Escher s Tessellations: The Symmetry of Wallpaper Patterns II. Symmetry II
|
|
- Sherilyn Elinor Shields
- 6 years ago
- Views:
Transcription
1 Escher s Tessellations: The Symmetry of Wallpaper Patterns II Symmetry II 1/27
2 Brief Review of the Last Class Last time we started to talk about the symmetry of wallpaper patterns. Recall that these are pictures with translational symmetry in two directions. Escher s tessellations are great examples. We discussed that there are certain movements of a picture (viewing it as a piece of an infinite picture) which, when made, superimpose the picture upon itself. The movements we discussed are called isometries. On Monday we discussed three types of isometries: translations, rotations, and reflections. Symmetry II 2/27
3 Translations Symmetry II 3/27
4 Rotations Symmetry II 4/27
5 Reflections Symmetry II 5/27
6 This picture has rotational symmetry. We can do a quarter turn rotation (90 ) and have the picture superimpose upon itself (if we ignore color). There are also half turns (180 ). There is no reflectional symmetry. Symmetry II 6/27
7 This picture has reflectional symmetry. The vertical lines through the backbones of the beetles are reflection lines. Symmetry II 7/27
8 What symmetry can we find in this picture? Symmetry II 8/27
9 Clicker Question What rotational symmetry is in this picture? A Quarter turn only B Half turn only C Quarter and half turn only D None E Something else Symmetry II 9/27
10 What about this picture? Symmetry II 10/27
11 Clicker Question Besides translational, what symmetry do you see? A Rotational only B Reflectional only C Rotational and reflectional Symmetry II 11/27
12 Rotations versus Reflections Sometimes it is difficult to distinguish between rotations and reflections. One way to distinguish them is that reflections switch orientation; that is, right and left are switched. Rotations do not switch orientation. Think about looking into a mirror. If you hold something in your right hand, in the mirror it looks like you are holding it in your left hand. Symmetry II 12/27
13 Homer Rotated The Homer on the right was obtained by rotating the Homer on the left. Symmetry II 13/27
14 Homer Reflected The Homer on the right was obtained by reflecting the Homer on the left. The program I used also made Homer look upside down. Symmetry II 14/27
15 Here is another reflection of Homer. Symmetry II 15/27
16 In the original and rotated images, Homer is holding the donut in his right hand. In each of the reflected images, he is holding the donut in his left hand. Original and Rotation Original and Reflection Symmetry II 16/27
17 Clicker Question Was the Bart on the right obtained from the Bart on the left by A a rotation? B a reflection? Symmetry II 17/27
18 Combining Isometries Another way to build isometries is to perform two consecutively. One example is to do a reflection followed by a translation. This is important enough to be named. It is called a glide reflection. Symmetry II 18/27
19 Glide Reflections Symmetry II 19/27
20 If we perform two isometries consecutively, using any of the four types above, the end result will again be one of the four types. Thus, any isometry is one of the four types: translations, rotations, reflections, glide reflections. Escher made heavy use of glide reflections as we will illustrate with several pictures. There are some mathematical ideas behind glide reflections that Escher had to discover in order to draw pictures demonstrating glides. Note that in the pictures below, there are glide reflections, which are built from a reflection and a translation, in which neither the reflection nor the translation is a symmetry of the picture, only the combination. Symmetry II 20/27
21 Symmetry II 21/27
22 If you reflect the picture vertically and then shift an appropriate amount, the picture will superimpose upon itself. The resulting glide reflection is a symmetry of the picture, while the vertical reflection or the translation are not symmetries of the picture. The amount of shift in the glide reflection is shown in the next picture. We can view the reflection as being along the vertical line connecting the white horsemen s chins. The symmetry in the following pictures is probably the most common in Escher s tessellations. Symmetry II 22/27
23 Symmetry II 23/27
24 This picture has the same symmetry as the previous one, in that there are translational and glide reflectional symmetry and nothing else. Symmetry II 24/27
25 In each of these three pictures Escher used a glide reflection starting with a vertical reflection. Symmetry II 25/27
26 The amount of vertical shift in the glide is exactly half of the smallest vertical translation. This can be proven mathematically, and Escher had to discover this to make his drawings. Symmetry II 26/27
27 Next Time On Friday we will conclude our discussion of Escher s Tessellations and the classification of these pictures. We ll discuss briefly the broad mathematical ideas used to obtain the classification. We ll also see examples of all 17 symmetry types. Escher drew pictures representing 16 of the 17 symmetry types. We ll see these pictures. Symmetry II 27/27
Escher s Tessellations: The Symmetry of Wallpaper Patterns. 30 January 2012
Escher s Tessellations: The Symmetry of Wallpaper Patterns 30 January 2012 Symmetry I 30 January 2012 1/32 This week we will discuss certain types of drawings, called wallpaper patterns, and how mathematicians
More informationPart XI. Classifying Frieze Patterns. The goal for this part is to classify frieze patterns.
Part XI Classifying Frieze Patterns The goal for this part is to classify frieze patterns. Observations about frieze patterns I True or False: A rotation symmetry of a strip must be a 1/2 turn. I True
More informationDownloaded from
Symmetry 1 1.Find the next figure None of these 2.Find the next figure 3.Regular pentagon has line of symmetry. 4.Equlilateral triangle has.. lines of symmetry. 5.Regular hexagon has.. lines of symmetry.
More informationBasic Mathematics Review 5232
Basic Mathematics Review 5232 Symmetry A geometric figure has a line of symmetry if you can draw a line so that if you fold your paper along the line the two sides of the figure coincide. In other words,
More informationA web source that was used to assist in the development is:
Logo Transformations The objective of this lesson is to analyze line and reflection symmetry and transformations in logos. The activity has three different parts. The first explores the symmetry of the
More informationSymmetry: A Visual Presentation
Symmetry: A Visual Presentation Line Symmetry Shape has line symmetry when one half of it is the mirror image of the other half. Symmetry exists all around us and many people see it as being a thing of
More informationPrinting by Rolling Möbius Band Stencils: Glide Reflection Embodied in Physical Action
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture Printing by Rolling Möbius Band Stencils: Glide Reflection Embodied in Physical Action Simon Morgan Data Constructs Twickenham,
More informationlines of weakness building for the future All of these walls have a b c d Where are these lines?
All of these walls have lines of weakness a b c d Where are these lines? A standard British brick is twice as wide as it is tall. Using British bricks, make a rectangle that does not have any lines of
More informationUnit 5 Shape and space
Unit 5 Shape and space Five daily lessons Year 4 Summer term Unit Objectives Year 4 Sketch the reflection of a simple shape in a mirror line parallel to Page 106 one side (all sides parallel or perpendicular
More informationTextile Journal. Figure 2: Two-fold Rotation. Figure 3: Bilateral reflection. Figure 1: Trabslation
Conceptual Developments in the Analysis of Patterns Part One: The Identification of Fundamental Geometrical Elements by M.A. Hann, School of Design, University of Leeds, UK texmah@west-01.novell.leeds.ac.uk
More informationContents. Congruent Triangles. Additional Practice Answers to Check Your Work. Section
Contents Section Congruent Triangles Flip, Turn, Resize, and Slide 1 Transformed Triangles 2 Constructing Parallel Lines 5 Transformations 6 Reflections 7 Rotations 10 Summary 13 Check Your Work 14 Additional
More informationUnit 6 Quadrilaterals
Unit 6 Quadrilaterals ay lasswork ay Homework Monday Properties of a Parallelogram 1 HW 6.1 11/13 Tuesday 11/14 Proving a Parallelogram 2 HW 6.2 Wednesday 11/15 Thursday 11/16 Friday 11/17 Monday 11/20
More informationNCERT Solution Class 7 Mathematics Symmetry Chapter: 14. Copy the figures with punched holes and find the axes of symmetry for the following:
Downloaded from Q.1) Exercise 14.1 NCERT Solution Class 7 Mathematics Symmetry Chapter: 14 Copy the figures with punched holes and find the axes of symmetry for the following: Sol.1) S.No. Punched holed
More informationThe first task is to make a pattern on the top that looks like the following diagram.
Cube Strategy The cube is worked in specific stages broken down into specific tasks. In the early stages the tasks involve only a single piece needing to be moved and are simple but there are a multitude
More informationGrade 7/8 Math Circles. Visual Group Theory
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles October 25 th /26 th Visual Group Theory Grouping Concepts Together We will start
More informationThe Droste-effect and the exponential transform
The Droste-effect and the exponential transform Bart de Smit Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden The Netherlands Email: desmit@math.leidenuniv.nl January, 2005 Abstract
More informationUK SENIOR MATHEMATICAL CHALLENGE
UK SENIOR MATHEMATICAL CHALLENGE Tuesday 8 November 2016 Organised by the United Kingdom Mathematics Trust and supported by Institute and Faculty of Actuaries RULES AND GUIDELINES (to be read before starting)
More informationPart VIII. Rosettes. The goal for this part is to identify and classify rosette patterns.
Part VIII Rosettes The goal for this part is to identify and classify rosette patterns. Classify For each picture, decide if it goes in the left pile or the right. Rosettes A rosette pattern is a pattern
More informationNovember 8, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol
More informationDownloaded from
Symmetry 1.Can you draw a figure whose mirror image is identical to the figure itself? 2.Find out if the figure is symmetrical or not? 3.Count the number of lines of symmetry in the figure. 4.A line
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 101 Practice Second Midterm MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A small country consists of four states. The population of State
More informationLiberty Pines Academy Russell Sampson Rd. Saint Johns, Fl 32259
Liberty Pines Academy 10901 Russell Sampson Rd. Saint Johns, Fl 32259 M. C. Escher is one of the world s most famous graphic artists. He is most famous for his so called impossible structure and... Relativity
More informationCounting Cube Colorings with the Cauchy-Frobenius Formula and Further Friday Fun
Counting Cube Colorings with the Cauchy-Frobenius Formula and Further Friday Fun Daniel Frohardt Wayne State University December 3, 2010 We have a large supply of squares of in 3 different colors and an
More informationEquilateral k-isotoxal Tiles
Equilateral k-isotoxal Tiles R. Chick and C. Mann October 26, 2012 Abstract In this article we introduce the notion of equilateral k-isotoxal tiles and give of examples of equilateral k-isotoxal tiles
More informationDisplaying Distributions with Graphs
Displaying Distributions with Graphs Recall that the distribution of a variable indicates two things: (1) What value(s) a variable can take, and (2) how often it takes those values. Example 1: Weights
More informationLessons in Duality and Symmetry from M.C. Escher
Lessons in Duality and Symmetry from M.C. Escher Doris Schattschneider Mathematics Department, Moravian College 1200 Main St., Bethlehem, PA 18018-6650 USA E-mail: schattdo@moravian.edu Abstract The Dutch
More information- Chapter 1: "Symmetry and Surface Area" -
Mathematics 9 C H A P T E R Q U I Z Form P - Chapter 1: "Symmetry and Surface Area" - Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In the figure, the
More information# 1. As shown, the figure has been divided into three identical parts: red, blue, and green. The figures are identical because the blue and red
# 1. As shown, the figure has been divided into three identical parts: red, blue, and green. The figures are identical because the blue and red figures are already in the correct orientation, and the green
More informationGeometry Vocabulary Book
Geometry Vocabulary Book Units 2-4 Page 1 Unit 2 General Geometry Point Characteristics: Line Characteristics: Plane Characteristics: RELATED POSTULATES: Through any two points there exists exactly one
More informationA = 5; B = 4; C = 3; B = 2; E = 1; F = 26; G = 25; H = 24;.; Y = 7; Z = 6 D
1. message is coded from letters to numbers using this code: = 5; B = 4; = 3; B = 2; E = 1; F = 26; G = 25; H = 24;.; Y = 7; Z = 6 When the word MISSISSIPPI is coded, what is the sum of all eleven numbers?.
More informationStep 1 - Introducing the Maurits Cornelis Escher Slideshow Guide
Step 1 - Introducing the Maurits Cornelis Escher Slideshow Guide BEGIN READING HERE MOTIVATION Raise your hand if you like to put puzzles together. Are you good at doing puzzles? On what kind of puzzles
More informationClass VI Mathematics (Ex. 13.1) Questions
Class VI Mathematics (Ex. 13.1) Questions 1. List any four symmetrical from your home or school. 2. For the given figure, which one is the mirror line, l 1 or l 2? 3. Identify the shapes given below. Check
More informationMeasuring and Drawing Angles and Triangles
NME DTE Measuring and Drawing ngles and Triangles Measuring an angle 30 arm origin base line 0 180 0 If the arms are too short to reach the protractor scale, lengthen them. Step 1: lace the origin of the
More informationMind Ninja The Game of Boundless Forms
Mind Ninja The Game of Boundless Forms Nick Bentley 2007-2008. email: nickobento@gmail.com Overview Mind Ninja is a deep board game for two players. It is 2007 winner of the prestigious international board
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 101 Practice Second Midterm MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A small country consists of four states. The population of State
More informationDownloaded from
Symmetry 1 1.A line segment is Symmetrical about its ---------- bisector (A) Perpendicular (B) Parallel (C) Line (D) Axis 2.How many lines of symmetry does a reactangle have? (A) Four (B) Three (C)
More informationLecture 6: Latin Squares and the n-queens Problem
Latin Squares Instructor: Padraic Bartlett Lecture 6: Latin Squares and the n-queens Problem Week 3 Mathcamp 01 In our last lecture, we introduced the idea of a diagonal Latin square to help us study magic
More informationCrossing Game Strategies
Crossing Game Strategies Chloe Avery, Xiaoyu Qiao, Talon Stark, Jerry Luo March 5, 2015 1 Strategies for Specific Knots The following are a couple of crossing game boards for which we have found which
More informationPERFORMANCE TASK. SYMMETRY, TRANSLATIONS & CONGRUENCE Scaff 2014
PERFORMANCE TASK SYMMETRY, TRANSLATIONS & CONGRUENCE Scaff 2014 Click on the link below, to watch a video on symmetry & translations http://www.linkslearning.k12.wa.us/kids/1_m ath/2_illustrated_lessons/4_line_symmetry/i
More informationCH 54 SPECIAL LINES. Ch 54 Special Lines. Introduction
479 CH 54 SPECIAL LINES Introduction Y ou may have noticed that all the lines we ve seen so far in this course have had slopes that were either positive or negative. You may also have observed that every
More informationLine Op Art. In the mid-20th century, artists such as Josef Albers, Victor Vasarely, and M.C. Escher experimented with Optical Art.
Intro Basic line, space Line Op Art Concept/Skill:When lines are close together in patterns, colors, motion or other optical illusions are created. OHS line - A1, A2, C4, C5, D3 Objective: The learner
More informationMathematics Competition Practice Session 6. Hagerstown Community College: STEM Club November 20, :00 pm - 1:00 pm STC-170
2015-2016 Mathematics Competition Practice Session 6 Hagerstown Community College: STEM Club November 20, 2015 12:00 pm - 1:00 pm STC-170 1 Warm-Up (2006 AMC 10B No. 17): Bob and Alice each have a bag
More informationUK JUNIOR MATHEMATICAL CHALLENGE. April 26th 2012
UK JUNIOR MATHEMATICAL CHALLENGE April 6th 0 SOLUTIONS These solutions augment the printed solutions that we send to schools. For convenience, the solutions sent to schools are confined to two sides of
More informationTwenty-sixth Annual UNC Math Contest First Round Fall, 2017
Twenty-sixth Annual UNC Math Contest First Round Fall, 07 Rules: 90 minutes; no electronic devices. The positive integers are,,,,.... Find the largest integer n that satisfies both 6 < 5n and n < 99..
More informationVariations on the Two Envelopes Problem
Variations on the Two Envelopes Problem Panagiotis Tsikogiannopoulos pantsik@yahoo.gr Abstract There are many papers written on the Two Envelopes Problem that usually study some of its variations. In this
More informationCambridge International Examinations Cambridge Primary Checkpoint
Cambridge International Examinations Cambridge Primary Checkpoint MATHEMATICS 0845/02 Paper 2 For Examination from 2014 SPECIMEN PAPER Candidates answer on the Question Paper. Additional Materials: Pen
More informationMANIPULATIVE MATHEMATICS FOR STUDENTS
MANIPULATIVE MATHEMATICS FOR STUDENTS Manipulative Mathematics Using Manipulatives to Promote Understanding of Elementary Algebra Concepts Lynn Marecek MaryAnne Anthony-Smith This file is copyright 07,
More information2 An n-person MK Proportional Protocol
Proportional and Envy Free Moving Knife Divisions 1 Introduction Whenever we say something like Alice has a piece worth 1/2 we mean worth 1/2 TO HER. Lets say we want Alice, Bob, Carol, to split a cake
More informationHANDS-ON TRANSFORMATIONS: DILATIONS AND SIMILARITY (Poll Code 44273)
HANDS-ON TRANSFORMATIONS: DILATIONS AND SIMILARITY (Poll Code 44273) Presented by Shelley Kriegler President, Center for Mathematics and Teaching shelley@mathandteaching.org Fall 2014 8.F.1 8.G.3 8.G.4
More informationA Tour of Tilings in Thirty Minutes
A Tour of Tilings in Thirty Minutes Alexander F. Ritter Mathematical Institute & Wadham College University of Oxford Wadham College Mathematics Alumni Reunion Oxford, 21 March, 2015. For a detailed tour
More informationp. 2 21st Century Learning Skills
Contents: Lesson Focus & Standards p. 1 Review Prior Stages... p. 2 Vocabulary..... p. 2 Lesson Content... p. 3-7 Math Connection.... p. 8-9 Review... p. 10 Trivia. p. 10 21st Century Learning Skills Learning
More informationGrade 7/8 Math Circles. Visual Group Theory
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles October 25 th /26 th Visual Group Theory Grouping Concepts Together We will start
More informationCalculations: Recording Addition (page 1 of 2) 2. Adding two digits (tens & ones)
Updated August 205 . Adding single digits (ones) Using a number line: Example: 7 + 4 = 0 2 3 4 5 6 7 8 9 0 2 Calculations: Recording Addition (page of 2) 2. Adding two digits (tens & ones) Using a number
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand HW 8
CS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand HW 8 1 Sundry Before you start your homewor, write down your team. Who else did you wor with on this homewor? List names and
More informationAll Levels. Solving the Rubik s Cube
Solving the Rubik s Cube All Levels Common Core: Objectives: Mathematical Practice Standards: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct
More informationThe Parkland Federation. February 2016
The Parkland Federation February 206 EYFS/KS Calculations: Recording Addition (page of ). Aggregation/combining 2. Augmentation/counting on 3. Counting Contexts: + + + + Pupils physically combining groups
More informationTo Explore the Properties of Parallelogram
Exemplar To Explore the Properties of Parallelogram Objective To explore the properties of parallelogram Dimension Measures, Shape and Space Learning Unit Quadrilaterals Key Stage 3 Materials Required
More informationTEST (a) Write these numbers in order of increasing size. 12, 7, 15, 4, 1, 10, Circle all the odd numbers.
1 TEST 5 1. Complete the picture so that it has 7 dots. 2. What is the number shown? 0 5 10 3. Fill in the missing numbers. 2 + 3 = 4 1 = (c) 3 + 4 = (d) 4 + = 9 (e) 8 = 3 (f) + 7 = 7 4. Write these numbers
More informationSUDOKU1 Challenge 2013 TWINS MADNESS
Sudoku1 by Nkh Sudoku1 Challenge 2013 Page 1 SUDOKU1 Challenge 2013 TWINS MADNESS Author : JM Nakache The First Sudoku1 Challenge is based on Variants type from various SUDOKU Championships. The most difficult
More informationLESSON 2: THE INCLUSION-EXCLUSION PRINCIPLE
LESSON 2: THE INCLUSION-EXCLUSION PRINCIPLE The inclusion-exclusion principle (also known as the sieve principle) is an extended version of the rule of the sum. It states that, for two (finite) sets, A
More informationThe Sine Function. Precalculus: Graphs of Sine and Cosine
Concepts: Graphs of Sine, Cosine, Sinusoids, Terminology (amplitude, period, phase shift, frequency). The Sine Function Domain: x R Range: y [ 1, 1] Continuity: continuous for all x Increasing-decreasing
More informationA Method to Generate Polyominoes and Polyiamonds for Tilings with Rotational Symmetry
A Method to Generate Polyominoes and Polyiamonds for Tilings with Rotational Symmetry Hiroshi Fukuda 1, Nobuaki Mutoh 1, Gisaku Nakamura 2, Doris Schattschneider 3 1 School of Administration and Informatics,
More informationSTAT 311 (Spring 2016) Worksheet: W3W: Independence due: Mon. 2/1
Name: Group 1. For all groups. It is important that you understand the difference between independence and disjoint events. For each of the following situations, provide and example that is not in the
More information1 /4. (One-Half) (One-Quarter) (Three-Eighths)
LESSON 4: Fractions: A fraction is a part of a whole. Slice a pizza, and you will have fractions: 1 /2 1 /4 3 /8 (One-Half) (One-Quarter) (Three-Eighths) The top number tells how many slices you have and
More informationMathematics in the Modern World
Mathematics as a Tool Geometric Designs 1/17 Mathematics in the Modern World Geometric Designs Joel Reyes Noche, Ph.D. jnoche@gbox.adnu.edu.ph Department of Mathematics Ateneo de Naga University Council
More information9.5 symmetry 2017 ink.notebook. October 25, Page Symmetry Page 134. Standards. Page Symmetry. Lesson Objectives.
9.5 symmetry 2017 ink.notebook Page 133 9.5 Symmetry Page 134 Lesson Objectives Standards Lesson Notes Page 135 9.5 Symmetry Press the tabs to view details. 1 Lesson Objectives Press the tabs to view details.
More informationGLOSSARY. a * (b * c) = (a * b) * c. A property of operations. An operation * is called associative if:
Associativity A property of operations. An operation * is called associative if: a * (b * c) = (a * b) * c for every possible a, b, and c. Axiom For Greek geometry, an axiom was a 'self-evident truth'.
More information4 th Grade Mathematics Learning Targets By Unit
INSTRUCTIONAL UNIT UNIT 1: WORKING WITH WHOLE NUMBERS UNIT 2: ESTIMATION AND NUMBER THEORY PSSA ELIGIBLE CONTENT M04.A-T.1.1.1 Demonstrate an understanding that in a multi-digit whole number (through 1,000,000),
More informationSome Monohedral Tilings Derived From Regular Polygons
Some Monohedral Tilings Derived From Regular Polygons Paul Gailiunas 25 Hedley Terrace, Gosforth Newcastle, NE3 1DP, England email: p-gailiunas@argonet.co.uk Abstract Some tiles derived from regular polygons
More information1. Use the following directions to draw a figure in the box to the right. a. Draw two points: and. b. Use a straightedge to draw.
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Homework 4 Name Date 1. Use the following directions to draw a figure in the box to the right. a. Draw two points: and. b. Use a straightedge to draw. c.
More informationHIGH SCHOOL MATHEMATICS CONTEST Sponsored by THE MATHEMATICS DEPARTMENT of WESTERN CAROLINA UNIVERSITY. LEVEL I TEST March 23, 2017
HIGH SCHOOL MATHEMATICS CONTEST Sponsored by THE MATHEMATICS DEPARTMENT of WESTERN CAROLINA UNIVERSITY LEVEL I TEST March 23, 2017 Prepared by: John Wagaman, Chairperson Nathan Borchelt DIRECTIONS: Do
More informationMathematics Enhancement Programme TEACHING SUPPORT: Year 3
Mathematics Enhancement Programme TEACHING UPPORT: Year 3 1. Question and olution Write the operations without brackets if possible so that the result is the same. Do the calculations as a check. The first
More informationBurnside s Lemma. Keywords : Burnside s counting theorem, formula, Permutation, Orbit, Invariant, Equivalence, Equivalence class
Osaka Keidai onshu, Vol. 6 No. July 0 urnside s Lemma utaka Nishiyama Abstract There is a famous problem which involves discriminating the faces of a die using colors: how many different patterns can be
More information4 th Grade Mathematics Instructional Week 30 Geometry Concepts Paced Standards: 4.G.1: Identify, describe, and draw parallelograms, rhombuses, and
4 th Grade Mathematics Instructional Week 30 Geometry Concepts Paced Standards: 4.G.1: Identify, describe, and draw parallelograms, rhombuses, and trapezoids using appropriate tools (e.g., ruler, straightedge
More information14.7 Maximum and Minimum Values
CHAPTER 14. PARTIAL DERIVATIVES 115 14.7 Maximum and Minimum Values Definition. Let f(x, y) be a function. f has a local max at (a, b) iff(a, b) (a, b). f(x, y) for all (x, y) near f has a local min at
More informationTechniques for Generating Sudoku Instances
Chapter Techniques for Generating Sudoku Instances Overview Sudoku puzzles become worldwide popular among many players in different intellectual levels. In this chapter, we are going to discuss different
More informationEducat o C. Thelvy LEAGUE. Math Kangaroo 2016 in USA. International Competition in Mathematics Thursday, March 17, 2016.
Thelvy LEAGUE Educat o C Math Kangaroo 2016 March 17, 2016 Levels 5 and 6 Mathematics Promotion Society K angourou Sans Frontieres Math Kangaroo in USA Math Kangaroo 2016 in USA International Competition
More information1. Use the following directions to draw a figure in the box to the right. a. Draw two points: and. b. Use a straightedge to draw.
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Problem Set 4 Name Date 1. Use the following directions to draw a figure in the box to the right. a. Draw two points: and. b. Use a straightedge to draw.
More informationClass 8: Factors and Multiples (Lecture Notes)
Class 8: Factors and Multiples (Lecture Notes) If a number a divides another number b exactly, then we say that a is a factor of b and b is a multiple of a. Factor: A factor of a number is an exact divisor
More informationSequential Dynamical System Game of Life
Sequential Dynamical System Game of Life Mi Yu March 2, 2015 We have been studied sequential dynamical system for nearly 7 weeks now. We also studied the game of life. We know that in the game of life,
More informationMAS336 Computational Problem Solving. Problem 3: Eight Queens
MAS336 Computational Problem Solving Problem 3: Eight Queens Introduction Francis J. Wright, 2007 Topics: arrays, recursion, plotting, symmetry The problem is to find all the distinct ways of choosing
More informationWhat s a Widget? EXAMPLE A L E S S O N 1.3
Page 1 of 7 L E S S O N 1.3 What s a Widget? Good definitions are very important in geometry. In this lesson you will write your own geometry definitions. Which creatures in the last group are Widgets?
More informationNOTES ON A DERIVED NEGATIVE SUPPLY
ELECTRONOTES WEBNOTE 10/15/2012 ENWN7 NOTES ON A DERIVED NEGATIVE SUPPLY Recently my attention came back to an old app note: An Op-Amp Supply Based on a 12.6V Filament Transformer, AN-136, June 15, 1979.
More informationMultiple : The product of a given whole number and another whole number. For example, some multiples of 3 are 3, 6, 9, and 12.
1.1 Factor (divisor): One of two or more whole numbers that are multiplied to get a product. For example, 1, 2, 3, 4, 6, and 12 are factors of 12 1 x 12 = 12 2 x 6 = 12 3 x 4 = 12 Factors are also called
More informationNovember 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More informationProblem of the Month: Between the Lines
Problem of the Month: Between the Lines Overview: In the Problem of the Month Between the Lines, students use polygons to solve problems involving area. The mathematical topics that underlie this POM are
More information36 th NEW BRUNSWICK MATHEMATICS COMPETITION
UNIVERSITY OF NEW BRUNSWICK UNIVERSITÉ DE MONCTON 36 th NEW BRUNSWICK MATHEMATICS COMPETITION Thursday, May 3 rd, 2018 GRADE 8 INSTRUCTIONS TO THE STUDENT: 1. Do not start the examination until you are
More informationKS1. Learning Outcomes. Andy Goldsworthy Flour prints
NUMERACY NATURE SHAPE Curriculum Links Art: () Use a range of materials creatively to design and make products; to develop a wide range of art and design techniques in using colour, pattern, texture, line,
More informationarxiv: v2 [math.gt] 21 Mar 2018
Tile Number and Space-Efficient Knot Mosaics arxiv:1702.06462v2 [math.gt] 21 Mar 2018 Aaron Heap and Douglas Knowles March 22, 2018 Abstract In this paper we introduce the concept of a space-efficient
More informationSymmetries of Cairo-Prismatic Tilings
Rose-Hulman Undergraduate Mathematics Journal Volume 17 Issue 2 Article 3 Symmetries of Cairo-Prismatic Tilings John Berry Williams College Matthew Dannenberg Harvey Mudd College Jason Liang University
More informationLearn to use translations, reflections, and rotations to transform geometric shapes.
Learn to use translations, reflections, and rotations to transform geometric shapes. Insert Lesson Title Here Vocabulary transformation translation rotation reflection line of reflection A rigid transformation
More informationNumberSense Companion Workbook Grade 4
NumberSense Companion Workbook Grade 4 Sample Pages (ENGLISH) Working in the NumberSense Companion Workbook The NumberSense Companion Workbooks address measurement, spatial reasoning (geometry) and data
More informationUniversity of Maryland Department of Physics
Spring 2002 University of Maryland Department of Physics Laura Lising Physics 122 May 8, 2003 Makeup Exam #2 Solutions Multiple choice questions. Just the answer counts for these. (8 points each) screen
More informationAn able class have been studying the coordinate geometry of the circle and have covered equations of circles and their properties.
Example 5 An able class have been studying the coordinate geometry of the circle and have covered equations of circles and their properties. Understand and use the coordinate geometry of the circle including
More information7.1 INTRODUCTION TO PERIODIC FUNCTIONS
7.1 INTRODUCTION TO PERIODIC FUNCTIONS Ferris Wheel Height As a Function of Time The London Eye Ferris Wheel measures 450 feet in diameter and turns continuously, completing a single rotation once every
More informationUnderlying Tiles in a 15 th Century Mamluk Pattern
Bridges Finland Conference Proceedings Underlying Tiles in a 15 th Century Mamluk Pattern Ron Asherov Israel rasherov@gmail.com Abstract An analysis of a 15 th century Mamluk marble mosaic pattern reveals
More informationLESSON PLAN: Symmetry
LESSON PLAN: Symmetry Subject Mathematics Content Area Space and Shape Topic Symmetry Concept Recognise and draw line of symmetry in 2-D geometrical and non geometrical shapes Determine line of symmetry
More informationAngle Measure and Plane Figures
Grade 4 Module 4 Angle Measure and Plane Figures OVERVIEW This module introduces points, lines, line segments, rays, and angles, as well as the relationships between them. Students construct, recognize,
More informationParallelograms and Symmetry
square Parallelograms and Symmetry The drawings below show how four dots can be connected to make a parallelogram. These are the only general possibilities. All four sides may be equal length (top 3 drawings)
More informationChapter 4 Number Theory
Chapter 4 Number Theory Throughout the study of numbers, students Á should identify classes of numbers and examine their properties. For example, integers that are divisible by 2 are called even numbers
More information