Cantor Set, Sierpinski Triangle, Koch Snowflake, Fractal Dimension
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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. -
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. -
On the Structures of Generating Iterated Function Systems of Cantor Sets
ON THE STRUCTURES OF GENERATING ITERATED FUNCTION SYSTEMS OF CANTOR SETS DE-JUN FENG AND YANG WANG Abstract. A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under the open set condition (OSC). If dimH F < 1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study. 1. Introduction N d In this paper, a family of contractive affine maps Φ = fφjgj=1 in R is called an iterated function system (IFS). According to Hutchinson [12], there is a unique non-empty compact d SN F = FΦ ⊂ R , which is called the attractor of Φ, such that F = j=1 φj(F ). Furthermore, FΦ is called a self-similar set if Φ consists of similitudes. It is well known that the standard middle-third Cantor set C is the attractor of the iterated function system (IFS) fφ0; φ1g where 1 1 2 (1.1) φ (x) = x; φ (x) = x + : 0 3 1 3 3 A natural question is: Is it possible to express C as the attractor of another IFS? Surprisingly, the general question whether the attractor of an IFS can be expressed as the attractor of another IFS, which seems a rather fundamental question in fractal geometry, has 1991 Mathematics Subject Classification. -
Fractal Curves and Complexity
Perception & Psychophysics 1987, 42 (4), 365-370 Fractal curves and complexity JAMES E. CUTI'ING and JEFFREY J. GARVIN Cornell University, Ithaca, New York Fractal curves were generated on square initiators and rated in terms of complexity by eight viewers. The stimuli differed in fractional dimension, recursion, and number of segments in their generators. Across six stimulus sets, recursion accounted for most of the variance in complexity judgments, but among stimuli with the most recursive depth, fractal dimension was a respect able predictor. Six variables from previous psychophysical literature known to effect complexity judgments were compared with these fractal variables: symmetry, moments of spatial distribu tion, angular variance, number of sides, P2/A, and Leeuwenberg codes. The latter three provided reliable predictive value and were highly correlated with recursive depth, fractal dimension, and number of segments in the generator, respectively. Thus, the measures from the previous litera ture and those of fractal parameters provide equal predictive value in judgments of these stimuli. Fractals are mathematicalobjectsthat have recently cap determine the fractional dimension by dividing the loga tured the imaginations of artists, computer graphics en rithm of the number of unit lengths in the generator by gineers, and psychologists. Synthesized and popularized the logarithm of the number of unit lengths across the ini by Mandelbrot (1977, 1983), with ever-widening appeal tiator. Since there are five segments in this generator and (e.g., Peitgen & Richter, 1986), fractals have many curi three unit lengths across the initiator, the fractionaldimen ous and fascinating properties. Consider four. sion is log(5)/log(3), or about 1.47. -
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. -
Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle. -
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. -
Fractal Dimension of Self-Affine Sets: Some Examples
FRACTAL DIMENSION OF SELF-AFFINE SETS: SOME EXAMPLES G. A. EDGAR One of the most common mathematical ways to construct a fractal is as a \self-similar" set. A similarity in Rd is a function f : Rd ! Rd satisfying kf(x) − f(y)k = r kx − yk for some constant r. We call r the ratio of the map f. If f1; f2; ··· ; fn is a finite list of similarities, then the invariant set or attractor of the iterated function system is the compact nonempty set K satisfying K = f1[K] [ f2[K] [···[ fn[K]: The set K obtained in this way is said to be self-similar. If fi has ratio ri < 1, then there is a unique attractor K. The similarity dimension of the attractor K is the solution s of the equation n X s (1) ri = 1: i=1 This theory is due to Hausdorff [13], Moran [16], and Hutchinson [14]. The similarity dimension defined by (1) is the Hausdorff dimension of K, provided there is not \too much" overlap, as specified by Moran's open set condition. See [14], [6], [10]. REPRINT From: Measure Theory, Oberwolfach 1990, in Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, numero 28, anno 1992, pp. 341{358. This research was supported in part by National Science Foundation grant DMS 87-01120. Typeset by AMS-TEX. G. A. EDGAR I will be interested here in a generalization of self-similar sets, called self-affine sets. In particular, I will be interested in the computation of the Hausdorff dimension of such sets. -
Fractal Geometry and Applications in Forest Science
ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals .................................................................................... -
Review of the Software Packages for Estimation of the Fractal Dimension
ICT Innovations 2015 Web Proceedings ISSN 1857-7288 Review of the Software Packages for Estimation of the Fractal Dimension Elena Hadzieva1, Dijana Capeska Bogatinoska1, Ljubinka Gjergjeska1, Marija Shumi- noska1, and Risto Petroski1 1 University of Information Science and Technology “St. Paul the Apostle” – Ohrid, R. Macedonia {elena.hadzieva, dijana.c.bogatinoska, ljubinka.gjergjeska}@uist.edu.mk {marija.shuminoska, risto.petroski}@cse.uist.edu.mk Abstract. The fractal dimension is used in variety of engineering and especially medical fields due to its capability of quantitative characterization of images. There are different types of fractal dimensions, among which the most used is the box counting dimension, whose estimation is based on different methods and can be done through a variety of software packages available on internet. In this pa- per, ten open source software packages for estimation of the box-counting (and other dimensions) are analyzed, tested, compared and their advantages and dis- advantages are highlighted. Few of them proved to be professional enough, reli- able and consistent software tools to be used for research purposes, in medical image analysis or other scientific fields. Keywords: Fractal dimension, box-counting fractal dimension, software tools, analysis, comparison. 1 Introduction The fractal dimension offers information relevant to the complex geometrical structure of an object, i.e. image. It represents a number that gauges the irregularity of an object, and thus can be used as a criterion for classification (as well as recognition) of objects. For instance, medical images have typical non-Euclidean, fractal structure. This fact stimulated the scientific community to apply the fractal analysis for classification and recognition of medical images, and to provide a mathematical tool for computerized medical diagnosing. -
An Investigation of Fractals and Fractal Dimension
An Investigation of Fractals and Fractal Dimension Student: Ian Friesen Advisor: Dr. Andrew J. Dean April 10, 2018 Contents 1 Introduction 2 1.1 Fractals in Nature . 2 1.2 Mathematically Constructed Fractals . 2 1.3 Motivation and Method . 3 2 Basic Topology 5 2.1 Metric Spaces . 5 2.2 String Spaces . 7 2.3 Hausdorff Metric . 8 3 Topological Dimension 9 3.1 Refinement . 9 3.2 Zero-dimensional Spaces . 9 3.3 Covering Dimension . 10 4 Measure Theory 12 4.1 Outer/Inner Lebesgue Measure . 12 4.2 Carath´eodory Measurability . 13 4.3 Two dimensional Lebesgue Measure . 14 5 Self-similarity 14 5.1 Ratios . 14 5.2 Examples and Calculations . 15 5.3 Chaos Game . 17 6 Fractal Dimension 18 6.1 Hausdorff Measure . 18 6.2 Packing Measure . 21 6.3 Fractal Definition . 24 7 Fractals and the Complex Plane 25 7.1 Julia Sets . 25 7.2 Mandelbrot Set . 26 8 Conclusion 27 1 1 Introduction 1.1 Fractals in Nature There are many fractals in nature. Most of these have some level of self-similarity, and are called self-similar objects. Basic examples of these include the surface of cauliflower, the pat- tern of a fern, or the edge of a snowflake. Benoit Mandelbrot (considered to be the father of frac- tals) thought to measure the coastline of Britain and concluded that using smaller units of mea- surement led to larger perimeter measurements. Mathematically, as the unit of measurement de- creases in length and becomes more precise, the perimeter increases in length.