FRACTAL GEOMETRY. Introduction to Fractal Geometry
|
|
- Juniper McCarthy
- 7 years ago
- Views:
Transcription
1 Introduction to Fractal Geometry FRACTAL GEOMETRY Fractal geometry is based on the idea of self-similar forms. To be selfsimilar, a shape must be able to be divided into parts that are smaller copies which are more or less similar to the whole. Because of the smaller similar divisions of fractals, they appear similar at all magnifications. However, while all fractals are self-similar, not all self-similar forms are fractals. (For example, a straight Euclidean line and a tessellation are self-similar, but are not fractals because they do not appear similar at all magnifications). Many times, fractals are defined by recursive formulas. Fractals often have a finite boundary that determines the area that it can take up, but the perimeter of the fractal continuously grows and is infinite. The Cantor Set, introduced by George Cantor, a German mathematician, in 1883, is one of the easiest ways to see the divisions similar to the whole after being magnified. The Cantor Set is a series of line segments in which the middle third is removed. After the middle third is removed, the first third and the last third remain. Of each of those segments, the middle third is removed leaving the first and last thirds. This goes on forever, removing the middle third of each new segment that is created. One of the most famous fractals is the Mandelbrot Set. French mathematician, Benoit Mandelbrot, began to study self-similarity in the 1960 s, and by 1980 was interested in graphing complex numbers. He used the recursive formula z z^2+c, where c is some real number and z is a complex number such as a+bi. Depending on the number put in, Mandelbrot discovered that some get larger and go off to infinity, while some get smaller and closer to zero. Mandelbrot then set the computer up to color the pixels for each number, or point on the complex plane. If the number got smaller and closer to zero, the computer colored it black. If it got larger it would get a different color. The colors depended on how quickly the number approached infinity. The picture he got turned out to be the most famous examples of fractal geometry. Fractal Geometry in Nature The basic definition of a fractal is something with a shape that gets smaller and repeats infinitely (i.e. if you were to zoom in on any one area of the object, you would be looking at the original picture). Fractals can be found in many forms. In nature and in the environment, approximate fractals are found everywhere, including some of the vegetables we eat. Below are some examples of fractals found in nature.
2 This website shows an animated version of how fractal geometry can be found in mountains and mountain ranges. The quartz stone is one example of a fractal image in nature. It has triangular points and would look similar if you were to zoom in on any part of the stone. The fern is a very good example of fractal geometry. It is very easy to demonstrate the zooming in idea.
3 This photo shows how a sea anemone and how even sea creatures can show fractal geometry. TF-8%26ni%3D20%26fr%3Dyfp-t-501- s%26b%3d181&w=500&h=400&imgurl=static.flickr.com%2f226%2f _0cfb8139e4_m.jp g&rurl=http%3a%2f%2fwww.flickr.com%2fphotos%2fseractal%2f %2f&size=91.6kb &name= _0cfb8139e4.jpg&p=fractal&type=jpeg&no=196&tt=357,067&oid=6de15e5d268 5f0c8&fusr=Seractal&tit=Fractal+anemone&hurl=http%3A%2F%2Fwww.flickr.com%2Fphotos%2Fs eractal%2f&ei=utf-8&src=p This picture shows a cross between broccoli and cauliflower that is a very good example of fractal geometry.
4 Trees also demonstrate the concept of fractals. Tree branches usually grow and split and then grow and split and continue the pattern. Therefore, at any point, you can take a picture and zoom in and you will be looking at a very similar picture. **http%3a// Queen Anne s Lace is also a very good example of fractal geometry. Each of the petals look like a smaller version of the flower.
5 History of Fractal Geometry It is necessary to include some essential information about the history of fractal geometry, considering this is a history of mathematics course. Fractal geometry is very new area of mathematics. It allows us to show shapes and structures using formulas. Fractal geometry actually began in the 17 th century with philosopher Leibniz contemplating self-similarity. About a century later, Karl Weierstrass showed a function that had the property of being non-intuitive. This means it was continuous everywhere, but not differentiable. Today his graph would be considered fractal. Another very famous person in the world of fractal geometry is that of Helge Von Koch, who presented a more geometric definition. Following is the Koch snowflake: Also worth mentioning is the work of Waclaw Sierpinski, who in 1915 constructed what is now know as the Sierpinski Triangle and Sierpinski Carpet, both of which are pictured below. Both are 2D, even though once considered curves.
6 Sierpinski Triangle Sierpinski Carpet Furthermore, one individual, Paul Piere Levy, investigated self-similar curves even further and came up with what is today known as the Levy C curve. A picture can be viewed below or an animation version can be accessed at: Many others explored iterated functions in the complex plane. However, until computer graphics, they were unable to visualize the many objects they discovered. Levy C Curve Most notable in fractal geometry is Benoit B. Mandelbrot, mentioned earlier. He is responsible for the name, Fractal Geometry. He also discovered the Mandelbrot set in 1980 after working with Gaston Julia s theorems. These theorems were published in 1917 and now that we have super computers to
7 make the millions of calculation necessary, his theorems could be tested. In conclusion, fractal geometry is just being to take off and already we have found practical application, such as reducing file size of images and greatly enhancing their resolution. There is no doubt that the fractal geometry will continue to become more important in mathematics, science, and technology. Hands-On Fractal Demonstration Compact Disc Mirror Class Activity: Koch Snowflake Materials: straight edge, pencils Procedure: Make sure each student has a straight edge, pencil, and paper. Have them create an equilateral triangle. Then divide each side of the triangle in half and make a dot. Connect the dots to create a triangle within. Continue this process as many times as possible. Below are some pictures to help better understand the process:
8 WORKS CITED Scratching the Surface: What are Fractals and Fractal Geometry. Growth Factors. ml Sierpinski Carpet. Fractal. Moles Fractal: levy c curve. Koch Snowflake. tion.html Point Symmetry: 7.5 mm Square Paper. Cantor Set. Wikipedia. 19 November Wikimedia Foundation, Inc. 3 December < The Math of Fractals. Cool Math CoolMath.com, Inc. 3 December < Fractal. Wikipedia. 2 December Wikimedia Foundation, Inc. 3 December < DlimitR. Fractals Mandelbrot. YouTube. 17 June YouTube, LLC. 3 December < Lanius, Cynthia. Sierpinski s Triangle December < The Sierpinski Triangle. Animated Mountain Quartz Crystal Photo Fern Photo Sea Anemone Photo %2Fimages.search.yahoo.com%2Fsearch%2Fimages%3F_adv_prop%3 Dimage%26va%3Dfractal%26sz%3Dall%26ei%3DUTF- 8%26ni%3D20%26fr%3Dyfp-t-501- s%26b%3d181&w=500&h=400&imgurl=static.flickr.com%2f226%2f _0cfb8139e4_m.jpg&rurl=http%3A%2F%2Fwww.flickr.com%2Fphot os%2fseractal%2f %2f&size=91.6kb&name= _0cf b8139e4.jpg&p=fractal&type=jpeg&no=196&tt=357,067&oid=6de15e5d26
9 85f0c8&fusr=Seractal&tit=Fractal+anemone&hurl=http%3A%2F%2Fwww.f lickr.com%2fphotos%2fseractal%2f&ei=utf-8&src=p Broccoli/Cauliflower Photo Tree Branch Photo 3/EXP= /**http%3A// 870/ Queen Anne s Lace Photo r/exp= /**http%3a// /
Patterns in Pascal s Triangle
Pascal s Triangle Pascal s Triangle is an infinite triangular array of numbers beginning with a at the top. Pascal s Triangle can be constructed starting with just the on the top by following one easy
More informationMathematical Ideas that Shaped the World. Chaos and Fractals
Mathematical Ideas that Shaped the World Chaos and Fractals Plan for this class What is chaos? Why is the weather so hard to predict? Do lemmings really commit mass suicide? How do we measure the coastline
More information8 Fractals: Cantor set, Sierpinski Triangle, Koch Snowflake, fractal dimension.
8 Fractals: Cantor set, Sierpinski Triangle, Koch Snowflake, fractal dimension. 8.1 Definitions Definition If every point in a set S has arbitrarily small neighborhoods whose boundaries do not intersect
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationFilling Space with Random Fractal Non-Overlapping Simple Shapes
Filling Space with Random Fractal Non-Overlapping Simple Shapes John Shier 6935 133rd Court Apple Valley, MN, 55124, USA E-mail: johnart@frontiernet.net Abstract We present an algorithm that randomly places
More informationMaking tessellations combines the creativity of an art project with the challenge of solving a puzzle.
Activities Grades 6 8 www.exploratorium.edu/geometryplayground/activities EXPLORING TESSELLATIONS Background: What is a tessellation? A tessellation is any pattern made of repeating shapes that covers
More informationINTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE
INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE ABSTRACT:- Vignesh Palani University of Minnesota - Twin cities e-mail address - palan019@umn.edu In this brief work, the existing formulae
More informationGrade 7/8 Math Circles November 3/4, 2015. M.C. Escher and Tessellations
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Tiling the Plane Grade 7/8 Math Circles November 3/4, 2015 M.C. Escher and Tessellations Do the following
More informationAdapted from activities and information found at University of Surrey Website http://www.mcs.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.
12: Finding Fibonacci patterns in nature Adapted from activities and information found at University of Surrey Website http://www.mcs.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.html Curriculum connections
More informationObjectives: Students will understand what the Fibonacci sequence is; and understand how the Fibonacci sequence is expressed in nature.
Assignment Discovery Online Curriculum Lesson title: Numbers in Nature Grade level: 9-12 Subject area: Mathematics Duration: Two class periods Objectives: Students will understand what the Fibonacci sequence
More informationGrade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
More informationThe GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationGEOMETRY. Constructions OBJECTIVE #: G.CO.12
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
More informationThe Basics of Physics with Calculus. AP Physics C
The Basics of Physics with Calculus AP Physics C Pythagoras started it all 6 th Century Pythagoras first got interested in music when he was walking past a forge and heard that the sounds of the blacksmiths'
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationChapter 13: Fibonacci Numbers and the Golden Ratio
Chapter 13: Fibonacci Numbers and the Golden Ratio 13.1 Fibonacci Numbers THE FIBONACCI SEQUENCE 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, The sequence of numbers shown above is called the Fibonacci
More informationGeometry of Minerals
Geometry of Minerals Objectives Students will connect geometry and science Students will study 2 and 3 dimensional shapes Students will recognize numerical relationships and write algebraic expressions
More information(beauty) nature. patterns in
patterns in nature (beauty) People have long seen beauty in the geometric shapes and patterns found in tulips and other flowers. In this lesson, students will observe and categorize these shapes, and discuss
More informationStanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in
More informationExploring Geometric Probabilities with Buffon s Coin Problem
Exploring Geometric Probabilities with Buffon s Coin Problem Kady Schneiter Utah State University kady.schneiter@usu.edu Published: October 2012 Overview of Lesson Investigate Buffon s coin problem using
More informationReflection and Refraction
Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,
More informationColour by Numbers Image Representation
Activity 2 Colour by Numbers Image Representation Summary Computers store drawings, photographs and other pictures using only numbers. The following activity demonstrates how they can do this. Curriculum
More informationVALLIAMMAI ENGNIEERING COLLEGE SRM Nagar, Kattankulathur 603203.
VALLIAMMAI ENGNIEERING COLLEGE SRM Nagar, Kattankulathur 603203. DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Year & Semester : III Year, V Semester Section : CSE - 1 & 2 Subject Code : CS6504 Subject
More informationINTRODUCTION TO EUCLID S GEOMETRY
78 MATHEMATICS INTRODUCTION TO EUCLID S GEOMETRY CHAPTER 5 5.1 Introduction The word geometry comes form the Greek words geo, meaning the earth, and metrein, meaning to measure. Geometry appears to have
More informationAssessment For The California Mathematics Standards Grade 4
Introduction: Summary of Goals GRADE FOUR By the end of grade four, students understand large numbers and addition, subtraction, multiplication, and division of whole numbers. They describe and compare
More information39 Symmetry of Plane Figures
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
More informationSECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationWhich two rectangles fit together, without overlapping, to make a square?
SHAPE level 4 questions 1. Here are six rectangles on a grid. A B C D E F Which two rectangles fit together, without overlapping, to make a square?... and... International School of Madrid 1 2. Emily has
More informationDiscovering Math: Exploring Geometry Teacher s Guide
Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationSession 6 Number Theory
Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple
More informationDrawing Lines of Symmetry Grade Three
Ohio Standards Connection Geometry and Spatial Sense Benchmark H Identify and describe line and rotational symmetry in two-dimensional shapes and designs. Indicator 4 Draw lines of symmetry to verify symmetrical
More information12-1 Representations of Three-Dimensional Figures
Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationWhy do mathematicians make things so complicated?
Why do mathematicians make things so complicated? Zhiqin Lu, The Math Department March 9, 2010 Introduction What is Mathematics? Introduction What is Mathematics? 24,100,000 answers from Google. Introduction
More informationPlease see the Global Visions module description. Sequence
Overview In Satellite Eyes, students will explore the ways in which satellite images provide details of the Earth s surface. By using lenses, satellites are capable of taking digital images of the Earth
More informationA Fractal-based Printed Slot Antenna for Multi-band Wireless Applications
Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 1047 A Fractal-based Printed Slot Antenna for Multi-band Wireless Applications Seevan F. Abdulkareem, Ali
More informationLecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS
1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps
More informationNJ ASK PREP. Investigation: Mathematics. Paper Airplanes & Measurement. Grade 3 Benchmark 3 Geometry & Measurement
S E C T I O N 4 NJ ASK PREP Mathematics Investigation: Paper Airplanes & Measurement Grade 3 Benchmark 3 Geometry & Measurement This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs
More informationMechanics 1: Vectors
Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized
More informationHow To Find The Area Of A Shape
9 Areas and Perimeters This is is our next key Geometry unit. In it we will recap some of the concepts we have met before. We will also begin to develop a more algebraic approach to finding areas and perimeters.
More informationGRADES 7, 8, AND 9 BIG IDEAS
Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for
More informationLesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations
Math Buddies -Grade 4 13-1 Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations Goal: Identify congruent and noncongruent figures Recognize the congruence of plane
More informationChapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold
Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional
More informationTessellating with Regular Polygons
Tessellating with Regular Polygons You ve probably seen a floor tiled with square tiles. Squares make good tiles because they can cover a surface without any gaps or overlapping. This kind of tiling is
More informationYear 12 Pure Mathematics. C1 Coordinate Geometry 1. Edexcel Examination Board (UK)
Year 1 Pure Mathematics C1 Coordinate Geometry 1 Edexcel Examination Board (UK) Book used with this handout is Heinemann Modular Mathematics for Edexcel AS and A-Level, Core Mathematics 1 (004 edition).
More informationTennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes
Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical
More informationSenior Phase Grade 9 Today Planning Pack MATHEMATICS
M780636110304 Senior Phase Grade 9 Today Planning Pack MATHEMATICS Contents: Work Schedule: Page Grade 9 2 Lesson Plans: Grade 9 4 Rubrics: Rubric 1: Recognising, classifying and representing numbers 19
More informationLinking Math With Art Through The Elements of Design
2007 Asilomar Mathematics Conference Linking Math With Art Through The Elements of Design Presented by Renée Goularte Thermalito Union School District Oroville, California www.share2learn.com share2learn@sbcglobal.net
More informationThird Grade Shapes Up! Grade Level: Third Grade Written by: Jill Pisman, St. Mary s School East Moline, Illinois Length of Unit: Eight Lessons
Third Grade Shapes Up! Grade Level: Third Grade Written by: Jill Pisman, St. Mary s School East Moline, Illinois Length of Unit: Eight Lessons I. ABSTRACT This unit contains lessons that focus on geometric
More informationDrawing Lines with Pixels. Joshua Scott March 2012
Drawing Lines with Pixels Joshua Scott March 2012 1 Summary Computers draw lines and circles during many common tasks, such as using an image editor. But how does a computer know which pixels to darken
More informationA Correlation of Pearson Texas Geometry Digital, 2015
A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations
More informationAn Application of Analytic Geometry to Designing Machine Parts--and Dresses
Electronic Proceedings of Undergraduate Mathematics Day, Vol. 3 (008), No. 5 An Application of Analytic Geometry to Designing Machine Parts--and Dresses Karl Hess Sinclair Community College Dayton, OH
More informationBasic Understandings
Activity: TEKS: Exploring Transformations Basic understandings. (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential to understanding underlying
More informationSection 7.2 Area. The Area of Rectangles and Triangles
Section 7. Area The Area of Rectangles and Triangles We encounter two dimensional objects all the time. We see objects that take on the shapes similar to squares, rectangle, trapezoids, triangles, and
More informationMA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
More informationSpecular reflection. Dielectrics and Distribution in Ray Tracing. Snell s Law. Ray tracing dielectrics
Specular reflection Dielectrics and Distribution in Ray Tracing CS 465 Lecture 22 Smooth surfaces of pure materials have ideal specular reflection (said this before) Metals (conductors) and dielectrics
More information1. The volume of the object below is 186 cm 3. Calculate the Length of x. (a) 3.1 cm (b) 2.5 cm (c) 1.75 cm (d) 1.25 cm
Volume and Surface Area On the provincial exam students will need to use the formulas for volume and surface area of geometric solids to solve problems. These problems will not simply ask, Find the volume
More informationGrade 8 Mathematics Geometry: Lesson 2
Grade 8 Mathematics Geometry: Lesson 2 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes and all information outside
More informationFractals and Human Biology
Fractals and Human Biology We are fractal. Our lungs, our circulatory system, our brains are like trees. They are fractal structures. Fractal geometry allows bounded curves of infinite length, and closed
More informationDELAWARE MATHEMATICS CONTENT STANDARDS GRADES 9-10. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))
Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 9-10) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS
More informationNEW MEXICO Grade 6 MATHEMATICS STANDARDS
PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical
More informationMemoir on iterations of rational functions ( 1 ) Mémoire sur l itération des fonctions rationnelles ( 2 )
Memoir on iterations of rational functions ( ) Mémoire sur l itération des fonctions rationnelles ( ) written by Gaston Maurice Julia Translated in English by Alessandro Rosa Alessandro Rosa Home address
More informationPUZZLES AND GAMES THE TOOTHPICK WAY
PUZZLES AND GAMES THE TOOTHPICK WAY Many thinking skills go into solving math problems. The more advanced the mathematics, the more skills you need. You rely less on straight memorization and more on your
More informationLIGHT SECTION 6-REFRACTION-BENDING LIGHT From Hands on Science by Linda Poore, 2003.
LIGHT SECTION 6-REFRACTION-BENDING LIGHT From Hands on Science by Linda Poore, 2003. STANDARDS: Students know an object is seen when light traveling from an object enters our eye. Students will differentiate
More informationCELLULAR AUTOMATA AND APPLICATIONS. 1. Introduction. This paper is a study of cellular automata as computational programs
CELLULAR AUTOMATA AND APPLICATIONS GAVIN ANDREWS 1. Introduction This paper is a study of cellular automata as computational programs and their remarkable ability to create complex behavior from simple
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
More informationVolume of Pyramids and Cones
Volume of Pyramids and Cones Objective To provide experiences with investigating the relationships between the volumes of geometric solids. www.everydaymathonline.com epresentations etoolkit Algorithms
More informationObjectives. Materials
Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways
More informationWhat You ll Learn. Why It s Important
These students are setting up a tent. How do the students know how to set up the tent? How is the shape of the tent created? How could students find the amount of material needed to make the tent? Why
More informationCalculating Area, Perimeter and Volume
Calculating Area, Perimeter and Volume You will be given a formula table to complete your math assessment; however, we strongly recommend that you memorize the following formulae which will be used regularly
More informationSTRAND: Number and Operations Algebra Geometry Measurement Data Analysis and Probability STANDARD:
how August/September Demonstrate an understanding of the place-value structure of the base-ten number system by; (a) counting with understanding and recognizing how many in sets of objects up to 50, (b)
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationLogo Symmetry Learning Task. Unit 5
Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to
More informationEstimation of Fractal Dimension: Numerical Experiments and Software
Institute of Biomathematics and Biometry Helmholtz Center Münhen (IBB HMGU) Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences, Novosibirsk
More informationGlow-in-the-Dark Geometry
The Big Idea Glow-in-the-Dark Geometry This week you ll make geometric shapes out of glow sticks. The kids will try all sizes and shapes of triangles and quadrilaterials, then lay out sticks in mystical
More informationSurface Area Quick Review: CH 5
I hope you had an exceptional Christmas Break.. Now it's time to learn some more math!! :) Surface Area Quick Review: CH 5 Find the surface area of each of these shapes: 8 cm 12 cm 4cm 11 cm 7 cm Find
More informationLateral and Surface Area of Right Prisms
CHAPTER A Lateral and Surface Area of Right Prisms c GOAL Calculate lateral area and surface area of right prisms. You will need a ruler a calculator Learn about the Math A prism is a polyhedron (solid
More informationBUILDING THREE-DIMENSIONAL (3D) STRUCTURES
Activities Grades 6 8 www.exploratorium.edu/geometryplayground/activities BUILDING THREE-DIMENSIONAL (3D) STRUCTURES Draw a 3D structure in two dimensions. [60 minutes] Materials: Six cubes (any cubes,
More informationMaking tessellations combines the creativity of an art project with the challenge of solving a puzzle.
Activities Grades 3 5 www.exploratorium.edu/geometryplayground/activities EXPLORING TESSELLATIONS Background: What is a tessellation? A tessellation is any pattern made of repeating shapes that covers
More informationUsing 3D Computer Graphics Multimedia to Motivate Teachers Learning of Geometry and Pedagogy
Using 3D Computer Graphics Multimedia to Motivate Teachers Learning of Geometry and Pedagogy Tracy Goodson-Espy Associate Professor goodsonespyt@appstate.edu Samuel L. Espy Viz Multimedia Boone, NC 28607
More informationInv 1 5. Draw 2 different shapes, each with an area of 15 square units and perimeter of 16 units.
Covering and Surrounding: Homework Examples from ACE Investigation 1: Questions 5, 8, 21 Investigation 2: Questions 6, 7, 11, 27 Investigation 3: Questions 6, 8, 11 Investigation 5: Questions 15, 26 ACE
More informationMODERN APPLICATIONS OF PYTHAGORAS S THEOREM
UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented
More informationDYNAMIC DOMAIN CLASSIFICATION FOR FRACTAL IMAGE COMPRESSION
DYNAMIC DOMAIN CLASSIFICATION FOR FRACTAL IMAGE COMPRESSION K. Revathy 1 & M. Jayamohan 2 Department of Computer Science, University of Kerala, Thiruvananthapuram, Kerala, India 1 revathysrp@gmail.com
More informationALGEBRA. Find the nth term, justifying its form by referring to the context in which it was generated
ALGEBRA Pupils should be taught to: Find the nth term, justifying its form by referring to the context in which it was generated As outcomes, Year 7 pupils should, for example: Generate sequences from
More informationGrade 7/8 Math Circles Sequences and Series
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered
More informationHigh School Functions Interpreting Functions Understand the concept of a function and use function notation.
Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.
More informationGraph Theory Origin and Seven Bridges of Königsberg -Rhishikesh
Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh Graph Theory: Graph theory can be defined as the study of graphs; Graphs are mathematical structures used to model pair-wise relations between
More informationBy the end of this set of exercises, you should be able to:
BASIC GEOMETRIC PROPERTIES By the end of this set of exercises, you should be able to: find the area of a simple composite shape find the volume of a cube or a cuboid find the area and circumference of
More informationActivity Set 4. Trainer Guide
Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES
More informationShapes & Designs Notes
Problem 1.1 Definitions: regular polygons - polygons in which all the side lengths and angles have the same measure edge - also referred to as the side of a figure tiling - covering a flat surface with
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationChapter 17. Geometric Thinking and Geometric Concepts. T he geometry curriculum in grades K 8 should provide an
Chapter 17 Geometric Thinking and Geometric Concepts T he geometry curriculum in grades K 8 should provide an opportunity to experience shapes in as many different forms as possible. These should include
More informationToday in Physics 217: the method of images
Today in Physics 17: the method of images Solving the Laplace and Poisson euations by sleight of hand Introduction to the method of images Caveats Example: a point charge and a grounded conducting sphere
More informationFurther Steps: Geometry Beyond High School. Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu
Further Steps: Geometry Beyond High School Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu Geometry the study of shapes, their properties, and the spaces containing
More information