DOCSLIB.ORG

Fractal Geometry Isa S

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fractal Geometry Isa S Fractal Geometry Isa S. Jubran – [email protected] – http://web.cortland.edu/jubrani A Fractal is a geometric object whose dimension is fractional. Most fractals are self similar, that is when any small part of a fractal is magnified the result resembles the original fractal. Examples of fractals are the Koch curve, the Seirpinski gasket, and the Menger sponge. Similarity Dimension: ² Take a line segment of length one unit and divide it into N equal parts. let old length 1 S = new length , then N = S . N = 3;S = 3 ² Take a square (dimension 2)p of area one square unit and divide it into N congruent parts. Then S = N and N = S2. N = 4;S = 2 ² Take a cube (dimension 3) of volume one cubic unit and divide it into N 1 3 congruent parts. Then S = N 3 and N = S . ......................... .... .... .... .... ...................... ..... ..... N = 8;S = 2 ..... ..... .... ..... ..................... .... .... ... ......................... .... .... .... .... .... ...................... .... .... .... .... So given set of dimension d consisting of N parts congruent part then N = Sd, old length where S = new length . Solving for d yields ln N d = ln S 1 Exercise 1: T1 = limn!1 Tn is called the Seirpinski Triangle or Gasket. 1. What is the perimeter of Tn? 2. What is the area of Tn? 3. Compute the fractal dimension of T1. T0 T1 T2 Solution: 3 n 1. Pn = ( 2 ) P0. 3 n 2. An = ( 4 ) A0. ln3 3. N = 3 and S = 2, hence d = ln2 . Exercise 2: C1 = limn!1 Cn is called the Seirpinski Carpet. C0 C1 C2 1. What is the perimeter of Cn? 2. What is the area of Cn? 3. Compute the fractal dimension of C1. Solution: 1 8 8n¡1 1. Pn = P0 + 3 P0 + 32 P0 + ¢ ¢ ¢ + 3n P0. 8 n 2. An = ( 9 ) A0. ln8 3. N = 8 and S = 3, hence d = ln3 . 2 Exercise 3: M1 = limn!1 Mn is called the Menger Sponge. 2 0 M0 2 M1 M2 1. What is the volume of Mn. 2. What is the surface area of Mn? 3. Compute the fractal dimension of M1. Solution: 3 3 1. V0 = 2 = 8 and Vn = 20 ¢ (1=3) ¢ (Vn¡1). 2 2 2 2. A0 = 6 ¢ 2 = 24 and An = 20 ¢ (1=3) ¢ (An¡1) ¡ 24 ¢ (1=3) ¢ (Atn¡1) ¢ 2 n¡1 = (20=9) ¢ Area(Cn¡1) ¡ (64=3) ¢ (8=9) where Area(Cn¡1) is the area of the n ¡ 1 stage in the construction of the Sierpinski carpet. Hence An = (8=9)n ¢ 16 + (20=9)n ¢ 8. ln20 3. N = 20 and S = 3, hence d = ln3 . 3 Exercise 4: Draw a fractal with dimension 2 . Solution: 6 .................... ........... .......... .................... .................... ................... ........... - . ......... .......... .......... ........... ? Demonstrate the applet ”Fractal Dimension” at: http://www.shodor.org/interactivate/activities/fracdim/index.html 3 Box Dimension: Box dimension is another way to measure fractal dimension, it is defined as follows: If X is a bounded subset of the Euclidean space, and ² ¸ 0. Let N(X; ²) be the minimal number of boxes in the grid of side length ² which are required to cover X. We say that X has box dimension D if the following limit exists and has value D. ln(N(X; ²)) lim 1 ²!0+ ln( ² ) Consider for example the Seirpinski gasket: 6 .................................................. 1 . .................................................. .................................................. .................................................. .................................................. .................................................. .- 0 1 ² N(X; ²) 1 1 1 2 3 1 4 9 . 1 3n 2n ln(4 ¢ 3n) ln 3 Hence D = lim = n!1 ln(2n) ln 2 Demonstrate the applet ”Box Counting Dimension” at: http://classes.yale.edu/fractals/Software/boxdim.html 4 Geometrically, if you plot the results on a graph with ln N(X; ²) on the vertical axis and ln(1=²) on the horizontal axis, then the slope of the best fit line of the data will be an approximation of Box dimension of the fractal. 6 (ln(4); ln(9)) ...................... ...................... ...................... (ln(2); ln(3))....................... ........................ - (ln(1); ln(1)) Discuss computing the fractal dimension of a bunch of broccoli. http://hypertextbook.com/facts/2002/broccoli.shtml My Results: 1 1 Ball Size N ² ln( ² ) ln(N) 6x5x2.5 10 1/4.5 -1.504 2.303 4x4x2.5 19 1/3.5 -1.253 2.94 3.5x3x2 28 1/2.83 -1.04 3.33 3x2.5x2 41 1/2.5 -.916 3.71 2.5x2.5x2 64 1/2.33 -.847 4.158 Use the linear regression applet at http://math.hws.edu/javamath/ryan/Regression.html to plot the data points and find the slope of the best fit line. Fractal dimension = slope = 2.62. If you throw the 1st point out you get fractal dimension=slope =2.82. 5 Divider Dimension: A fractal curve has fractal (or divider) dimension D if its length L when measured with rods of length l is given by L = C ¢ l1¡D C is a constant that is a certain measure of the apparent length, and the equation above must be true for several different values of l. Consider the Koch curve: K2 l L 1 1 1 4 3 3 1 16 9 9 . 1 4 n 3n ( 3 ) 4 n 1 n It appears to have length L = ( 3 ) when measured with rods of length l = ( 3 ) . 4 1 Hence; ( )n = C ¢ ( )n(1¡D) 3 3 4 1 This implies that n ln 3 = ln(C) + n(1 ¡ D) ln 3 . In order for the equation to be satisfied for all n, ln C must be zero, hence C = 1 ln 4 and so D = ln 3 . Geometrically, if you graph ln(L) versus ln(l) the D = 1 ¡ Slope of line. Exercise: The west coast of Britain when measured with rods of length 10 km is 3020 km. But when measured with rods of length 100 km it is 1700 km. What is the fractal dimension of the west coast of Britain? Demonstrate the applet ”Coastline of Hong Kong” at: http://www.eie.polyu.edu.hk/»cktse/movies/pheno/map.html 6 Affine transformations: An affine transformation is of the form à ! à ! à ! à ! x a b x e f( ) = ¢ + y c d y f or à ! à ! à ! à ! x r cos(θ) ¡s sin(Á) x e f( ) = ¢ + y r sin(θ) s cos(Á) y f Where r and s are the scaling factors in the x and y directions respectively. θ and Á measure rotation of horizontal and vertical lines resp. e and f measure horizontal and vertical translations resp. It can be easily shown that r2 = a2 + c2, s2 = b2 + d2, θ = arctan(c=a) and Á = arctan(¡b=d). 6 b 6 c d Á θ - - a ? ? 6 B 6 B C. ..................... C. ..................... ..................... Similarity . Affinity . B0 . B0 C0. ....................... C0................................. ....................... ........................ - ....................... ....................... - D A0 A D A0 A x0 = 1 x x0 = 3 x 0 12 0 14 y = 2 y y = 2 y A reflection about the y-axis would be given by à ! ¡1 0 x0 = ¡x and y0 = y ¡¡¡¡¡¡¡ matrix form 0 1 A reflection about the x-axis would be given by à ! 1 0 x0 = x and y0 = ¡y ¡¡¡¡¡¡¡ matrix form 0 ¡1 7 Rotation is represented as follows: 0 B.. .... ... 6 6 ... .. ... ... ... ... ... B ... B C. C. 0 . 0 . C ............................... B. 0 ... ... ...A ... ... ... ... ... ... .. .. .. ... ... ... ... ... θ = 0 . ... θ = 45 ... ... ... ... ... Á = 45 . ... Á = 0 ... ... ... ................................................. - ......................... ..................... - D A D A x0 = x ¡ p1 y x0 = p1 x 2 2 y0 = p1 y y0 = p1 x + y 2 2 à ! à ! cos(θ) ¡ sin(Á) a b Matrixform : = sin(θ) cos(Á) c d 6 . 0 .B..... ... ... ... ... ... ... ... ... ... ... B ... C ... ... ... 0 .. .. C ..... ... 0 ... ...A ... ... ... ... .. .. ... ... ... ... θ = 45 ... ... ... ... Á = 45 ... ... ........................... ..................... - D A x0 = p1 x ¡ p1 y 2 2 y0 = p1 x + p1 y 2 2 8 Shears are represented as follows: B0 6 6 . ..................... 0 ...................... ..................... ...................... .A . 0 ...............................................B ... ....................... ........................B .C 0 C. C. B . ....................... ........................ - ...................... ........................ - D A D A x0 = x x0 = x + 2y y0 = 2x + y y0 = y Shear in the y direction Shear in the x direction 9 Iterated Function Systems: 2 An Iterated Function System in R is a collection fF1;F2;:::;Fng of contraction 2 mappings on R .(Fi is a contraction mapping if given x; y then d(Fi(x);Fi(y)) · sd(x; y) where 0 · s < 1) The Collage Theorem says that there is a unique nonempty compact subset A ½ R2 called the attractor such that A = F1(A) [ F2(A) [ ::: [ Fn(A) Sketch of Proof: Given a complete metric space X, let H(X) be the collection of nonempty compact subsets of X. Given A and B in H(X) define d(A; B) to be the smallest number r such that each point of in A is within r of some point in B and vice versa. If X is complete so is H(X). Given an IFS on X, define G : H(X) ! H(X) by G(K) = F1(K) [ ::: [ Fn(K). If each Fi is a contraction then so is G. The contraction mapping theorem says that a contraction mapping on a complete space has a unique fixed point. Hence there is A 2 H(X) such that G(A) = A = F1(A) [ F2(A) [ ::: [ Fn(A). 2 n Given K0 ½ R , let Kn = G (K0). Then sid(K ;G(K )) d(A; K ) · 0 0 i 1 ¡ s Hence Ki is a very good approximation of A for large enough i. Deterministic Algorithm: (Slow!) n Given K0, let Kn = G (K0). Now Kn ! A. Random Algorithm: (Fast!) Choose P0 2 K0 and let Pi+1 = Fj(Pi) where Fj is chosen at random from all of the Fi. Clearly Pi 2 Ki and for sufficiently large i,Pi is arbitrarily close to some point in A.
Load more

Recommended publications
  • Using Fractal Dimension for Target Detection in Clutter
    KIM T. CONSTANTIKES USING FRACTAL DIMENSION FOR TARGET DETECTION IN CLUTTER The detection of targets in natural backgrounds requires that we be able to compute some characteristic of target that is distinct from background clutter. We assume that natural objects are fractals and that the irregularity or roughness of the natural objects can be characterized with fractal dimension estimates. Since man-made objects such as aircraft or ships are comparatively regular and smooth in shape, fractal dimension estimates may be used to distinguish natural from man-made objects. INTRODUCTION Image processing associated with weapons systems is fractal. Falconer1 defines fractals as objects with some or often concerned with methods to distinguish natural ob­ all of the following properties: fine structure (i.e., detail jects from man-made objects. Infrared seekers in clut­ on arbitrarily small scales) too irregular to be described tered environments need to distinguish the clutter of with Euclidean geometry; self-similar structure, with clouds or solar sea glint from the signature of the intend­ fractal dimension greater than its topological dimension; ed target of the weapon. The discrimination of target and recursively defined. This definition extends fractal from clutter falls into a category of methods generally into a more physical and intuitive domain than the orig­ called segmentation, which derives localized parameters inal Mandelbrot definition whereby a fractal was a set (e.g.,texture) from the observed image intensity in order whose "Hausdorff-Besicovitch dimension strictly exceeds to discriminate objects from background. Essentially, one its topological dimension.,,2 The fine, irregular, and self­ wants these parameters to be insensitive, or invariant, to similar structure of fractals can be experienced firsthand the kinds of variation that the objects and background by looking at the Mandelbrot set at several locations and might naturally undergo because of changes in how they magnifications.
    [Show full text]
  • Fractal 3D Magic Free
    FREE FRACTAL 3D MAGIC PDF Clifford A. Pickover | 160 pages | 07 Sep 2014 | Sterling Publishing Co Inc | 9781454912637 | English | New York, United States Fractal 3D Magic | Banyen Books & Sound Option 1 Usually ships in business days. Option 2 - Most Popular! This groundbreaking 3D showcase offers a rare glimpse into the dazzling world of computer-generated fractal art. Prolific polymath Clifford Pickover introduces the collection, which provides background on everything from Fractal 3D Magic classic Mandelbrot set, to the infinitely porous Menger Sponge, to ethereal fractal flames. The following eye-popping gallery displays mathematical formulas transformed into stunning computer-generated 3D anaglyphs. More than intricate designs, visible in three dimensions thanks to Fractal 3D Magic enclosed 3D glasses, will engross math and optical illusions enthusiasts alike. If an item you have purchased from us is not working as expected, please visit one of our in-store Knowledge Experts for free help, where they can solve your problem or even exchange the item for a product that better suits your needs. If you need to return an item, simply bring it back to any Micro Center store for Fractal 3D Magic full refund or exchange. All other products may be returned within 30 days of purchase. Using the software may require the use of a computer or other device that must meet minimum system requirements. It is recommended that you familiarize Fractal 3D Magic with the system requirements before making your purchase. Software system requirements are typically found on the Product information specification page. Aerial Drones Micro Center is happy to honor its customary day return policy for Aerial Drone returns due to product defect or customer dissatisfaction.
    [Show full text]
  • Copyright by Timothy Alexander Cousins 2016
    Copyright by Timothy Alexander Cousins 2016 The Thesis Committee for Timothy Alexander Cousins Certifies that this is the approved version of the following thesis: Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media APPROVED BY SUPERVISING COMMITTEE: Supervisor: Hugh Daigle Maša Prodanović Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media by Timothy Alexander Cousins, B. S. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin August 2016 Dedication I would like to dedicate this to my parents, Michael and Joanne Cousins. Acknowledgements I would like to thank the continuous support of Professor Hugh Daigle over these last two years in guiding throughout my degree and research. I would also like to thank Behzad Ghanbarian for being a great mentor and guide throughout the entire research process, and for constantly giving me invaluable insight and advice, both for the research and for life in general. I would also like to thank my parents for the consistent support throughout my entire life. I would also like to acknowledge Edmund Perfect (Department of Earth and Planetary, University of Tennessee) and Jung-Woo Kim (Radioactive Waste Disposal Research Division, Korea Atomic Energy Research Institute) for providing Lacunarity MATLAB code used in this study. v Abstract Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media Timothy Alexander Cousins, M.S.E.
    [Show full text]
  • Measuring the Fractal Dimensions of Empirical Cartographic Curves
    MEASURING THE FRACTAL DIMENSIONS OF EMPIRICAL CARTOGRAPHIC CURVES Mark C. Shelberg Cartographer, Techniques Office Aerospace Cartography Department Defense Mapping Agency Aerospace Center St. Louis, AFS, Missouri 63118 Harold Moellering Associate Professor Department of Geography Ohio State University Columbus, Ohio 43210 Nina Lam Assistant Professor Department of Geography Ohio State University Columbus, Ohio 43210 Abstract The fractal dimension of a curve is a measure of its geometric complexity and can be any non-integer value between 1 and 2 depending upon the curve's level of complexity. This paper discusses an algorithm, which simulates walking a pair of dividers along a curve, used to calculate the fractal dimensions of curves. It also discusses the choice of chord length and the number of solution steps used in computing fracticality. Results demonstrate the algorithm to be stable and that a curve's fractal dimension can be closely approximated. Potential applications for this technique include a new means for curvilinear data compression, description of planimetric feature boundary texture for improved realism in scene generation and possible two-dimensional extension for description of surface feature textures. INTRODUCTION The problem of describing the forms of curves has vexed researchers over the years. For example, a coastline is neither straight, nor circular, nor elliptic and therefore Euclidean lines cannot adquately describe most real world linear features. Imagine attempting to describe the boundaries of clouds or outlines of complicated coastlines in terms of classical geometry. An intriguing concept proposed by Mandelbrot (1967, 1977) is to use fractals to fill the void caused by the absence of suitable geometric representations.
    [Show full text]
  • Connectivity Calculus of Fractal Polyhedrons
    Pattern Recognition 48 (2015) 1150–1160 Contents lists available at ScienceDirect Pattern Recognition journal homepage: www.elsevier.com/locate/pr Connectivity calculus of fractal polyhedrons Helena Molina-Abril a,n, Pedro Real a, Akira Nakamura b, Reinhard Klette c a Universidad de Sevilla, Spain b Hiroshima University, Japan c The University of Auckland, New Zealand article info abstract Article history: The paper analyzes the connectivity information (more precisely, numbers of tunnels and their Received 27 December 2013 homological (co)cycle classification) of fractal polyhedra. Homology chain contractions and its combi- Received in revised form natorial counterparts, called homological spanning forest (HSF), are presented here as an useful 6 April 2014 topological tool, which codifies such information and provides an hierarchical directed graph-based Accepted 27 May 2014 representation of the initial polyhedra. The Menger sponge and the Sierpiński pyramid are presented as Available online 6 June 2014 examples of these computational algebraic topological techniques and results focussing on the number Keywords: of tunnels for any level of recursion are given. Experiments, performed on synthetic and real image data, Connectivity demonstrate the applicability of the obtained results. The techniques introduced here are tailored to self- Cycles similar discrete sets and exploit homology notions from a representational point of view. Nevertheless, Topological analysis the underlying concepts apply to general cell complexes and digital images and are suitable for Tunnels Directed graphs progressing in the computation of advanced algebraic topological information of 3-dimensional objects. Betti number & 2014 Elsevier Ltd. All rights reserved. Fractal set Menger sponge Sierpiński pyramid 1. Introduction simply-connected sets and those with holes (i.e.
    [Show full text]
  • Fractals, Self-Similarity & Structures
    © Landesmuseum für Kärnten; download www.landesmuseum.ktn.gv.at/wulfenia; www.biologiezentrum.at Wulfenia 9 (2002): 1–7 Mitteilungen des Kärntner Botanikzentrums Klagenfurt Fractals, self-similarity & structures Dmitry D. Sokoloff Summary: We present a critical discussion of a quite new mathematical theory, namely fractal geometry, to isolate its possible applications to plant morphology and plant systematics. In particular, fractal geometry deals with sets with ill-defined numbers of elements. We believe that this concept could be useful to describe biodiversity in some groups that have a complicated taxonomical structure. Zusammenfassung: In dieser Arbeit präsentieren wir eine kritische Diskussion einer völlig neuen mathematischen Theorie, der fraktalen Geometrie, um mögliche Anwendungen in der Pflanzen- morphologie und Planzensystematik aufzuzeigen. Fraktale Geometrie behandelt insbesondere Reihen mit ungenügend definierten Anzahlen von Elementen. Wir meinen, dass dieses Konzept in einigen Gruppen mit komplizierter taxonomischer Struktur zur Beschreibung der Biodiversität verwendbar ist. Keywords: mathematical theory, fractal geometry, self-similarity, plant morphology, plant systematics Critical editions of Dean Swift’s Gulliver’s Travels (see e.g. SWIFT 1926) recognize a precise scale invariance with a factor 12 between the world of Lilliputians, our world and that one of Brobdingnag’s giants. Swift sarcastically followed the development of contemporary science and possibly knew that even in the previous century GALILEO (1953) noted that the physical laws are not scale invariant. In fact, the mass of a body is proportional to L3, where L is the size of the body, whilst its skeletal rigidity is proportional to L2. Correspondingly, giant’s skeleton would be 122=144 times less rigid than that of a Lilliputian and would be destroyed by its own weight if L were large enough (cf.
    [Show full text]
  • S27 Fractals References
    S27 FRACTALS S27 Fractals References: Supplies: Geogebra (phones, laptops) • Business cards or index cards for building Sierpinski’s tetrahedron or Menger’s • sponge. 407 What is a fractal? S27 FRACTALS What is a fractal? Definition. A fractal is a shape that has 408 What is a fractal? S27 FRACTALS Animation at Wikimedia Commons 409 What is a fractal? S27 FRACTALS Question. Do fractals necessarily have reflection, rotation, translation, or glide reflec- tion symmetry? What kind of symmetry do they have? What natural objects approximate fractals? 410 Fractals in the world S27 FRACTALS Fractals in the world A fractal is formed when pulling apart two glue-covered acrylic sheets. 411 Fractals in the world S27 FRACTALS High voltage breakdown within a block of acrylic creates a fractal Lichtenberg figure. 412 Fractals in the world S27 FRACTALS What happens to a CD in the microwave? 413 Fractals in the world S27 FRACTALS A DLA cluster grown from a copper sulfate solution in an electrodeposition cell. 414 Fractals in the world S27 FRACTALS Anatomy 415 Fractals in the world S27 FRACTALS 416 Fractals in the world S27 FRACTALS Natural tree 417 Fractals in the world S27 FRACTALS Simulated tree 418 Fractals in the world S27 FRACTALS Natural coastline 419 Fractals in the world S27 FRACTALS Simulated coastline 420 Fractals in the world S27 FRACTALS Natural mountain range 421 Fractals in the world S27 FRACTALS Simulated mountain range 422 Fractals in the world S27 FRACTALS The Great Wave o↵ Kanagawa by Katsushika Hokusai 423 Fractals in the world S27 FRACTALS Patterns formed by bacteria grown in a petri dish.
    [Show full text]
  • On the Topological Convergence of Multi-Rule Sequences of Sets and Fractal Patterns
    Soft Computing https://doi.org/10.1007/s00500-020-05358-w FOCUS On the topological convergence of multi-rule sequences of sets and fractal patterns Fabio Caldarola1 · Mario Maiolo2 © The Author(s) 2020 Abstract In many cases occurring in the real world and studied in science and engineering, non-homogeneous fractal forms often emerge with striking characteristics of cyclicity or periodicity. The authors, for example, have repeatedly traced these characteristics in hydrological basins, hydraulic networks, water demand, and various datasets. But, unfortunately, today we do not yet have well-developed and at the same time simple-to-use mathematical models that allow, above all scientists and engineers, to interpret these phenomena. An interesting idea was firstly proposed by Sergeyev in 2007 under the name of “blinking fractals.” In this paper we investigate from a pure geometric point of view the fractal properties, with their computational aspects, of two main examples generated by a system of multiple rules and which are enlightening for the theme. Strengthened by them, we then propose an address for an easy formalization of the concept of blinking fractal and we discuss some possible applications and future work. Keywords Fractal geometry · Hausdorff distance · Topological compactness · Convergence of sets · Möbius function · Mathematical models · Blinking fractals 1 Introduction ihara 1994; Mandelbrot 1982 and the references therein). Very interesting further links and applications are also those The word “fractal” was coined by B. Mandelbrot in 1975, between fractals, space-filling curves and number theory but they are known at least from the end of the previous (see, for instance, Caldarola 2018a; Edgar 2008; Falconer century (Cantor, von Koch, Sierpi´nski, Fatou, Hausdorff, 2014; Lapidus and van Frankenhuysen 2000), or fractals Lévy, etc.).
    [Show full text]
  • Fractal Initialization for High-Quality Mapping with Self-Organizing Maps
    Neural Comput & Applic DOI 10.1007/s00521-010-0413-5 ORIGINAL ARTICLE Fractal initialization for high-quality mapping with self-organizing maps Iren Valova • Derek Beaton • Alexandre Buer • Daniel MacLean Received: 15 July 2008 / Accepted: 4 June 2010 Ó Springer-Verlag London Limited 2010 Abstract Initialization of self-organizing maps is typi- 1.1 Biological foundations cally based on random vectors within the given input space. The implicit problem with random initialization is Progress in neurophysiology and the understanding of brain the overlap (entanglement) of connections between neu- mechanisms prompted an argument by Changeux [5], that rons. In this paper, we present a new method of initiali- man and his thought process can be reduced to the physics zation based on a set of self-similar curves known as and chemistry of the brain. One logical consequence is that Hilbert curves. Hilbert curves can be scaled in network size a replication of the functions of neurons in silicon would for the number of neurons based on a simple recursive allow for a replication of man’s intelligence. Artificial (fractal) technique, implicit in the properties of Hilbert neural networks (ANN) form a class of computation sys- curves. We have shown that when using Hilbert curve tems that were inspired by early simplified model of vector (HCV) initialization in both classical SOM algo- neurons. rithm and in a parallel-growing algorithm (ParaSOM), Neurons are the basic biological cells that make up the the neural network reaches better coverage and faster brain. They form highly interconnected communication organization. networks that are the seat of thought, memory, con- sciousness, and learning [4, 6, 15].
    [Show full text]
  • FRACTAL CURVES 1. Introduction “Hike Into a Forest and You Are Surrounded by Fractals. the In- Exhaustible Detail of the Livin
    FRACTAL CURVES CHELLE RITZENTHALER Abstract. Fractal curves are employed in many different disci- plines to describe anything from the growth of a tree to measuring the length of a coastline. We define a fractal curve, and as a con- sequence a rectifiable curve. We explore two well known fractals: the Koch Snowflake and the space-filling Peano Curve. Addition- ally we describe a modified version of the Snowflake that is not a fractal itself. 1. Introduction \Hike into a forest and you are surrounded by fractals. The in- exhaustible detail of the living world (with its worlds within worlds) provides inspiration for photographers, painters, and seekers of spiri- tual solace; the rugged whorls of bark, the recurring branching of trees, the erratic path of a rabbit bursting from the underfoot into the brush, and the fractal pattern in the cacophonous call of peepers on a spring night." Figure 1. The Koch Snowflake, a fractal curve, taken to the 3rd iteration. 1 2 CHELLE RITZENTHALER In his book \Fractals," John Briggs gives a wonderful introduction to fractals as they are found in nature. Figure 1 shows the first three iterations of the Koch Snowflake. When the number of iterations ap- proaches infinity this figure becomes a fractal curve. It is named for its creator Helge von Koch (1904) and the interior is also known as the Koch Island. This is just one of thousands of fractal curves studied by mathematicians today. This project explores curves in the context of the definition of a fractal. In Section 3 we define what is meant when a curve is fractal.
    [Show full text]
  • Turtlefractalsl2 Old
    Lecture 2: Fractals from Recursive Turtle Programs use not vain repetitions... Matthew 6: 7 1. Fractals Pictured in Figure 1 are four fractal curves. What makes these shapes different from most of the curves we encountered in the previous lecture is their amazing amount of fine detail. In fact, if we were to magnify a small region of a fractal curve, what we would typically see is the entire fractal in the large. In Lecture 1, we showed how to generate complex shapes like the rosette by applying iteration to repeat over and over again a simple sequence of turtle commands. Fractals, however, by their very nature cannot be generated simply by repeating even an arbitrarily complicated sequence of turtle commands. This observation is a consequence of the Looping Lemmas for Turtle Graphics. Sierpinski Triangle Fractal Swiss Flag Koch Snowflake C-Curve Figure 1: Four fractal curves. 2. The Looping Lemmas Two of the simplest, most basic turtle programs are the iterative procedures in Table 1 for generating polygons and spirals. The looping lemmas assert that all iterative turtle programs, no matter how many turtle commands appear inside the loop, generate shapes with the same general symmetries as these basic programs. (There is one caveat here, that the iterating index is not used inside the loop; otherwise any turtle program can be simulated by iteration.) POLY (Length, Angle) SPIRAL (Length, Angle, Scalefactor) Repeat Forever Repeat Forever FORWARD Length FORWARD Length TURN Angle TURN Angle RESIZE Scalefactor Table 1: Basic procedures for generating polygons and spirals via iteration. Circle Looping Lemma Any procedure that is a repetition of the same collection of FORWARD and TURN commands has the structure of POLY(Length, Angle), where Angle = Total Turtle Turning within the Loop Length = Distance from Turtle’s Initial Position to Turtle’s Final Position within the Loop That is, the two programs have the same boundedness, closing, and symmetry.
    [Show full text]
  • Introduction to Fractal Geometry: Definition, Concept, and Applications
    University of Northern Iowa UNI ScholarWorks Presidential Scholars Theses (1990 – 2006) Honors Program 1992 Introduction to fractal geometry: Definition, concept, and applications Mary Bond University of Northern Iowa Let us know how access to this document benefits ouy Copyright ©1992 - Mary Bond Follow this and additional works at: https://scholarworks.uni.edu/pst Part of the Geometry and Topology Commons Recommended Citation Bond, Mary, "Introduction to fractal geometry: Definition, concept, and applications" (1992). Presidential Scholars Theses (1990 – 2006). 42. https://scholarworks.uni.edu/pst/42 This Open Access Presidential Scholars Thesis is brought to you for free and open access by the Honors Program at UNI ScholarWorks. It has been accepted for inclusion in Presidential Scholars Theses (1990 – 2006) by an authorized administrator of UNI ScholarWorks. For more information, please contact [email protected]. INTRODUCTION TO FRACTAL GEOMETRY: DEFINITI0N7 CONCEPT 7 AND APP LI CA TIO NS MAY 1992 BY MARY BOND UNIVERSITY OF NORTHERN IOWA CEDAR F ALLS7 IOWA • . clouds are not spheres, mountains are not cones~ coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line, . • Benoit B. Mandelbrot For centuries, geometers have utilized Euclidean geometry to describe, measure, and study the world around them. The quote from Mandelbrot poses an interesting problem when one tries to apply Euclidean geometry to the natural world. In 1975, Mandelbrot coined the term ·tractal. • He had been studying several individual ·mathematical monsters.· At a young age, Mandelbrot had read famous papers by G. Julia and P . Fatou concerning some examples of these monsters. When Mandelbrot was a student at Polytechnique, Julia happened to be one of his teachers.
    [Show full text]