1 0 1 0 square root of −Id. This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction. 1. Introduction and main results Let Mn(a1; : : : ; an) 2 SL(2; Z) be the matrix defined by the product ! ! ! an −1 an−1 −1 a1 −1 (1.1) Mn(a1; : : : ; an) := ··· ; 1 0 1 0 1 0 where (a1; a2; : : : ; an) are positive integers. In terms of the generators of SL(2; Z) 0 −1 ! 1 1 ! S = ;T = ; 1 0 0 1 an an−1 a1 the matrix (1.1) reads: Mn(a1; : : : ; an) = T ST S ··· T S. Every matrix A 2 SL(2; Z) can be written in the form (1.1) in many different ways. The goal of this paper is to describe all solutions of the following three equations Mn(a1; : : : ; an) = Id; (Problem I) Mn(a1; : : : ; an) = −Id; (Problem II) 2 Mn(a1; : : : ; an) = −Id: (Problem III) Problem II, with a certain total positivity restriction, was studied in [8, 7] under the name of \frieze patterns". The theorem of Conway and Coxeter [7] establishes a one-to-one correspondence between the arXiv:1710.02996v5 [math.CO] 21 May 2018 solutions of Problem II such that a1 + a2 + ··· + an = 3n − 6, and triangulations of n-gons.