FRACTAL CURVES 1. Introduction “Hike Into a Forest and You Are Surrounded by Fractals. the In- Exhaustible Detail of the Livin
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Infinite Perimeter of the Koch Snowflake And
The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area Yaroslav D. Sergeyev∗ y Abstract The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Nu- merical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be in- finite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An − Ak results to be infinitesimal. Finally, snowflakes con- structed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed. -
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. -
Harmonious Hilbert Curves and Other Extradimensional Space-Filling Curves
Harmonious Hilbert curves and other extradimensional space-filling curves∗ Herman Haverkorty November 2, 2012 Abstract This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d0 < d, the d-dimensional curve is compatible with the d0-dimensional curve with respect to the order in which the curves visit the points of any d0-dimensional axis-parallel space that contains the origin. Similar generalizations to arbitrary dimensions are described for several variants of Peano's curve (the original Peano curve, the coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional harmonious Hilbert curves and the Meurthe curves have neutral orientation: as compared to the curve as a whole, arbitrary pieces of the curve have each of d! possible rotations with equal probability. Thus one could say these curves are `statistically invariant' under rotation|unlike the Peano curves, the coil curves, the half-coil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of R-trees, this paper shows how to construct a 2d-dimensional generalized Hilbert or Peano curve that traverses the points of a certain d-dimensional diagonally placed subspace in the order of a given d-dimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper. 1 Introduction Space-filling curves A space-filling curve in d dimensions is a continuous, surjective mapping from R to Rd. -
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. -
WHY the KOCH CURVE LIKE √ 2 1. Introduction This Essay Aims To
p WHY THE KOCH CURVE LIKE 2 XANDA KOLESNIKOW (SID: 480393797) 1. Introduction This essay aims to elucidate a connection between fractals and irrational numbers, two seemingly unrelated math- ematical objects. The notion of self-similarity will be introduced, leading to a discussion of similarity transfor- mations which are the main mathematical tool used to show this connection. The Koch curve and Koch island (or Koch snowflake) are thep two main fractals that will be used as examples throughout the essay to explain this connection, whilst π and 2 are the two examples of irrational numbers that will be discussed. Hopefully,p by the end of this essay you will be able to explain to your friends and family why the Koch curve is like 2! 2. Self-similarity An object is self-similar if part of it is identical to the whole. That is, if you are to zoom in on a particular part of the object, it will be indistinguishable from the entire object. Many objects in nature exhibit this property. For example, cauliflower appears self-similar. If you were to take a picture of a whole cauliflower (Figure 1a) and compare it to a zoomed in picture of one of its florets, you may have a hard time picking the whole cauliflower from the floret. You can repeat this procedure again, dividing the floret into smaller florets and comparing appropriately zoomed in photos of these. However, there will come a point where you can not divide the cauliflower any more and zoomed in photos of the smaller florets will be easily distinguishable from the whole cauliflower. -
Redalyc.Self-Similarity of Space Filling Curves
Ingeniería y Competitividad ISSN: 0123-3033 [email protected] Universidad del Valle Colombia Cardona, Luis F.; Múnera, Luis E. Self-Similarity of Space Filling Curves Ingeniería y Competitividad, vol. 18, núm. 2, 2016, pp. 113-124 Universidad del Valle Cali, Colombia Available in: http://www.redalyc.org/articulo.oa?id=291346311010 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Ingeniería y Competitividad, Volumen 18, No. 2, p. 113 - 124 (2016) COMPUTATIONAL SCIENCE AND ENGINEERING Self-Similarity of Space Filling Curves INGENIERÍA DE SISTEMAS Y COMPUTACIÓN Auto-similaridad de las Space Filling Curves Luis F. Cardona*, Luis E. Múnera** *Industrial Engineering, University of Louisville. KY, USA. ** ICT Department, School of Engineering, Department of Information and Telecommunication Technologies, Faculty of Engineering, Universidad Icesi. Cali, Colombia. [email protected]*, [email protected]** (Recibido: Noviembre 04 de 2015 – Aceptado: Abril 05 de 2016) Abstract We define exact self-similarity of Space Filling Curves on the plane. For that purpose, we adapt the general definition of exact self-similarity on sets, a typical property of fractals, to the specific characteristics of discrete approximations of Space Filling Curves. We also develop an algorithm to test exact self- similarity of discrete approximations of Space Filling Curves on the plane. In addition, we use our algorithm to determine exact self-similarity of discrete approximations of four of the most representative Space Filling Curves. -
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. -
Lasalle Academy Fractal Workshop – October 2006
LaSalle Academy Fractal Workshop – October 2006 Fractals Generated by Geometric Replacement Rules Sierpinski’s Gasket • Press ‘t’ • Choose ‘lsystem’ from the menu • Choose ‘sierpinski2’ from the menu • Make the order ‘0’, press ‘enter’ • Press ‘F4’ (This selects the video mode. This mode usually works well. Fractint will offer other options if you press the delete key. ‘Shift-F6’ gives more colors and better resolution, but doesn’t work on every computer.) You should see a triangle. • Press ‘z’, make the order ‘1’, ‘enter’ • Press ‘z’, make the order ‘2’, ‘enter’ • Repeat with some other orders. (Orders 8 and higher will take a LONG time, so don’t choose numbers that high unless you want to wait.) Things to Ponder Fractint repeats a simple rule many times to generate Sierpin- ski’s Gasket. The repetition of a rule is called an iteration. The order 5 image, for example, is created by performing 5 iterations. The Gasket is only complete after an infinite number of iterations of the rule. Can you figure out what the rule is? One of the characteristics of fractals is that they exhibit self-similarity on different scales. For example, consider one of the filled triangles inside of Sierpinski’s Gasket. If you zoomed in on this triangle it would look identical to the entire Gasket. You could then find a smaller triangle inside this triangle that would also look identical to the whole fractal. Sierpinski imagined the original triangle like a piece of paper. At each step, he cut out the middle triangle. How much of the piece of paper would be left if Sierpinski repeated this procedure an infinite number of times? Von Koch Snowflake • Press ‘t’, choose ‘lsystem’ from the menu, choose ‘koch1’ • Make the order ‘0’, press ‘enter’ • Press ‘z’, make the order ‘1’, ‘enter’ • Repeat for order 2 • Look at some other orders that are less than 8 (8 and higher orders take more time) Things to Ponder What is the rule that generates the von Koch snowflake? Is the von Koch snowflake self-similar in any way? Describe how. -
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]. -
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. -
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. -
Efficient Neighbor-Finding on Space-Filling Curves
Universitat¨ Stuttgart Efficient Neighbor-Finding on Space-Filling Curves Bachelor Thesis Author: David Holzm¨uller* Degree: B. Sc. Mathematik Examiner: Prof. Dr. Dominik G¨oddeke, IANS Supervisor: Prof. Dr. Miriam Mehl, IPVS October 18, 2017 arXiv:1710.06384v3 [cs.CG] 2 Nov 2019 *E-Mail: [email protected], where the ¨uin the last name has to be replaced by ue. Abstract Space-filling curves (SFC, also known as FASS-curves) are a useful tool in scientific computing and other areas of computer science to sequentialize multidimensional grids in a cache-efficient and parallelization-friendly way for storage in an array. Many algorithms, for example grid-based numerical PDE solvers, have to access all neighbor cells of each grid cell during a grid traversal. While the array indices of neighbors can be stored in a cell, they still have to be computed for initialization or when the grid is adaptively refined. A fast neighbor- finding algorithm can thus significantly improve the runtime of computations on multidimensional grids. In this thesis, we show how neighbors on many regular grids ordered by space-filling curves can be found in an average-case time complexity of (1). In 풪 general, this assumes that the local orientation (i.e. a variable of a describing grammar) of the SFC inside the grid cell is known in advance, which can be efficiently realized during traversals. Supported SFCs include Hilbert, Peano and Sierpinski curves in arbitrary dimensions. We assume that integer arithmetic operations can be performed in (1), i.e. independent of the size of the integer.