Fractals: Properties and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Fractals: properties and applications MATH CO-OP VERONICA CIOCANEL, BROWN UNIVERSITY Fractal ball experiment: DIY! How do we think of dimension? Conclusions: Fractal properties • Fractals exhibit fractal dimensions: all objects whose dimension is not an integer are fractals. • Fractals are self-similar. d = 1.2683 Fractals - mathematical objects Mandelbrot set Variation of a Mandelbrot set Fractals - around us Lake Mead coastline The Great Wave off Kanagawa - Hokusai 1. Fractal antennas Sierpinski Example of fractal triangle antenna • Fractal-shape antennas can respond to more frequencies than regular ones. • They can be ¼ the size of the regular ones: use in cellular phones and military communication hardware. • BUT: Not all fractal shapes are best suited for antennas. Koch curve fractal antenna 2. Coastlines Border length 987 km (reported by the Portuguese) • Portugal - Spain border 1214 km (reported by the Spanish) Measurements were using different scales! Returning to coastlines… South Africa Britain Approximating a smooth curve using straight lines – guaranteed to get closer to the true value of the curve length Can we say the same for the UK coastline? Perimeter/length: • Coastlines have fractal-like properties: complexity changes with measurement scale • A lot like the Koch curve • This curve has infinite length! • Length: makes little sense But, concept of fractal dimension makes sense! South Africa: d = 1.02 Britain: d = 1.25 • This is called the “Coastline paradox”: measured length of a stretch of coastline depends on the measurement scale • But for practical use, the ruler scale is not that fine: km’s are enough! • Approximating the coastline with an infinite fractal is thus not so useful in this case..

Copy Link

Recommended publications

Joint Session: Measuring Fractals Diversity in Mathematics, 2019
Joint Session: Measuring Fractals Diversity in Mathematics, 2019 Tongou Yang The University of British Columbia, Vancouver [email protected] July 2019 Tongou Yang (UBC) Fractal Lectures July 2019 1 / 12 Let's Talk Geography Which country has the longest coastline? What is the longest river on earth? Tongou Yang (UBC) Fractal Lectures July 2019 2 / 12 (a) Map of Canada (b) Map of Nunavut Which country has the longest coastline? List of countries by length of coastline Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12 (b) Map of Nunavut Which country has the longest coastline? List of countries by length of coastline (a) Map of Canada Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12 Which country has the longest coastline? List of countries by length of coastline (a) Map of Canada (b) Map of Nunavut Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12 What's the Longest River on Earth? A Youtube Video Tongou Yang (UBC) Fractal Lectures July 2019 4 / 12 Measuring a Smooth Curve Example: use a rope Figure: Boundary between CA and US on the Great Lakes Tongou Yang (UBC) Fractal Lectures July 2019 5 / 12 Measuring a Rugged Coastline Figure: Coast of Nova Scotia Tongou Yang (UBC) Fractal Lectures July 2019 6 / 12 Covering by Grids 4 3 2 1 0 0 1 2 3 4 Tongou Yang (UBC) Fractal Lectures July 2019 7 / 12 Counting Grids Number of Grids=15 Side Length ≈ 125km Coastline ≈ 15 × 125 = 1845km. Tongou Yang (UBC) Fractal Lectures July 2019 8 / 12 Covering by Finer Grids 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 Tongou Yang (UBC) Fractal Lectures July 2019 9 / 12 Counting Grids Number of Grids=34 Side Length ≈ 62km Coastline ≈ 34 × 62 = 2108km.
[Show full text]
Grade 6 Math Circles Fractals Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 6 Math Circles March 10/11 2020 Fractals Introduction Fractals are mathematical objects that are self-similar, meaning that they have the same structure when you zoom in on them. For example, this is a famous fractal called the Mandelbrot set: Image by Arnaud Cheritat, retrieved from: https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/File:MilMand_1000iter.png 1 The main shape, which looks a bit like a snowman, is shaded in the image below: As an exercise, shade in some of the smaller snowmen that are all around the largest one in the image above. This is an example of self-similarity, and this pattern continues into inﬁnity. Watch what happens when you zoom deeper and deeper into the Mandelbrot Set here: https://www.youtube.com/watch?v=pCpLWbHVNhk. Drawing Fractals Fractals like the Mandelbrot Set are generated using formulas and computers, but there are a number of simple fractals that we can draw ourselves. 2 Sierpinski Triangle Image by Beojan Stanislaus, retrieved from: https://commons.wikimedia.org/wiki/File:Sierpinski_triangle.svg Use the template and the steps below to draw this fractal: 1. Start with an equilateral triangle. 2. Connect the midpoints of all three sides. This will create one upside down triangle and three right-side up triangles. 3. Repeat from step 1 for each of the three right-side up triangles. 3 Apollonian Gasket Retrieved from: https://mathlesstraveled.com/2016/04/20/post-without-words-6/ Use the template and the steps below to draw this fractal: 1.
[Show full text]
Exploring Fractal Dimension
MATH2916 A Working Seminar University of Sydney School of Mathematics and Statistics Exploring Fractal Dimension Dibyendu Roy SID:480344308 June 2, 2019 MATH2916 Exploring Fractal Dimension Dibyendu Roy 1 Motivations We all have an intuitive understanding of roughness, and can easily tell that a jagged rock is rougher than a marble. We can similarly understand that the Koch curve is "rougher" than a straight line. To make this definition of "roughness" or "fractal-ness" more rigorous, we introduce the idea of fractal dimension. What would such a dimension be defined as? It makes sense to attempt to define such a "dimen- sion" to fill the gaps in the standard integers that we use to define dimension conventionally. We know that an object in one dimension is a line and an example of a two dimensional object is a square. But what would an example of a 1.26 dimensional object look like? In this text we will explore the nature of fractal dimension culminating in a rigorous definition that can be applied to any object. 2 Compass dimension 2.1 The coastline paradox Let us imagine that we are are a cartographer trying to find the length of the coastline of Britain. As the coastline is not a regular mathematically defined shape it would make sense for us to try and find a series of approximations that would converge to the true value after a large number of iterations. Let us consider the following method for evaluating the perimeter of a shape. We take a compass (or more appropriately a divider) and open it up to a fixed distance s.
[Show full text]
Are Fractals Really Just a Work of Art?
Are fractals really just a work of art? Fractals – a sophisticated blend of simplicity and complexity to produce dumbfounding patterns, appreciated by all mathematicians. A fractal is defined as a self-similar geometrical figure, where the pattern is duplicated every time it is zoomed in. These marvels can be created using computer graphics, or could even be painted. A brain could concentrate at a single fractal for a while, and this exposure can be said to reduce stress levels by up to 60 percent. Nevertheless, are fractals just appreciative patterns that we humans like to glare upon, or do they serve a mathematical purpose? The father of fractal geometry, Benoit Mandelbrot, was said to have a remarkable geometric intuition, which paved him into giving unique insight to mathematical problems. Whilst working for the IBM, he iterated the ‘z = z² + c’ equation, where he observed a bug-like structure repeating. This result famously became known as ‘The Mandelbrot set’, and this discovery founded modern fractal geometry. Mandelbrot’s conception of fractals was to model nature in a way that captures roughness; this is a conception that goes against some of modern maths, like calculus which evolves around smooth shapes. There are three techniques of generating fractals: escape time fractals (defined by a recurrence relation at each point in a space), iterated function fractals (which have a fixed geometric replacement rule) and random fractals (which are generated by sophisticated modelling). A Sierpinski triangle is an example of an iterated function fractal. It is an equilateral triangle subdivided recursively into smaller triangles. When originally looked at, you can see a large triangle split into three more identical triangles, which can go on forever.
[Show full text]
Generating a Fractal Tree
GENERATING A FRACTAL TREE STUDENT: JINGYI GAO ADVISOR: MITCHELL NEWBERRY 1. Abstract Inspired by a beautiful tree pattern carved into a stone screen, we want to generate self-similar tree pattern on any arbitrary shape. In this paper, we will discuss the concept of fractal, Hausdorff dimension, self-similarity, and scaling relationships. We will then explain in what sense this tree is and isn't a fractal, how does the scaling relationship correspond to its physical properties, and how do we determine these relationships of our tree. Finally, we will explain how to build fractal trees from recursive relationships. 2. History and Motivation Figure 1. The marble screen of Sidi Saiyyed mosque Image: CC-BY Vrajesh Jani The whole project is motivated by this beautiful tree pattern on the mar- ble screen of Sidi Saiyyed mosque in Ahmedabad, Gujarat, India, as shown in Figure 1, which is a notable piece of Islamic architecture, built in 1572AD. Date: August 25, 2020. 1 2 STUDENT: JINGYI GAO ADVISOR: MITCHELL NEWBERRY Marveling at these beautiful carve stone windows, we attempted to gen- erate this kind of tree pattern on our own. So the question becomes: if we want to decorate any shape of window, how would be generate a pattern like this? One answer is that we can definitely follow our artistic sense and draw it out using paper and pen. However, if I want to do it more efficiently, can I write an algorithm to help me do this? Or in other words, are there any mathematical ways to describe this kind of tree pattern? Could they be gen- erated by following some specific rules? Indeed, there are former researchers worked on interpreting art in a mathematical way.
[Show full text]
Fractals Everywhere St Paul’S Catholic School Geometry Workshop II Welcome to the Second Geometry Workshop at St Paul’S
Fractals everywhere St Paul’s Catholic School Geometry Workshop II Welcome to the second geometry workshop at St Paul’s. Today we’ll be looking at the maths of fractals, which are mathematical objects with very surprising properties! We start off with a simple question: How long is the coastline of Great Britain? This turns out to be difficult to answer because coastlines are examples of fractals. In this workshop we’ll look at the difficulties in measuring fractals and see how this related to their dimension. Then we can apply what we’ve learnt to study the length of the coastline of Great Britain. The coastline paradox and fractals The coastline paradox is the observation that the length of a coastline is ambiguous, or, in mathematical terms, not well-defined; the length of a coastline depends on the scale at which you measure it, and increases without limit as the scale increases. Why is this? It is because coastlines are examples of fractals. What is a fractal? These are examples of fractals Geometric fractals These are strictly self-similar: They are made up of smaller, exact copies of themselves. They can be created using a recursive rule: A geometric rule that you repeatedly apply to some initial shape. The image below shows the first four steps in the construction of a fractal called the Sierpinski carpet. Assume that the length of the outer square in each image is 1 unit. Fill in the following table. Step Number of squares Area of squares Area of image 1 2 3 n What is the area of this fractal? Fractal dimension The strange measurements of fractals occur because they are neither one-dimensional nor two-dimensional, but something in between.
[Show full text]
Fractal Geometry
Fractal geometry Fractals are basically never-ending patterns which are self-similar in all scales - it means that we can take a piece of a fractal and it looks the same as the whole shape. Their name comes from the fact that they don’t have whole number dimensions, instead they have fractional dimensions They are created by repeating the same process over and over again. 1. Fractals in nature Fractals are very frequent in nature and can be found between the familiar dimensions. Some phenomena that possess fractal features are coastlines, blood vessels, mountain ranges and snowflakes. They all seem quite random at first, but, as all fractals, have an underlying pattern which determines what they look like. 2. Chaos Theory Fractals are strictly connected with the theory of Chaos. It studies the unpredicted events and teaches how to cope with them. It deals with non-linear things which are normally impossible to predict or control, such as turbulence or the stock market. It states that within the most unpredictable complex systems, there are some certain features, which let us deal with such systems, such as self-similarity, self-organization, feedback loops, interconnectedness. Fractals can be used to show the images of dynamic systems (chaotic systems, pictures of Chaos) when they are driven by recursion (applied within their own definition). 3. Techniques of generation of fractals One of probably the most used methods for generating fractals is Iterated Function Systems (IFS) which uses fixed geometric replacement rules. It's used for example while generating the Koch Curve or Sierpinski Triangle which are both discussed in more detail below.
[Show full text]
The Center for Gifted 2019/20 Fall and Winter Weekday Wonders Syllabuses for the Great Mathematics Experience
The Center for Gifted 2019/20 Fall and Winter Weekday Wonders Syllabuses for The Great Mathematics Experience For Kindergarten–2nd grades - Mathemagicians: Math is magical! Employ your mathematical powers to make sense of problems, reason abstractly and quantitatively, and construct viable models for a variety of mathematical concepts. Becoming a Mathemagician – Finding the power within. o Examine oral and written mathematical codes and create your own codes. o Explore magic squares and other mathematical shapes using numbers. o Solve puzzles such as tangrams, tessellations, Set, 24, and Tenzi that incorporate mathematical concepts like tangrams and pentominoes. Math around the World – Geographic and cultural influences in mathematics. o Explore mathematical notation of the past in Central America, Egypt, India and Italy to better understand our number system. o Use classical depictions of mathematical stories from Africa, China, and India to understand how the mathematics affects our everyday lives. o After exploring geometric concepts from Japan, China, Italy, France, and North America, use a variety of materials to create geometric designs and structures. The Art of Math – Discover how mathematics is part of our everyday world and of the art around us. o Find symmetry in designs and nature using pattern blocks and other materials. o Use coordinate graphing to find treasure and create designs. o Tessellations: Why shapes tessellate; what shapes tessellate. Math on the Move – Explore ways mathematics and science complement each other. o Can Hot Wheels and Lego’s be used to explore the effects of height and length of an inclined plane on distance travelled? o Are spoons levers? How can spoons be used as simple catapults? Can we analyze results of items launched by simple catapults? o How can we use bubbles to explore mathematical concepts? Geometry 2D and 3D – Measurement, lines, points, angles, and figures.
[Show full text]
Grade 7/8 Math Circles Fractals Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles February 11/12/13 2020 Fractals Introduction Fractals are mathematical objects that are self-similar, meaning that they have the same structure when you zoom in on them. For example, this is a famous fractal called the Mandelbrot set: Image by Arnaud Cheritat, retrieved from: https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/File:MilMand_1000iter.png 1 The main shape, which looks a bit like a snowman, is shaded in the image below: As an exercise, shade in some of the smaller snowmen that are all around the largest one in the image above. This is an example of self-similarity, and this pattern continues into inﬁnity. Watch what happens when you zoom deeper and deeper into the Mandelbrot Set here: https://www.youtube.com/watch?v=pCpLWbHVNhk. Drawing Fractals Fractals like the Mandelbrot Set are generated using formulas and computers, but there are a number of simple fractals that we can draw ourselves. 2 Sierpinski Triangle Image by Beojan Stanislaus, retrieved from: https://commons.wikimedia.org/wiki/File:Sierpinski_triangle.svg Use the template and the steps below to draw this fractal: 1. Start with an equilateral triangle. 2. Connect the midpoints of all three sides. This will create one upside down triangle and three right-side up triangles. 3. Repeat from step 1 for each of the three right-side up triangles. 3 Apollonian Gasket Retrieved from: https://mathlesstraveled.com/2016/04/20/post-without-words-6/ Use the template and the steps below to draw this fractal: 1.
[Show full text]
Mathematics in the Sea Sonia Kovalevsky Day Before
STUDENT CHAPTER CORNER actively helping to promote the UIC Kovalevsky days. Together with the help of the Public School District ofces, Coordinator: Emily Sergel, [email protected] this year the registration included students from about 20 schools, and 93% of the girls registered had never been to a Mathematics in the Sea Sonia Kovalevsky Day before. On the day itself, after an introduction to the AWM Laura P. Schaposnik and James Unwin, University of Illinois for the students and teachers, there was a brief presentation at Chicago of Sonia Kovalevsky’s achievements and the obstacles she overcame in her life. Te students were then separated into On May 4th 2019, with the help of several volunteers, groups for the activities of the day. Te theme for this ffth we organized the ffth Sonia Kovalevsky Day for girls at the Sonia Kovalevsky Day at UIC was Mathematics in the Sea, University of Illinois at Chicago (UIC), in cooperation with and the students were taught three diferent lessons. the AWM. As we highlighted in the preceding AWM Newsletter, Fractal Coastlines. In the mid-20th century Lewis Sonia Kovalevsky Days have been organized nationwide by Fry Richardson pointed out that the question “How long is AWM members for almost 30 years. Sonia Kovalevsky Days the coastline of Great Britain?” has no explicit answer and consist of a program of workshops, talks, and problem-solving can only be estimated. Indeed, a coastline typically has a competitions for female high school and middle school fractal dimension, thus making the notion of length students and their teachers, both women and men.
[Show full text]
On Musical Self-Similarity
On Musical Self-Similarity Acta Semiotica Fennica XXXIX Approaches to Musical Semiotics Series Acta Semiotica Fennica Editor: Eero Tarasti Associate Editors: Paul Forsell Richard Littlefield Editorial Board: Honorary Member: Thomas A. Sebeok † Pertti Ahonen Henri Broms Jacques Fontanille André Helbo Altti Kuusamo Ilkka Niiniluoto Pekka Pesonen Hannu Riikonen Kari Salosaari Vilmos Voigt Gabriel Pareyon On Musical Self-Similarity Intersemiosis as Synecdoche and Analogy Acta Semiotica Fennica XXXIX Approaches to Musical Semiotics 13 International Semiotics Institute at Imatra Semiotic Society of Finland 2011 This book is a publication of The International Semiotics Institute • Imatra (ISI) http://www.isisemiotics.fi [email protected] tel. +358 20 617 6639 tel. +358 20 617 6700 fax +358 20 617 6696 Cover design by Gabriel Pareyon Cover picture by Matías Vidal Layout by Paul Forsell and the author Copyright 2011 by Gabriel Pareyon All rights reserved Printed by Yliopistopaino, Helsinki (April, 2011) ISSN 1235-479X Acta Semiotica Fennica XXXIX ISSN 1458-492 Approaches to Musical Semiotics 13 ISBN 978-952-5431-32-2 (paperback) ISBN 978-952-10-6970-3 (PDF) Abstract Self-similarity, a concept taken from mathematics, is gradually becoming a keyword in musicology. Although a polysemic term, self-similarity often refers to the multi-scalar feature repetition in a set of relationships, and it is commonly valued as an indication for musical ‘coherence’ and ‘consistency’. This investigation provides a theory of musical meaning formation in the context of intersemiosis, that is, the translation of meaning from one cognitive domain to another cognitive domain (e.g. from mathematics to music, or to speech or graphic forms).
[Show full text]
12 an Introduction to Fractals
81 12 An Introduction to Fractals The aim of this lecture is to give you a brief introduction to fractals, sufficient to be able to answer the assignment questions3. You may answer the assignment questions working in pairs and providing on set of answers containing both of your names. In answering the assignment questions you will find it useful to explore various webpages devoted to fractals. You can find a list of starting places at http://www.ma.adfa.oz.au/~mark/teaching.dir/fractals.html To give you time to get acquainted with these, and associated, web sites instead of having a lecture on Friday we are providing you with a free period. We advise you to use this time wisely. By the end of this section you should: ² Be able to give a non-technical explanation of what a fractal is. ² Explain how fractals can be generated by simple iterative processes. ² Understand some of the applications of fractals. ² Appreciate that fractals can arise in simple models of physical processes. 12.1 The Mandelbrot Set and Self-Similarity One of the most intricate and beautiful images in mathematics is the Mandelbrot set, discovered by Benoit Mandelbrot in 1980. Mandelbrot’s discovery and popularisation of this image is responsible for the development of fractal science. Figure 16 (a) shows the Mandelbrot Set. I won’t describe what the definition of the Mandelbrot Set is, that’s assignment question 1! For our purposes it suffices to note that the Mandelbrot Set divides the plane into two regions: an ‘inner region’, the black region in the figure, and an ‘outer region’.
[Show full text]