Stat 45: Our Fractal World? Topics for Today. Lecture 1: Getting Started. David Donoho Statistics Department Stanford University. Sierpinski Gaskets

Size: px
Start display at page:

Download "Stat 45: Our Fractal World? Topics for Today. Lecture 1: Getting Started. David Donoho Statistics Department Stanford University. Sierpinski Gaskets"

Transcription

1 Stat 45N: Our Fractal World? Lecture 1 1 Stat 45N: Our Fractal World? Lecture 1 2 Stat 45: Our Fractal World? Topics for Today Lecture 1: Getting Started What Are They? David Donoho Statistics Department Stanford University Where Do They Occur? Why are they Controversial? Real Ideas behind them... Look Ahead Stat 45N: Our Fractal World? Lecture 1 3 Stat 45N: Our Fractal World? Lecture 1 4 Sierpinski Gaskets Examples of Fractals Sierpinski Gasket Menger Sponge Koch Curve (a) Two-D (b) Three-D

2 Stat 45N: Our Fractal World? Lecture 1 5 Stat 45N: Our Fractal World? Lecture 1 6 W. Sierpinski Menger Sponge Stat 45N: Our Fractal World? Lecture 1 7 Koch Curve Stat 45N: Our Fractal World? Lecture 1 8 Some Properties Multiscale Recursive Fractional Dimensional

3 Stat 45N: Our Fractal World? Lecture 1 9 Stat 45N: Our Fractal World? Lecture 1 10 G. Peano Where do they occcur? Math Analysis Heroes: Cantor, Peano, Koch, Menger, Cesaro Curves with no tangent Space-Filling Curves Chaos Theory Computer Graphics Stat 45N: Our Fractal World? Lecture 1 11 Stat 45N: Our Fractal World? Lecture 1 12

4 Stat 45N: Our Fractal World? Lecture 1 13 Stat 45N: Our Fractal World? Lecture 1 14 Where are they thought to occcur? How Long is the Coast of Britain Coastline of Britain Lungs, Arteries Galaxy Distributions DNA Physics, Chemistry Internet Art(?) (a) Polygon Approx (b) Blowing-Up Stat 45N: Our Fractal World? Lecture 1 15 L.F. Richardson Stat 45N: Our Fractal World? Lecture 1 16 Related Concepts Chaos Wavelets Multifractals

5 Stat 45N: Our Fractal World? Lecture 1 17 Stat 45N: Our Fractal World? Lecture 1 18 Benoit Mandelbrot Why are They Controversial? Late 19th Century Monsters in Analysis Perrin Quote (Handout) 1970 s Mandelbrot Nature vs. Euclid Fractals Everywhere but education obscures them Revolutions across all sciences (Kuhn, Paradigm Shift) Stat 45N: Our Fractal World? Lecture 1 19 New Yorker magazine, July 31, 2000 GORE WITHOUT A SCRIPT Stat 45N: Our Fractal World? Lecture If you look at a map of the coastline of New Jersey, and then magnify that a thousand times, the basic design of the ins and outs of the coastline will be the same at every level of magnification. And they call that the self-sameness principle. I don t understand it. It s way beyond my depth. But I do believe there s something about our world that He began another long pause. I m searching for the right word here that manifests that self-sameness principle in a lot of different ways. And when we find a brand-new understanding of the world that comes out of a powerful new discovery in science, it often allows us to look at social and political matters and find ways toconnect the dots that haven t made sense before. THE 2000 CAMPAIGN: A TEST OF CHARACTER From the NY Times June 21, 2000 At one point in a conversation about some the big think questions that intrigue him, he began to moan, I can t say this, it s going to sound so weird. He went on that way for a minute, as if he were about to admit to something really twisted, before he finally revealed the big secret: Oh, O.K.! I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world.

6 Stat 45N: Our Fractal World? Lecture 1 21 Stat 45N: Our Fractal World? Lecture 1 22 Real Ideas? What will we use? Fractional Dimension Multiscale Analysis Book: Fractals: Images of Chaos Lauwerier Reserve: Mandelbrot, Fractal Geometry of Nature Recursive Synthesis Edgar, Measure, Topology, and Fractal Geometry Multiscale Processes in Nature Schröder, Fractals, Chaos, and Power Laws Mathematical Modelling Barnsley, Fractals Everywhere Data onslaught Computer Resources Scientific Articles Stat 45N: Our Fractal World? Lecture 1 23 Stat 45N: Our Fractal World? Lecture 1 24 Course Structure First 4 weeks: elementary properties of fractals Next 4 weeks: applications of fractals in science Next 4 weeks: cognate areas (wavelets, refinement schemes, multifractals, etc.) Not a math course Course Philosophy Mixture of Lecture and Discussion each meeting Readings each week Stanford since 1990 I Knew Nash Awards, Honors About Me

Session 27: Fractals - Handout

Session 27: Fractals - Handout Session 27: Fractals - Handout 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 Mandelbrot (1924-2010)

More information

FRACTALS The term fractal was coined by mathematician Benoit Mandelbrot A fractal object, unlike a circle or any regular object, has complexity at all scales Natural Fractal Objects Natural fractals

More information

Fractal Coding. CS 6723 Image Processing Fall 2013

Fractal Coding. CS 6723 Image Processing Fall 2013 Fractal Coding CS 6723 Image Processing Fall 2013 Fractals and Image Processing The word Fractal less than 30 years by one of the history s most creative mathematician Benoit Mandelbrot Other contributors:

More information

Fractals in Nature and Mathematics: From Simplicity to Complexity

Fractals in Nature and Mathematics: From Simplicity to Complexity Fractals in Nature and Mathematics: From Simplicity to Complexity Dr. R. L. Herman, UNCW Mathematics & Physics Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 1/41 Outline

More information

Fractals. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.

Fractals. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna. Fractals Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ 2 Geometric Objects Man-made objects are geometrically simple (e.g., rectangles,

More information

Mathematics 350 Section 6.3 Introduction to Fractals

Mathematics 350 Section 6.3 Introduction to Fractals Mathematics 350 Section 6.3 Introduction to Fractals A fractal is generally "a rough or fragmented geometric shape that is self-similar, which means it can be split into parts, each of which is (at least

More information

Lecture 8: Modelling Urban Morphology:

Lecture 8: Modelling Urban Morphology: SCHOOL OF GEOGRAPHY Lecture 8: Modelling Urban Morphology: Fractal Geometry, Relations to CA, And Urban Form Outline What are Fractals? Definitions and Properties Scaling and Links to Fractal Patterns

More information

Scientific Calculation and Visualization

Scientific Calculation and Visualization Scientific Calculation and Visualization Topic Iteration Method for Fractal 2 Classical Electrodynamics Contents A First Look at Quantum Physics. Fractals.2 History of Fractal.3 Iteration Method for Fractal.4

More information

Fractal Geometry. LIACS Natural Computing Group Leiden University

Fractal Geometry. LIACS Natural Computing Group Leiden University Fractal Geometry Contents Introduction The Fractal Geometry of Nature Self-Similarity Some Pioneering Fractals Dimension and Fractal Dimension Cellular Automata Particle Systems Scope of Fractal Geometry

More information

Generation of 3D Fractal Images for Mandelbrot and Julia Sets

Generation of 3D Fractal Images for Mandelbrot and Julia Sets 178 Generation of 3D Fractal Images for Mandelbrot and Julia Sets Bulusu Rama #, Jibitesh Mishra * # Department of Computer Science and Engineering, MLR Institute of Technology Hyderabad, India 1 rama_bulusu@yahoo.com

More information

Fractal Geometry. Prof. Thomas Bäck Fractal Geometry 1. Natural Computing Group

Fractal Geometry. Prof. Thomas Bäck Fractal Geometry 1. Natural Computing Group Fractal Geometry Prof. Thomas Bäck Fractal Geometry 1 Contents Introduction The Fractal Geometry of Nature - Self-Similarity - Some Pioneering Fractals - Dimension and Fractal Dimension Scope of Fractal

More information

Fractals. Materials. Pencil Paper Grid made of triangles

Fractals. Materials. Pencil Paper Grid made of triangles Fractals Overview: Fractals are new on the mathematics scene, however they are in your life every day. Cell phones use fractal antennas, doctors study fractal-based blood flow diagrams to search for cancerous

More information

Fractal Dimension and the Cantor Set

Fractal Dimension and the Cantor Set Fractal Dimension and the Cantor Set Shailesh A Shirali Shailesh Shirali is Director of Sahyadri School (KFI), Pune, and also Head of the Community Mathematics Centre in Rishi Valley School (AP). He has

More information

Fun with Fractals and Functions. CHAMP at University of Houston March 2, 2015 Houston, Texas

Fun with Fractals and Functions. CHAMP at University of Houston March 2, 2015 Houston, Texas Fun with Fractals and Functions CHAMP at University of Houston March 2, 2015 Houston, Texas Alice Fisher afisher@rice.edu Director of Technology Applications & Integration at Rice University School Mathematics

More information

Mathematics Numbers: Percentages. Science and Mathematics Education Research Group

Mathematics Numbers: Percentages. Science and Mathematics Education Research Group F FA ACULTY C U L T Y OF O F EDUCATION E D U C A T I O N Department of Curriculum and Pedagogy Mathematics Numbers: Percentages Science and Mathematics Education Research Group Supported by UBC Teaching

More information

Exploring the Effect of Direction on Vector-Based Fractals

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

More information

Fractals Week 10, Lecture 19

Fractals Week 10, Lecture 19 CS 430/536 Computer Graphics I Fractals Week 0, Lecture 9 David Breen, William Regli and Maim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Dreel University http://gicl.cs.dreel.edu

More information

A TECHNOLOGY-ENHANCED FRACTAL/CHAOS COURSE. Taeil Yi University of Texas at Brownsville 80 Fort Brown Brownsville, TX

A TECHNOLOGY-ENHANCED FRACTAL/CHAOS COURSE. Taeil Yi University of Texas at Brownsville 80 Fort Brown Brownsville, TX A TECHNOLOGY-ENHANCED FRACTAL/CHAOS COURSE Taeil Yi University of Texas at Brownsville 80 Fort Brown Brownsville, TX 78520 tyi@utb.edu Abstract Easy construction of fractal figures is the most valuable

More information

BarnCamp The HTML 5 <canvas> element as seen through a fractal eye: a brief introduction

BarnCamp The HTML 5 <canvas> element as seen through a fractal eye: a brief introduction BarnCamp 2011 The HTML 5 element as seen through a fractal eye: a brief introduction Why? I like Fractals, they're pretty to look at and it amazes me how such beauty can come from a simple little

More information

Solid models and fractals

Solid models and fractals Solid models and fractals COM3404 Richard Everson School of Engineering, Computer Science and Mathematics University of Exeter R.M.Everson@exeter.ac.uk http://www.secamlocal.ex.ac.uk/studyres/com304 Richard

More information

CGT 581 G Procedural Methods Fractals

CGT 581 G Procedural Methods Fractals CGT 581 G Procedural Methods Fractals Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics Technology Procedural Techniques Model is generated by a piece of code. Model is not represented

More information

Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013

Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013 Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013 Background Fractal geometry is one of the most important developments in mathematics in the second half of the 20th century. Fractals

More information

Graphics in IT82. Representing Graphical Data. Graphics in IT82. Lectures Overview. Representing Graphical Data. Logical / Physical Representation

Graphics in IT82. Representing Graphical Data. Graphics in IT82. Lectures Overview. Representing Graphical Data. Logical / Physical Representation Graphics in IT82 What does computer graphics cover? Representing Graphical Data Chapman & Chapman, chapters 3,4,5 Richardson IT82 Input, output, and representation of graphical data Creation of graphics

More information

Hei nz-ottopeitgen. Hartmut Jürgens Dietmar Sau pe. Chaos and Fractals. New Frontiers of Science

Hei nz-ottopeitgen. Hartmut Jürgens Dietmar Sau pe. Chaos and Fractals. New Frontiers of Science Hei nz-ottopeitgen Hartmut Jürgens Dietmar Sau pe Chaos and Fractals New Frontiers of Science Preface Authors VU X I Foreword 1 Mitchell J. Feigenbaum Introduction: Causality Principle, Deterministic

More information

Fractals and Multi-Layer Coloring Algorithms

Fractals and Multi-Layer Coloring Algorithms Fractals and Multi-Layer Coloring Algorithms Javier Barrallo and Santiago Sanchez Mathematics, Physics and Computer Science The University of the Basque Country School of Architecture. Plaza Onati, 2.

More information

ARi. Amalgamated Research Inc. What are fractals?

ARi. Amalgamated Research Inc. What are fractals? ARi www.arifractal.com What are fractals? Amalgamated Research Inc. A key characteristic of fractals is self-similarity. This means that similar structure is observed at many scales. Figure 1 illustrates

More information

Discrete Dynamical Systems: A Pathway for Students to Become Enchanted with Mathematics

Discrete Dynamical Systems: A Pathway for Students to Become Enchanted with Mathematics Discrete Dynamical Systems: A Pathway for Students to Become Enchanted with Mathematics Robert L. Devaney, Professor Department of Mathematics Boston University Boston, MA 02215 USA bob@bu.edu Abstract.

More information

Graph Fractals. An Honors Thesis (Honrs 499) by Ben J. Kelly. Thesis Advisor. Ball State University Muncie, Indiana. May 1995

Graph Fractals. An Honors Thesis (Honrs 499) by Ben J. Kelly. Thesis Advisor. Ball State University Muncie, Indiana. May 1995 Graph Fractals An Honors Thesis (Honrs 499) by Ben J. Kelly Thesis Advisor Ball State University Muncie, Indiana May 1995 Expected Date Of Graduation: May 6, 1995 ~, 5fCol! rj,e5;s ~ 7) 2 J+B(). L~ if

More information

Fun with Fractals Saturday Morning Math Group

Fun with Fractals Saturday Morning Math Group Fun with Fractals Saturday Morning Math Group Alistair Windsor Fractals Fractals are amazingly complicated patterns often produced by very simple processes. We will look at two different types of fractals

More information

FRACTAL: A SET WHICH IS LARGER THAN THE UNIVERSE

FRACTAL: A SET WHICH IS LARGER THAN THE UNIVERSE ISSN 2320-9143 40 International Journal of Advance Research, IJOAR.org Volume 1, Issue 3, March 2013, Online: ISSN 2320-9143 FRACTAL: A SET WHICH IS LARGER THAN THE UNIVERSE Soumya Prakash Sahu, Indian

More information

Fractal Analysis. By: Mahnaz EtehadTavakol

Fractal Analysis. By: Mahnaz EtehadTavakol Fractal Analysis By: Mahnaz EtehadTavakol A fractal a non-regular geometric shape can be split into parts which posses self similarity Naturally Occurring Fractal A special type of broccoli, this cruciferous

More information

Computer Graphics (CS 543) Lecture 2c: Fractals. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics (CS 543) Lecture 2c: Fractals. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI) Computer Graphics (CS 543 Lecture c: Fractals Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI What are Fractals? Mathematical expressions to generate pretty pictures Evaluate

More information

Section 9.5. Tessellations. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 9.5. Tessellations. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 9.5 Tessellations What You Will Learn Tessellations 9.5-2 Tessellations A tessellation (or tiling) is a pattern consisting of the repeated use of the same geometric figures to entirely cover a

More information

Clouds, biological growth, and coastlines are

Clouds, biological growth, and coastlines are L A B 11 KOCH SNOWFLAKE Fractals Clouds, biological growth, and coastlines are examples of real-life phenomena that seem too complex to be described using typical mathematical functions or relationships.

More information

Iterated Functions Systems and Fractal Coding

Iterated Functions Systems and Fractal Coding Qing Jun He 90121047 Math 308 Essay Iterated Functions Systems and Fractal Coding 1. Introduction Fractal coding techniques are based on the theory of Iterated Function Systems (IFS) founded by Hutchinson

More information

Computer Graphics 4731 Lecture 5: Fractals. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics 4731 Lecture 5: Fractals. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI) Computer Graphics 4731 Lecture 5: Fractals Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI What are Fractals? Mathematical expressions to generate pretty pictures Evaluate

More information

Lecture 3: Some Strange Properties of Fractal Curves

Lecture 3: Some Strange Properties of Fractal Curves Lecture 3: Some Strange Properties of Fractal Curves I have been a stranger in a strange land. Exodus 2:22 1. Fractal Strangeness Fractals have a look and feel that is very different from ordinary curves.

More information

This paper provides an introduction to the Apollonian fractal, also known by some as

This paper provides an introduction to the Apollonian fractal, also known by some as An Introduction to the Apollonian Fractal Paul Bourke Email: pdb@swin.edu.au Swinburne University of Technology P. O. Box 218, Hawthorn Melbourne, Vic 3122, Australia. Abstract This paper provides an introduction

More information

Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND

Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 2 WHAT YOU WILL LEARN Transformational geometry,

More information

Images of some fractals

Images of some fractals Fun with Fractals Dr. Bori Mazzag Redwood Empire Mathematics Tournament March 25, 2006 Images of some fractals What are fractals, anyway? Important aspects of fractals: Self-similarity What are fractals,

More information

Fixed Point Iterative Techniques An Application to Fractals

Fixed Point Iterative Techniques An Application to Fractals Fixed Point Iterative Techniques An Application to Fractals Narayan Partap 1 and Prof. Renu Chugh 2 1 Amity Institute of Applied Sciences, Amity University, Noida, India 2 Department of Mathematics, M.D.

More information

Representing Graphical Data

Representing Graphical Data Representing Graphical Data Chapman & Chapman, chapters 3,4,5 Richardson 1 Graphics in IT82 What does computer graphics cover? IT82 Input, output, and representation of graphical data Creation of graphics

More information

MITOCW 2. IV: Consistency, Completeness, and Geometry

MITOCW 2. IV: Consistency, Completeness, and Geometry MITOCW 2. IV: Consistency, Completeness, and Geometry The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational

More information

Chapel Hill Math Circle: Symmetry and Fractals

Chapel Hill Math Circle: Symmetry and Fractals Chapel Hill Math Circle: Symmetry and Fractals 10/7/17 1 Introduction This worksheet will explore symmetry. To mathematicians, a symmetry of an object is, roughly speaking, a transformation that does not

More information

Module 6. Campaign Layering

Module 6.  Campaign Layering Module 6 Email Campaign Layering Slide 1 Hello everyone, it is Andy Mackow and in today s training, I am going to teach you a deeper level of writing your email campaign. I and I am calling this Email

More information

Fractals. University. 1 The approach and terminology used here are from Michael Frame s Fractal course taught for many years at Yale

Fractals. University. 1 The approach and terminology used here are from Michael Frame s Fractal course taught for many years at Yale Fractals The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained

More information

Fractals: a way to represent natural objects

Fractals: a way to represent natural objects Fractals: a way to represent natural objects In spatial information systems there are two kinds of entity to model: natural earth features like terrain and coastlines; human-made objects like buildings

More information

<The von Koch Snowflake Investigation> properties of fractals is self-similarity. It means that we can magnify them many times and after every

<The von Koch Snowflake Investigation> properties of fractals is self-similarity. It means that we can magnify them many times and after every Jiwon MYP 5 Math Ewa Puzanowska 18th of Oct 2012 About Fractal... In geometry, a fractal is a shape made up of parts that are the same shape as itself and are of

More information

Complexity is around us. Part one: the chaos game

Complexity is around us. Part one: the chaos game Complexity is around us. Part one: the chaos game Dawid Lubiszewski Complex phenomena like structures or processes are intriguing scientists around the world. There are many reasons why complexity is a

More information

MAADHYAM. Nurturing Gifted Minds. Printed Under Gifted Education Abhiyaan An Initiative By The Office Of Principal Scientific Advisor To The

MAADHYAM. Nurturing Gifted Minds. Printed Under Gifted Education Abhiyaan An Initiative By The Office Of Principal Scientific Advisor To The MAADHYAM Nurturing Gifted Minds Printed Under Gifted Education Abhiyaan An Initiative By The Office Of Principal Scientific Advisor To The 1 Government Of India INTRODUCTION TO FRACTALS When you see a

More information

Outline. Solid models and fractals. Constructive solid geometry. Constructive solid geometry COM3404. Richard Everson

Outline. Solid models and fractals. Constructive solid geometry. Constructive solid geometry COM3404. Richard Everson Outline Solid models and fractals COM School of Engineering, Computer Science and Mathematics University of Exeter Constructive solid geometry Fractals Dimension s Landscape generation L-systems R.M.Everson@exeter.ac.uk

More information

ITERATIVE OPERATIONS IN CONSTRUCTION CIRCULAR AND SQUARE FRACTAL CARPETS

ITERATIVE OPERATIONS IN CONSTRUCTION CIRCULAR AND SQUARE FRACTAL CARPETS ITERATIVE OPERATIONS IN CONSTRUCTION CIRCULAR AND SQUARE FRACTAL CARPETS Dr. Yusra Faisal Al-Irhaim, Marah Mohamed Taha University of Mosul, Iraq ABSTRACT: Carpet designing is not only a fascinating activity

More information

An Introduction to Fractals

An Introduction to Fractals An Introduction to Fractals Sarah Hardy December 10, 2018 Abstract Fractals can be defined as an infinitely complex pattern that is self-similar, that is contains replicas of itself of varying sizes, across

More information

Website.

Website. Admin stuff Questionnaire Name Email Math courses taken so far General academic trend (major) General interests What about Chaos interests you the most? What computing experience do you have? Website www.cse.ucsc.edu/classes/ams146/spring05/index.html

More information

A Review of Fractals Properties: Mathematical Approach

A Review of Fractals Properties: Mathematical Approach Science Journal of Applied Mathematics and Statistics 2017; 5(3): 98-105 http://www.sciencepublishinggroup.com/j/sjams doi: 10.11648/j.sjams.20170503.11 ISSN: 2376-9491 (Print); ISSN: 2376-9513 (Online)

More information

The UCSD Freshman Seminar on The Slide Rule

The UCSD Freshman Seminar on The Slide Rule The UCSD Freshman Seminar on The Slide Rule Joe Pasquale Department of Computer Science and Engineering University of California, San Diego Background Started seminar in 2003 For UCSD freshman 10 weeks,

More information

Introduction to fractal geometry: Definition, concept, and applications

Introduction to fractal geometry: Definition, concept, and applications University of Northern Iowa UNI ScholarWorks Presidential Scholars Theses (1990 2006) University Honors Program 1992 Introduction to fractal geometry: Definition, concept, and applications Mary Bond University

More information

lecture 9 Object hierarchies - call trees and GL_MODELVIEW stack - fractals - L systems

lecture 9 Object hierarchies - call trees and GL_MODELVIEW stack - fractals - L systems lecture 9 Object hierarchies - call trees and GL_MODELVIEW stack - fractals - L systems Last lecture: - hierarchy of bounding volumes of objects and scenes - spatial partition represented as a tree (BSP

More information

LET S GET STARTED WITH NUMBERS. Xin Li Math Circle, Spring 2018 University of Central Florida

LET S GET STARTED WITH NUMBERS. Xin Li Math Circle, Spring 2018 University of Central Florida LET S GET STARTED WITH NUMBERS Xin Li Math Circle, Spring 2018 University of Central Florida WHAT IS A NUMBER? Give me an example of a number Give me an example of a natural number Give me an example of

More information

CSC 015: FUNDAMENTALS OF COMPUTER SCIENCE I

CSC 015: FUNDAMENTALS OF COMPUTER SCIENCE I CSC 015: FUNDAMENTALS OF COMPUTER SCIENCE I Lecture 1: Class Introduction DR. BO TANG ASSISTANT PROFESSOR HOFSTRA UNIVERSITY 1 9/7/16 CSC15 - Python OUTLINE What is Computer Science? What is this Class

More information

Beautiful Repetitions

Beautiful Repetitions Beautiful Repetitions 5-minute introduction to Iterations & Fractals Gaurish Korpal (gaurish4math.wordpress.com) National Institute of Science Education and Research, Bhubaneswar March 28, 2015 Gaurish

More information

AN ALGORITHM TO GENERATE MODELS OF SNOWFLAKES

AN ALGORITHM TO GENERATE MODELS OF SNOWFLAKES AN ALGORITHM TO GENERATE MODELS OF SNOWFLAKES PHILIP CHUNG, COLIN BLOOMFIELD Abstract. In this paper we will describe our method of creating a computer algorithm to generate two-dimensional representations

More information

Women s Worlds 2011 Ottawa, CA

Women s Worlds 2011 Ottawa, CA Copyright Barbara Alice Mann, 2011. Women s Worlds 2011 Ottawa, CA The Fractal Binaries of the Gift 7 July 2011 by Barbara Alice Mann The base number of Western culture is ONE: One god, one life, one soul.

More information

COASTLINES AND FRACTAL GEOMETRY: ESTIMATING LENGTH AND GENERATING ISLANDS. Miranda Bradshaw Dallas Pullen Math 365 Wright 5/8/12

COASTLINES AND FRACTAL GEOMETRY: ESTIMATING LENGTH AND GENERATING ISLANDS. Miranda Bradshaw Dallas Pullen Math 365 Wright 5/8/12 COASTLINES AND FRACTAL GEOMETRY: ESTIMATING LENGTH AND GENERATING ISLANDS Miranda Bradshaw Dallas Pullen Math 365 Wright 5/8/12 Introduction The first connections that were made between coastlines and

More information

Chapter 1 Introduction

Chapter 1 Introduction Page 1 Chapter 1 Introduction 1.1 Introduction The twin subjects of fractal geometry and chaotic dynamics have been behind an enormous change in the way scientists and engineers perceive, and subsequently

More information

Survey of the Mathematics of Big Data

Survey of the Mathematics of Big Data Survey of the Mathematics of Big Data Issues with Big Data, Mathematics to the Rescue Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Math & Big Data Fall 2015 1 / 28 Introduction We survey some

More information

RAMSEY NUMBERS IN SIERPINSKI TRIANGLE. Vels University, Pallavaram Chennai , Tamil Nadu, INDIA

RAMSEY NUMBERS IN SIERPINSKI TRIANGLE. Vels University, Pallavaram Chennai , Tamil Nadu, INDIA International Journal of Pure and Applied Mathematics Volume 6 No. 4 207, 967-975 ISSN: 3-8080 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: 0.2732/ijpam.v6i4.3 PAijpam.eu

More information

COMPUTER ANALYSIS OF FRACTAL SETS

COMPUTER ANALYSIS OF FRACTAL SETS Proceedings of the Czech Japanese Seminar in Applied Mathematics 2006 Czech Technical University in Prague, September 14-17, 2006 pp. 1 8 COMPUTER ANALYSIS OF FRACTAL SETS PETR PAUŠ1 Abstract. This article

More information

Given four lines in space, how many lines meet all four?: The geometry, topology, and combinatorics of the Grassmannian

Given four lines in space, how many lines meet all four?: The geometry, topology, and combinatorics of the Grassmannian Atlanta January 5, 2005 Given four lines in space, how many lines meet all four?: The geometry, topology, and combinatorics of the Grassmannian Ravi Vakil, Stanford University http://math.stanford.edu/

More information

Usability Test Report: Bento results interface 1

Usability Test Report: Bento results interface 1 Usability Test Report: Bento results interface 1 Summary Emily Daly and Ian Sloat conducted usability testing on the functionality of the Bento results interface. The test was conducted at the temporary

More information

New Mandelbrot and Julia Sets for Transcendental Function

New Mandelbrot and Julia Sets for Transcendental Function New Mandelbrot and Julia Sets for Transcendental Function Suraj Singh Panwar #1, Mr.Pawan Kumar Mishra *2 # M.Tech. - CSE, Scholar, Faculty of Technology, Computer Science and Engineering Department, Uttarakhand

More information

pagina 1 van 5 Location: Food for Thought > The Particle > Unification into a fractal dimension Blaze Labs Research Menu Home Food for Thought EHD Thrusters New Energy Research Experiments Links Contact

More information

Word processing and spreadsheet applications are among the most

Word processing and spreadsheet applications are among the most In This Chapter Chapter 1 Starting Out with iwork 09 Leaving the past behind The iwork timesavers: Do it once, do it right, and reuse it Word processing and spreadsheet applications are among the most

More information

Making sense of chaos An evaluation of the current state of information architecture for the Web

Making sense of chaos An evaluation of the current state of information architecture for the Web Making sense of chaos An evaluation of the current state of information architecture for the Web Anne de Ridder UW 521 Winter Seminar Series, February 3, 2012 What you ll hear about today A bit about me

More information

New Escape Time Koch Curve in Complex Plane

New Escape Time Koch Curve in Complex Plane New Escape Time Koch Curve in Complex Plane Priti Dimri Associate Professor, Department of Computer Science and Engineering G.B Pant Engineering College Pauri Garhwal, 246001 Dharmendra Kumar Associate

More information

Uttarkhand Technical University, J.B.Institute of Technology, Uttarakhand Technical University, Dehradun, INDIA Dehradun, INDIA Dehradun, INDIA

Uttarkhand Technical University, J.B.Institute of Technology, Uttarakhand Technical University, Dehradun, INDIA Dehradun, INDIA Dehradun, INDIA Volume 3, Issue 12, December 2013 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Analysis of

More information

Fractal Data Modeling

Fractal Data Modeling Fractal Data Modeling Fractal geometry creates beautiful patterns from simple recursive algorithms. One of the things we find so appealing is their self- similarity at different scales. That is, as you

More information

4.3 Discovering Fractal Geometry in CAAD

4.3 Discovering Fractal Geometry in CAAD 4.3 Discovering Fractal Geometry in CAAD Francisco Garcia, Angel Fernandez*, Javier Barrallo* Facultad de Informatica. Universidad de Deusto Bilbao. SPAIN E.T.S. de Arquitectura. Universidad del Pais Vasco.

More information

A PRACTICAL GUIDE TO SHAREPOINT 2013: NO FLUFF! JUST PRACTICAL EXERCISES TO ENHANCE YOUR SHAREPOINT 2013 LEARNING! BY SAIFULLAH SHAFIQ

A PRACTICAL GUIDE TO SHAREPOINT 2013: NO FLUFF! JUST PRACTICAL EXERCISES TO ENHANCE YOUR SHAREPOINT 2013 LEARNING! BY SAIFULLAH SHAFIQ A PRACTICAL GUIDE TO SHAREPOINT 2013: NO FLUFF! JUST PRACTICAL EXERCISES TO ENHANCE YOUR SHAREPOINT 2013 LEARNING! BY SAIFULLAH SHAFIQ DOWNLOAD EBOOK : A PRACTICAL GUIDE TO SHAREPOINT 2013: NO FLUFF! SHAREPOINT

More information

The Menger Sponge in Google SketchUp

The Menger Sponge in Google SketchUp The Sierpinsky Carpet (shown below on the left) is a 2D fractal made from squares repeatedly divided into nine smaller squares. The Menger Sponge (shown below on the right) is the 3D version of this fractal.

More information

Fractals in Nature. Ivan Sudakov. Mathematics Undergraduate Colloquium University of Utah 09/03/2014.

Fractals in Nature. Ivan Sudakov. Mathematics Undergraduate Colloquium University of Utah 09/03/2014. Chesapeake Bay Mathematics Undergraduate Colloquium University of Utah 09/03/2014 Fractals in Nature Ivan Sudakov www.math.utah.edu/~sudakov www.mathclimate.org References 1. W. Szemplinska-Stupnicka.

More information

A Selection of Interesting Sets

A Selection of Interesting Sets A Selection of January 25, 2010 A Selection of Overview Overview Set, Oh joyous sets Overview Set, Oh joyous sets Some large, some small Overview Set, Oh joyous sets Some large, some small All of them

More information

List Building Starter Course. Lesson 2. Writing Your Campaign. Sean Mize

List Building Starter Course. Lesson 2. Writing Your  Campaign. Sean Mize List Building Starter Course Lesson 2 Writing Your Email Campaign 1 List Building Starter Course Lesson 2 Writing Your Email Campaign Mize List Building Starter Course Lesson 2 Writing Your Email Campaign

More information

Binary, Hexadecimal and Octal number system

Binary, Hexadecimal and Octal number system Binary, Hexadecimal and Octal number system Binary, hexadecimal, and octal refer to different number systems. The one that we typically use is called decimal. These number systems refer to the number of

More information

Managing Your Grade Book This lesson will show you how to set up your grade book columns and have Canvas calculate your final grades for you.

Managing Your Grade Book This lesson will show you how to set up your grade book columns and have Canvas calculate your final grades for you. Managing Your Grade Book This lesson will show you how to set up your grade book columns and have Canvas calculate your final grades for you. Activating the Grade Book Click on Settings at the bottom of

More information

Dr. Chuck Cartledge. 14 Jan. 2015

Dr. Chuck Cartledge. 14 Jan. 2015 CS-495/595 Big Data Lecture #1 Dr. Chuck Cartledge 14 Jan. 2015 1/37 Table of contents 1 Logistics Class mechanics What the class will cover? Assignments and projects Class participation Office hours Demographics

More information

Symmetric Fractals. Seeking Sangaku Ramanujan, Hardy, and Ono

Symmetric Fractals. Seeking Sangaku Ramanujan, Hardy, and Ono Symmetric Fractals Seeking Sangaku Ramanujan, Hardy, and Ono Published by the Mathematical Association of America : : November 2016 Figure 1. Clockwise from far left, the Sierpinski triangle, the Koch

More information

Exploring Fractals through Geometry and Algebra. Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss

Exploring Fractals through Geometry and Algebra. Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss Exploring Fractals through Geometry and Algebra Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss Learning Objective and skills practiced Students will: Learn the three criteria

More information

Self-Similar Snowflakes with Optional Fractal Extension

Self-Similar Snowflakes with Optional Fractal Extension Self-Similar Snowflakes with Optional Fractal Extension Elizabeth Untiedt Mathematics OBJECTIVE, BACKGROUND INFORMATION, & REFERENCES Standards Met: Algebra: Represent, describe, and analyze patterns and

More information

NEW GOSPER SPACE FILLING CURVES

NEW GOSPER SPACE FILLING CURVES NEW GOSPER SPACE FILLING CURVES HIROSHI FUKUDA MICHIO SHIMIZU GISAKU NAKAMURA School of Administration and Informatics, Department of Liberal Art, Research Institute of Education, University of Shizuoka,

More information

Subject: Top-Paying IT Certificates for 2015 (And Our New Courses)

Subject: Top-Paying IT Certificates for 2015 (And Our New Courses) ITProTV Emails What You Missed Email #1 Subject: Top-Paying IT Certificates for 2015 (And Our New Courses) If you re like me you re already thinking about your 2015 goals. So I thought I d share a few

More information

CS 543: Computer Graphics Lecture 3 (Part I): Fractals. Emmanuel Agu

CS 543: Computer Graphics Lecture 3 (Part I): Fractals. Emmanuel Agu CS 543: Computer Graphics Lecture 3 (Part I: Fractals Emmanuel Agu What are Fractals? Mathematical expressions Approach infinity in organized way Utilizes recursion on computers Popularized by Benoit Mandelbrot

More information

CS 4300 Computer Graphics. Prof. Harriet Fell Fall 2012 Lecture 28 November 8, 2012

CS 4300 Computer Graphics. Prof. Harriet Fell Fall 2012 Lecture 28 November 8, 2012 CS 4300 Computer Graphics Prof. Harriet Fell Fall 2012 Lecture 28 November 8, 2012 1 Today s Topics Fractals Mandelbrot Set Julia Sets L-Systems 2 Fractals The term fractal was coined in 1975 by Benoît

More information

Fractal Geometry: History and Theory. Classical Euclidean geometry cannot accurately represent the natural world; fractal geometry is

Fractal Geometry: History and Theory. Classical Euclidean geometry cannot accurately represent the natural world; fractal geometry is Geri Smith Smith 1 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

More information

Viewports. Peter-Paul Koch CSS Day, 4 June 2014

Viewports. Peter-Paul Koch   CSS Day, 4 June 2014 Viewports Peter-Paul Koch http://quirksmode.org http://twitter.com/ppk CSS Day, 4 June 2014 or: Why responsive design works Peter-Paul Koch http://quirksmode.org http://twitter.com/ppk CSS Day, 4 June

More information

Donald Knuth and Software Patents

Donald Knuth and Software Patents Donald Knuth and Software Patents http://swpat.ffii.org/players/knuth/index.en.html Workgroup\\swpatag@ffii.org 2003-12-15 Donald Knuth, pioneer and cult figure of informatics (computer science), author

More information

EECS 282 Information Systems Design and Programming. Atul Prakash Professor, Computer Science and Engineering University of Michigan

EECS 282 Information Systems Design and Programming. Atul Prakash Professor, Computer Science and Engineering University of Michigan EECS 282 Information Systems Design and Programming Atul Prakash Professor, Computer Science and Engineering University of Michigan 1 What is the Course About? A second programming course - but different

More information

1/16. Emergence in Artificial Life. Sebastian Marius Kirsch Back Close

1/16. Emergence in Artificial Life. Sebastian Marius Kirsch Back Close 1/16 Emergence in Artificial Life Sebastian Marius Kirsch skirsch@moebius.inka.de 2/16 Artificial Life not life as it is, but life as it could be very recent field of science first a-life conference in

More information

Algorithms in Systems Engineering IE170. Lecture 2. Dr. Ted Ralphs

Algorithms in Systems Engineering IE170. Lecture 2. Dr. Ted Ralphs Algorithms in Systems Engineering IE170 Lecture 2 Dr. Ted Ralphs IE170 Lecture 2 1 References for Today s Lecture Required reading CLRS Chapter 1 References D.E. Knuth, The Art of Computer Programming,

More information

EECS 282 Information Systems Design and Programming. Atul Prakash Professor, Computer Science and Engineering University of Michigan

EECS 282 Information Systems Design and Programming. Atul Prakash Professor, Computer Science and Engineering University of Michigan EECS 282 Information Systems Design and Programming Atul Prakash Professor, Computer Science and Engineering University of Michigan 1 What is the Course About? A second programming course - but different

More information