DOCSLIB.ORG

Review of the Software Packages for Estimation of the Fractal Dimension

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. We bring up to the attention that, throughout the papers we have analyzed, and the software tools we have examined, there is a significant inconsistency among the au- thors, concerning the dimension’s terminology. Sometimes the box-counting dimen- sion is called the Hausdorff dimension or the similarity dimension is called the box counting dimension. Here we will give only the definitions of fractal and box-counting fractal dimensions. 201 S. Loshkovska, S. Koceski (Editors): ICT Innovations 2015, Web Proceedings, ISSN 1857-7288 © ICT ACT http://ictinnovations.org/2015, 2015 ICT Innovations 2015 Web Proceedings ISSN 1857-7288 Definition by Barnsley [1]. Given 휀 > 0. Let 푁(휀) be the smallest number of closed balls of radius 휀>0 needed to cover the set F. The fractal dimension of the set F is the number ln⁡(푁(휀)) 퐷(퐹) = 푙푖푚 ⁡⁡⁡⁡⁡ (1) 휀→0 ln 휀 A theorem stated in [1] states that if instead of closed balls, boxes of side length 1/2n are used, the fractal dimension of the set F, D(F), is equal to the number ln⁡(푁(푛)) 퐷 (퐹) = 푙푖푚 (2) 퐵 푛→∞ ln 2푛 where N(n) is the number of boxes that have nonempty intersection with the set F. DB(F) is usually referred to as the box-counting fractal dimension of the set F. The software tools that we have analyzed, mainly compute the box-counting dimen- sion by the basic box-counting method, described by the equation (2). (Practically, the image is covered with the boxes of side length 1/2i, where i is a positive integer that has appropriately chosen maximum value n. Then, the approximation of the limit from the definition is the slope of the line that fits the finite set of data {(log2i, logN(i)) | i = 1, 2,…, n, n is a finite number}). Falconer in [6] gives excellent clarification of the different types of the dimensions. The fractal approach in medical diagnostics is noted as reasonable and many times reported in papers [9], [16], [14], [12], [13], [10]. The authors encounter many problems related to the image analysis, like: having images with different resolution (which af- fects the fractal dimension), the brightness-contrast enhancement of the imaging device, the expertise of the person that collects the images, the presence of noise, masking structures, the variability of shapes and tissues, ([4]; [15]). Additionally, they encoun- ter problems related to the fractal analysis: which software tool will be the most com- patible for their research, which type of the dimension will quantify the best their sub- jects of research, which method for estimation of particular fractal dimension to choose (if the software tool offers such options, at all), how to set the parameters included, does the tool accepts the images in particular format, and so on. Some of the ambigui- ties are detected by Lopes, Betrouni, Braverman and reported in [11], [2]. In this paper we will state the uncertainties related to the reliability of the existing software tools for estimation of the box counting fractal dimension, performed using the basic method. We will apply several software tools (will be listed later) to estimate the box-count- ing fractal dimension of three strict self-similar IFS fractals: the Sierpinski triangle, the Koch snowflake and the Minkovski sausage, and their parts, specially chosen: smaller copies of the fractal itself and one biomedical image of tinea corporis disease (see Fig. 1). 202 S. Loshkovska, S. Koceski (Editors): ICT Innovations 2015, Web Proceedings, ISSN 1857-7288 © ICT ACT http://ictinnovations.org/2015, 2015 ICT Innovations 2015 Web Proceedings ISSN 1857-7288 Fig. 1. Samples of fractals It is known (see for instance [6]) that the similarity, box counting and Hausdorff di- mensions of the Sierpinski triangle and Koch curve are equal, ln 3 DH (Sierp) = DB (Sierp) = DS (Sierp) = =1.58496 and ln 2 ln 4 DH (Koch curve) = DB (Koch curve)= DS (Koch curve) = =1.26186. ln 3 A theorem formulated in [1] can be used to show that DB (Koch snowflake) = DB (Koch curve)= 1.26186. Similarly, it can be shown that ln 8 DH (Mink. saus.)= DB (Mink. saus.)= DS (Mink. saus.) = = 1.5000. ln 4 Later (in section 3) we will compare the values of these box counting dimensions with our estimated results, so to characterize the reliability of the examined software tools. Additionally, we have decided to estimate the fractal dimensions of parts (actually smaller copies) of Sierpinski triangle and Koch curve, to check the consistency of the algorithms, since the dimensions of a self-similar fractal and its smaller copy, in ideal case, when infinite number of iterations are performed, are equal. We have used both types of black and white images: white image on black background (Minkowski sau- sage) and black image on white background (Sierpinski and Koch fractals), because some of the software tools have an option for choosing which pixels are “active”, black or white. We have also estimated the fractal dimension of a medical image, such as the image of the skin disease Tinea corporis (Fig. 1), so to test the software tools on real cases. In the remaining sections we will analyze the chosen open source available software tools used for computing the box-counting fractal dimension, test the software tools over the aforementioned strict self-similar fractals and a medical image, resume the results of testing and give the description and comparison of the tools. 203 S. Loshkovska, S. Koceski (Editors): ICT Innovations 2015, Web Proceedings, ISSN 1857-7288 © ICT ACT http://ictinnovations.org/2015, 2015 ICT Innovations 2015 Web Proceedings ISSN 1857-7288 2 Description and Analysis of the Available Software Tools The software tools illustrated in this section are based mostly on the basic method for computing the box-counting fractal dimension explained in the section 1. There are only two tools which are based on other methods and such aspect is emphasized in the description of the software tools. 2.1 FracLAC [Author: Audrey Karperien, http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm] FracLAC is an open source available plugin for ImageJ aimed for the detailed fractal analysis of images. FracLAC is also available as a standalone Java application, but the plugin is more up-to-date and has more features than standalone version. FracLAC for ImageJ is a user friendly and comprehensive toolset, widely used in many different disciplines, especially in medical image processing, in almost all medical fields: oph- thalmology and visual science, oncology, neurology, cardiology, in cancer research etc. A lot of papers from these and other scientific fields that use FracLAC are cited on its official web site. FracLAC is not only raw software for mathematical analysis, but it also contains a rich documentation explaining the concept of fractal analysis. It can be used for fractal and multifractal analysis over: entire open images, ROIs and multiple files. Images, of type jpg, tiff, png or gif, have to be opened with ImageJ. They have to be in grayscale or binary format (otherwise, conversion is necessary). Preprocessing (like thresholding, background subtraction, dilatation, edge detection) should be done in ImageJ or other image processing software. Preparation of the images depends on the aim and type of analysis and also on the type of image (grayscale or binary, contour or texture etc.). The user friendly interface offers a few options: Box-counting and Lacunarity Anal- ysis (which is analyzed within this paper), Sliding Box Lacunarity Analysis, Mass vs. Distance Analysis, Local Connected Fractal Dimension Analysis, Sub Sample and Par- ticle Analyser Analysis and Multifractal Analysis, with the possibility of setting a con- siderable number of parameters in all the aforementioned options. After reading the comprehensive explanation that is available in the interface itself, we have set the pa- rameters as if they were standards so that comparable results with other software pack- ages could be obtained (such settings are actually much below the wide web of possi- bilities offered through FracLAC).
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [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]
  • Fractals a Fractal Is a Shape That Seem to Have the Same Structure No Matter How Far You Zoom In, Like the figure Below
    Fractals A fractal is a shape that seem to have the same structure no matter how far you zoom in, like the figure below. Sometimes it's only part of the shape that repeats. In the figure below, called an Apollonian Gasket, no part looks like the whole shape, but the parts near the edges still repeat when you zoom in. Today you'll learn how to construct a few fractals: • The Snowflake • The Sierpinski Carpet • The Sierpinski Triangle • The Pythagoras Tree • The Dragon Curve After you make a few of those, try constructing some fractals of your own design! There's more on the back. ! Challenge Problems In order to solve some of the more difficult problems today, you'll need to know about the geometric series. In a geometric series, we add up a sequence of terms, 1 each of which is a fixed multiple of the previous one. For example, if the ratio is 2 , then a geometric series looks like 1 1 1 1 1 1 1 + + · + · · + ::: 2 2 2 2 2 2 1 12 13 = 1 + + + + ::: 2 2 2 The geometric series has the incredibly useful property that we have a good way of 1 figuring out what the sum equals. Let's let r equal the common ratio (like 2 above) and n be the number of terms we're adding up. Our series looks like 1 + r + r2 + ::: + rn−2 + rn−1 If we multiply this by 1 − r we get something rather simple. (1 − r)(1 + r + r2 + ::: + rn−2 + rn−1) = 1 + r + r2 + ::: + rn−2 + rn−1 − ( r + r2 + ::: + rn−2 + rn−1 + rn ) = 1 − rn Thus 1 − rn 1 + r + r2 + ::: + rn−2 + rn−1 = : 1 − r If we're clever, we can use this formula to compute the areas and perimeters of some of the shapes we create.
    [Show full text]
  • Exploring the Effect of Direction on Vector-Based Fractals
    Exploring the Effect of Direction on Vector-Based Fractals Magdy Ibrahim and Robert J. Krawczyk College of Architecture Illinois Institute of Technology 3360 S. State St. Chicago, IL, 60616, USA Email: [email protected], [email protected] Abstract This paper investigates an approach to begin the study of fractals in architectural design. Vector-based fractals are studied to determine if the modification of vector direction in either the generator or the initiator will develop alternate fractal forms. The Koch Snowflake is used as the demonstrating fractal. Introduction A fractal is an object or quantity that displays self-similarity on all scales. The object need not exhibit exactly the same structure at all scales, but the same “type” of structures must appear on all scales [7]. Fractals were first discussed by Mandelbrot in the 1970s [4], but the idea was identified as early as 1925. Fractals have been investigated for their visual qualities as art, their relationship to explain natural processes, music, medicine, and in mathematics [5]. Javier Barrallo classified fractals [2] into six main groups depending on their type: 1. Fractals derived from standard geometry by using iterative transformations on an initial common figure. 2. IFS (Iterated Function Systems), this is a type of fractal introduced by Michael Barnsley. 3. Strange attractors. 4. Plasma fractals. Created with techniques like fractional Brownian motion. 5. L-Systems, also called Lindenmayer systems, were not invented to create fractals but to model cellular growth and interactions. 6. Fractals created by the iteration of complex polynomials. From the mathematical point of view, we can classify fractals into three major categories.
    [Show full text]
  • Fractals Dalton Allan and Brian Dalke College of Science, Engineering & Technology Nominated by Amy Hlavacek, Associate Professor of Mathematics
    Fractals Dalton Allan and Brian Dalke College of Science, Engineering & Technology Nominated by Amy Hlavacek, Associate Professor of Mathematics Dalton is a dual-enrolled student at Saginaw Arts and Sciences Academy and Saginaw Valley State University. He has taken mathematics classes at SVSU for the past four years and physics classes for the past year. Dalton has also worked in the Math and Physics Resource Center since September 2010. Dalton plans to continue his study of mathematics at the Massachusetts Institute of Technology. Brian is a part-time mathematics and English student. He has a B.S. degree with majors in physics and chemistry from the University of Wisconsin--Eau Claire and was a scientific researcher for The Dow Chemical Company for more than 20 years. In 2005, he was certified to teach physics, chemistry, mathematics and English and subsequently taught mostly high school mathematics for five years. Currently, he enjoys helping SVSU students as a Professional Tutor in the Math and Physics Resource Center. In his spare time, Brian finds joy playing softball, growing roses, reading, camping, and spending time with his wife, Lorelle. Introduction For much of the past two-and-one-half millennia, humans have viewed our geometrical world through the lens of a Euclidian paradigm. Despite knowing that our planet is spherically shaped, math- ematicians have developed the mathematics of non-Euclidian geometry only within the past 200 years. Similarly, only in the past century have mathematicians developed an understanding of such common phenomena as snowflakes, clouds, coastlines, lightning bolts, rivers, patterns in vegetation, and the tra- jectory of molecules in Brownian motion (Peitgen 75).
    [Show full text]
  • New Escape Time Koch Curve in Complex Plane
    International Journal of Computer Applications (0975 – 8887) Volume 58– No.8, November 2012 New Escape Time Koch Curve in Complex Plane Priti Dimri Dharmendra Kumar Ashish Negi Associate Professor, Associate Professor, Associate Professor, Department of Computer Department of Computer Department of Computer Science and Engineering Science Science and Engineering G.B Pant Engineering College Institute of Management G.B Pant Engineering College Pauri Garhwal, 246001 Studies Dehradun Pauri Garhwal, 246001 ABSTRACT that of f(x) = |x| at x = 0, where no tangent is defined.[15] Von Koch curves allow numerous variations and have Because of its unique construction the Koch curve has a inspired many researchers and fractal artists to produce noninteger dimensionality. In the Koch curve, a fractal pattern amazing pieces of art. In this paper we present a new of 60-degree-to-base line segments one-third the length of the algorithm for plotting the Koch curve using complex previous line is repeated. The portion of the base line under a variables. Further we have applied various coloring newly formed triangle is deleted. Such geometric algorithms to generate complex Koch fractals. manipulation continued indefinitely, forms the Koch Curve. [23] Keywords The Von Koch Curve shows the self-similar property of a Koch curves, Fractal Coloring, Escape Time Algorithm. fractal. The same pattern appears everywhere along the curve 1. INTRODUCTION in different scale, from visible to infinitesimal. The Koch curve has a Hausdorff dimension of log(4)/log(3), i.e. A fractal is an object or quantity that displays self-similarity approximately 1.2619 and is enclosed in a finite area [8].
    [Show full text]
  • THE KOCH CURVE Image Starting with an Equilateral Triangle and Replacing Each Edge by a Zig-Zag Curve Made up of 4 Pieces
    AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE TEACHER NETWORK THE KOCH CURVE Image starting with an equilateral triangle and replacing each edge by a zig-zag curve made up of 4 pieces. Each of the 4 pieces is one third of the length of the line segment it replaces so it looks as if equilateral triangles have been attached to the shape. Now imagine repeating this process over and over again. What would happen? The edges of the triangle at the first stage are one unit in length. Here are the first four stages. The Koch curve is sometimes called the snowflake curve. This curve is the outer perimeter of the shape formed by the outer edges when the process is repeated infinitely often. 1. The table shows that the snowflake construction produces three types of sequences A, B and C. The last row gives the nth term in the sequence. Describe the type of sequence in each case giving two properties of the sequence. 2. What happens to the perimeter as n tends to infinity? The next table shows some of the calculations of the area enclosed by the Koch curve. 3. Show that the sum of the increases in area in column C is 1 ! !!! 1 + ! + ! + ⋯ + ! 3 ! ! ! Find the sum of this series. In studying what happens to the area inside the curve in this iteration it is convenient to take the area of the original triangle as 1 square unit for the calculations. To relate this calculation to the previous table where the edge of the triangle has length 1 unit use the scale factor √3/4.] Investigate the increase in area of the Von Koch snowflake at successive stages.
    [Show full text]
  • Fractal Geometry: History and Theory
    Smith 1 Geri Smith MATH H324 College Geometry Dr. Kent Honors Research Paper April 26 th , 2011 Fractal Geometry: History and Theory Classical Euclidean geometry cannot accurately represent the natural world; fractal geometry is the geometry of nature. Fractal geometry can be described as an extension of Euclidean geometry and can create concrete models of the various physical structures within nature. In short, fractal geometry and fractals are characterized by self-similarity and recursion, which entails scaling patterns, patterns within patterns, and symmetry across every scale. Benoit Mandelbrot, the “father” of fractal geometry, coined the term “fractal,” in the1970s, from the Latin “Fractus” (broken), to describe these infinitely complex scaling shapes. Yet, the basic ideas behind fractals were explored as far back as the seventeenth century; however, the oldest fractal is considered to be the nineteenth century's Cantor set. A variety of mathematical curiosities, “pathological monsters,” like the Cantor set, the Julia set, and the Peano space-filling curve, upset 19 th century standards and confused mathematicians, who believed this demonstrated the ability to push mathematics out of the realm of the natural world. It was not until fractal geometry was developed in the 1970s that what past mathematicians thought to be unnatural was shown to be truly representative of natural phenomena. In the late 1970s and throughout the 1980s, fractal geometry captured world-wide interest, even among non-mathematicians (probably due to the fact fractals make for pretty pictures), and was a popular topic- conference sprung up and Mandelbrot's treatise on fractal geometry, The Fractal Geometry of Nature, was well-read even outside of the math community.
    [Show full text]
  • Exploring the Effect of Direction on Vector-Based Fractals
    BRIDGES Mathematical Connections in Art, Music, and Science Exploring the Effect of Direction on Vector-Based Fractals Magdy Ibrahim and Robert J. Krawczyk College of Architecture Dlinois Institute of Technology 3360 S. State St. Chicago, lL, 60616, USA Email: [email protected], [email protected] Abstract This paper investigates an approach to begin the study of fractals in architectural design. Vector-based fractals are studied to determine if the modification of vector direction in either the generator or the initiator will develop alternate fractal forms. The Koch Snowflake is used as the demonstrating fractal. Introduction A fractal is an object or quantity that displays self-similarity on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales [7]. Fractals were first discussed by Mandelbrot in the 1970s [4], but the idea was identified as early as 1925. Fractals have been investigated for their visual qualities as art, their relationship to explain natural processes, music, medicine, and in mathematics [5]. Javier Barrallo classified fractals [2] into six main groups depending on their type: 1. Fractals derived from standard geometry by using iterative transformations on an initial common figure. 2. IFS (Iterated Function Systems), this is a type of fractal introduced by Michael Barnsley. 3. Strange attractors. 4. Plasma fractals. Created with techniques like fractional Brownian motion. 5. L-Systems, also called Lindenmayer systems, were not invented to create fractals but to model cellular growth and interactions. 6. Fractals created by the iteration of complex polynomials. From the mathematical point of view, we can classify fractals into three major categories.
    [Show full text]
  • Koch and Levy Curves
    Koch and Levy Curves The next fractal you will create is the Koch curve, invented by Koch in 1904. The Koch curve is the self-similar fractal formed from the Initiator and Generator shown below. Notice that the Generator is formed by dividing the Initiator in thirds and replacing the middle third with two legs of an equilateral triangle whose base is the size of the removed piece. Scale copies of the Koch curve Initiator and Generator, on isometric dot paper, are included on the last page of this lesson. Koch curve initiator and generator. 1. Create stage 2 of the Koch curve. 2. How many line segments make up Stage 2? If we set our scale so the length of the Initiator is 1 unit, how long is each of the line segments that make up Stage 2? So what is the overall length of Stage 2 of the Koch curve? 3. Create Stage 3 of the Koch curve. 4. How many line segments make up Stage 3? How long is each line segment? So what is the overall length of Stage 3 of the Koch curve? 5. Create stage 4 of the Koch curve. 6. How many line segments make up Stage 4? How long is each line segment? So what is the overall length of Stage 4 of the Koch curve? 7. Find an algebraic expression for the length of Stage n of the Koch curve. 8. Return to the Initiator and Generator. Explain how interpret your expression for the length of Stage n of the Koch curve in terms of the geometry of the Initiator and Generator.
    [Show full text]