Escher s Circle Limit Anneke Bart Saint Louis University Introduction

Size: px
Start display at page:

Download "Escher s Circle Limit Anneke Bart Saint Louis University Introduction"

Transcription

1 Escher s Circle Limit Anneke Bart Saint Louis University Introduction What are some of the most fundamental things we do in geometry? In the beginning we mainly look at lines, segments, polygons and circles. This is even reflected in the axioms ( ground rules ) of geometry. Euclidean Geometry (Planar Geometry) We look at polygons and tessellations we can create with these polygons. A tessellation is a covering of the plane by shapes, called tiles, so that there are no empty spaces and no overlapped tiles. Tessellations are also called tilings. Spherical Geometry What happens when we try to draw polygons on a sphere, or if we try to tile a sphere? We would soon realize that the axioms of planar geometry do not work. We actually get a different geometry.

2 This begs the question: Are there any other geometries? If so, what do they look like? Hyperbolic Geometry What does it look like? This is a drawing of hyperbolic space. This space is an infinite space. There is some distortion present. We need to interpret all the fish as being the same size. The ones near the edge are not smaller. They are far away (compare to perspective drawing!) To travel fast in this geometry we travel along semi-circles perpendicular to the edge.

3 On the left we see examples of geodesics (hyperbolic straight lines) On the right we see a stick figure walking off into the distance. We will now explore hyperbolic geometry by looking at Escher s Circle Limits. All M.C. Escher works Cordon Art BV - Baarn - the Netherlands. All M.C. Escher works (c) 2007 The M.C. Escher Company - the Netherlands. All rights reserved. Used by permission.

4 Escher's Circle Limit Exploration - EscherMath Escher's Circle Limit Exploration From EscherMath Objective: Explore Escher's Circle Limit prints to develop an intuition for hyperbolic geometry Materials Printed copies of Circle Limit I, Circle Limit II, Circle Limit III, and Circle Limit IV (Heaven and Hell). All four Circle Limit prints, dimmed: Image:Four-dim-circle-limits.pdf Circle Limit I Recall that in Spherical Geometry, the shortest path between two points is along a great circle. These shortest paths are called geodesics, and the geodesics play the same role as do straight lines in Euclidean geometry. Escher's Circle Limit prints are based on a new kind of geometry, Hyperbolic Geometry. The red lines shown on Circle Limit I are the geodesics in this new geometry. These curves will play the role of straight lines. Each red line follows the spines of a line of fish. 1. There are two types of red line marked in the Circle Limit I figure. Describe them. Draw more geodesics by following the spines of other rows of fish. Describe the curves that result. In these pictures of hyperbolic geometry, geodesics come in two forms, either straight lines through the center of the disk, or arcs of circles that meet the disk's edge at 90. Segments of geodesics form the sides of polygons. Polygons in hypebolic geometry will look "pinched" to our Euclidean eyes. 2. What type of polygons do you see in this figure? 3. Compare the angle sum of one of these polygons to the corresponding angle sum for Euclidean geometry. Circle Limit I is a picture of a surface called "hyperbolic space", but it is a distorted picture. In actual hyperbolic space, these fish would all have the same size and shape. M.C. Escher, Circle Limit I (1958) with geodesics in red What is the highest order of rotation symmetry for this print? Describe the geometric tessellation underlying Circle Limit I. 1 of 3 4/21/08 11:14 AM

5 Escher's Circle Limit Exploration - EscherMath Circle Limits II and IV M.C. Escher, Circle Limit II (1959) M.C. Escher, Circle Limit IV (Heaven and Hell) (1960) For Circle Limit II: What is the highest order of rotation? Draw geodesics in this figure. Describe the underlying geometric tessellation. For Circle Limit IV: 8. What is the highest order of rotation? What other orders of rotation are present? 9. Draw geodesics in this figure. Describe the underlying geometric tessellation. 10. Draw a geodesic NOT passing through the center point. 11. How many geodesics can you draw through the center point, so that the new geodesic does not meet the geodesic you picked in the previous question? Another way to ask the same question: How many geodesics pass through the point so that the new geodesic is parallel to the first geodesic? Circle Limit III 2 of 3 4/21/08 11:14 AM

6 Escher's Circle Limit Exploration - EscherMath This Circle Limit is the most subtle. The white lines look like the geodesics in the other Circle Limit prints, but they are not the same. A closer look shows that these white lines are not geodesics at all. 12. Pick a triangle and determine its corner angles by considering the number of polygons at a vertex. Assume all angles at each vertex are equal (they are, though the distortion makes this harder to believe). 13. What is the sum of the angles in the triangle? Is this possible in hyperbolic geometry? 14. Look at where the white lines meet the boundary of the disk. At what angle do the white lines seem to meet the boundary of the disk? Why is this a "problem"? M.C. Escher, Circle Limit III, Handin: Marked up Circle Limit prints and a sheet with answers to all questions. Retrieved from " Category: Exploration This page was last modified 16:32, 17 April of 3 4/21/08 11:14 AM

7 Circle Limit I Circle Limit II Circle Limit III Circle Limit IV

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

Mathematical mysteries: Strange Geometries

Mathematical mysteries: Strange Geometries 1997 2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non commercial use. For other uses, including electronic redistribution,

More information

COMPUTER DESIGN OF REPEATING HYPERBOLIC PATTERNS

COMPUTER DESIGN OF REPEATING HYPERBOLIC PATTERNS COMPUTER DESIGN OF REPEATING HYPERBOLIC PATTERNS Douglas Dunham University of Minnesota Duluth Department of Computer Science 1114 Kirby Drive Duluth, Minnesota 55812-2496 USA ddunham@d.umn.edu Abstract:

More information

Mathematics and the prints of M.C. Escher. Joe Romano Les Houches Summer School 23 July 2018

Mathematics and the prints of M.C. Escher. Joe Romano Les Houches Summer School 23 July 2018 Mathematics and the prints of M.C. Escher Joe Romano Les Houches Summer School 23 July 2018 Possible topics projective geometry non-euclidean geometry topology & knots ambiguous perspective impossible

More information

AMS Sectional Meeting, Richmond VA Special Session on Mathematics and the Arts

AMS Sectional Meeting, Richmond VA Special Session on Mathematics and the Arts AMS Sectional Meeting, Richmond VA Special Session on Mathematics and the Arts Hyperbolic Truchet Tilings: First Steps Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA Outline A brief

More information

MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY. Timeline. 10 minutes Exercise session: Introducing curved spaces

MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY. Timeline. 10 minutes Exercise session: Introducing curved spaces MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY Timeline 10 minutes Introduction and History 10 minutes Exercise session: Introducing curved spaces 5 minutes Talk: spherical lines and polygons 15 minutes

More information

WUSTL Math Circle Sept 27, 2015

WUSTL Math Circle Sept 27, 2015 WUSTL Math Circle Sept 7, 015 The K-1 geometry, as we know it, is based on the postulates in Euclid s Elements, which we take for granted in everyday life. Here are a few examples: 1. The distance between

More information

A Family of Butterfly Patterns Inspired by Escher Douglas Dunham University of Minnesota Duluth Duluth, Minnesota

A Family of Butterfly Patterns Inspired by Escher Douglas Dunham University of Minnesota Duluth Duluth, Minnesota 15 th International Conference on Geometry and Graphics A Family of Butterfly Patterns Inspired by Escher Douglas Dunham University of Minnesota Duluth Duluth, Minnesota Outline Families of patterns -

More information

Board Tiling, Chocolate Breaking with a Hint of Fibonacci. Part I By Harry Main-Luu

Board Tiling, Chocolate Breaking with a Hint of Fibonacci. Part I By Harry Main-Luu Board Tiling, Chocolate Breaking with a Hint of Fibonacci Part I By Harry Main-Luu General Overview Part 1: Tiling a Plane Part 2: Tiling a Board Part 3: Breaking and Sharing Chocolate Some overarching

More information

Hyperbolic Geometry. Thomas Prince. Imperial College London. 21 January 2017

Hyperbolic Geometry. Thomas Prince. Imperial College London. 21 January 2017 Hyperbolic Geometry Thomas Prince Imperial College London 21 January 2017 Thomas Prince (Imperial College London) Hyperbolic Planes 21 January 2017 1 / 31 Introducing Geometry What does the word geometry

More information

Patterned Triply Periodic Polyhedra

Patterned Triply Periodic Polyhedra Patterned Triply Periodic Polyhedra Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/ ddunham/

More information

Creating Escher-Style Tessellations

Creating Escher-Style Tessellations Creating Escher-Style Tessellations Focus on After this lesson, you will be able to... create tessellations from combinations of regular and irregular polygons describe the tessellations in terms of the

More information

Symmetry: The Quarterly of the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry) Volume 9, Numbers 2 o 4.

Symmetry: The Quarterly of the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry) Volume 9, Numbers 2 o 4. Symmetry: The Quarterly of the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry) Volume 9, Numbers 2 o 4. i998 Symmetry: Culture and Sctence 199 Vol 9, Nos. 2-4, 199-205,

More information

Escher and Coxeter. A Mathematical Conversation

Escher and Coxeter. A Mathematical Conversation A Mathematical Conversation 5 June 2017 M. C. Escher (1898 1972) Hand with Reflecting Sphere (Lithograph, 1935) Education I Early education in Arnhem. I Admitted 1919 to School for Architecture and Decorative

More information

Lesson 10. Unit 3. Creating Designs. Transformational Designs. Reflection

Lesson 10. Unit 3. Creating Designs. Transformational Designs. Reflection Lesson 10 Transformational Designs Creating Designs M.C. Escher was an artist that made remarkable pieces of art using geometric transformations. He was first inspired by the patterns in mosaic tiles.

More information

The Use of Repeating Patterns to Teach Hyperbolic Geometry Concepts

The Use of Repeating Patterns to Teach Hyperbolic Geometry Concepts The Use of Repeating Patterns to Teach Hyperbolic Geometry Concepts Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web

More information

SubTile 2013, Marseille. An Algorithm to Create Hyperbolic Escher Tilings

SubTile 2013, Marseille. An Algorithm to Create Hyperbolic Escher Tilings SubTile 2013, Marseille An Algorithm to Create Hyperbolic Escher Tilings Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA Email: ddunham@d.umn.edu Web: http://www.d.umn.edu/~ddunham

More information

Joint Mathematics Meetings 2014

Joint Mathematics Meetings 2014 Joint Mathematics Meetings 2014 Patterns with Color Symmetry on Triply Periodic Polyhedra Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA Outline Background Triply periodic polyhedra

More information

Tessellations. Irena Swanson Reed College, Portland, Oregon. MathPath, Lewis & Clark College, Portland, Oregon, 24 July 2018

Tessellations. Irena Swanson Reed College, Portland, Oregon. MathPath, Lewis & Clark College, Portland, Oregon, 24 July 2018 Tessellations Irena Swanson Reed College, Portland, Oregon MathPath, Lewis & Clark College, Portland, Oregon, 24 July 2018 What is a tessellation? A tiling or a tessellation of the plane is a covering

More information

Bending Circle Limits

Bending Circle Limits Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture Bending Circle Limits Vladimir Bulatov Corvallis Oregon, USA info@bulatov.org Abstract M.C.Escher s hyperbolic tessellations

More information

274 Curves on Surfaces, Lecture 5

274 Curves on Surfaces, Lecture 5 274 Curves on Surfaces, Lecture 5 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 5 Ideal polygons Previously we discussed three models of the hyperbolic plane: the Poincaré disk, the upper half-plane,

More information

ISAMA 2011 Columbia College, Chicago IL. Enumerations of Hyperbolic Truchet Tiles Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA

ISAMA 2011 Columbia College, Chicago IL. Enumerations of Hyperbolic Truchet Tiles Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA ISAMA 2011 Columbia College, Chicago IL Enumerations of Hyperbolic Truchet Tiles Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA Outline A brief history of Truchet tilings Examples

More information

Grade 7/8 Math Circles November 3/4, M.C. Escher and Tessellations

Grade 7/8 Math Circles November 3/4, 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 information

Quantitative Literacy: Thinking Between the Lines

Quantitative Literacy: Thinking Between the Lines Quantitative Literacy: Thinking Between the Lines Crauder, Evans, Johnson, Noell Chapter 9: Geometry 2013 W. H. Freeman & Co. 1 Lesson Plan Perimeter, area, and volume: How do I measure? Proportionality

More information

Aspects of Geometry. Finite models of the projective plane and coordinates

Aspects of Geometry. Finite models of the projective plane and coordinates Review Sheet There will be an exam on Thursday, February 14. The exam will cover topics up through material from projective geometry through Day 3 of the DIY Hyperbolic geometry packet. Below are some

More information

Zome Symmetry & Tilings

Zome Symmetry & Tilings Zome Symmetry & Tilings Tia Baker San Francisco State tiab@mail.sfsu.edu 1 Introduction Tessellations also known as tilings are a collection of polygons that fill the plane with no overlaps or gaps. There

More information

Mathematics As A Liberal Art

Mathematics As A Liberal Art Math 105 Fall 2015 BY: 2015 Ron Buckmire Mathematics As A Liberal Art Class 26: Friday November 13 Fowler 302 MWF 10:40am- 11:35am http://sites.oxy.edu/ron/math/105/15/ Euclid, Geometry and the Platonic

More information

Bridges 2011 University of Coimbra, Portugal. Hyperbolic Truchet Tilings Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA

Bridges 2011 University of Coimbra, Portugal. Hyperbolic Truchet Tilings Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA Bridges 2011 University of Coimbra, Portugal Hyperbolic Truchet Tilings Douglas Dunham University of Minnesota Duluth Duluth, Minnesota USA Outline A brief history of Truchet tilings Truchet s investigation

More information

2. Draw a non-isosceles triangle. Now make a template of this triangle out of cardstock or cardboard.

2. Draw a non-isosceles triangle. Now make a template of this triangle out of cardstock or cardboard. Tessellations The figure at the left shows a tiled floor. Because the floor is entirely covered by the tiles we call this arrangement a tessellation of the plane. A regular tessellation occurs when: The

More information

Straight and Angle on Non-Planar Surfaces: Non-Euclidean Geometry Introduction A module in the Algebra Project high school curriculum

Straight and Angle on Non-Planar Surfaces: Non-Euclidean Geometry Introduction A module in the Algebra Project high school curriculum Straight and Angle on Non-Planar Surfaces: Non-Euclidean Geometry Introduction A module in the Algebra Project high school curriculum David W. Henderson, lead writer Notes to teachers: pg 2 NE1. Straight

More information

Artistic Patterns in Hyperbolic Geometry

Artistic Patterns in Hyperbolic Geometry Artistic Patterns in Hyperbolic Geometry Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-2496, USA E-mail: ddunha.m.(qd. umn. edu BRIDGES Mathematical Connections

More information

Optimal Möbius Transformation for Information Visualization and Meshing

Optimal Möbius Transformation for Information Visualization and Meshing Optimal Möbius Transformation for Information Visualization and Meshing Marshall Bern Xerox Palo Alto Research Ctr. David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science

More information

Tessellations: Wallpapers, Escher & Soccer Balls. Robert Campbell

Tessellations: Wallpapers, Escher & Soccer Balls. Robert Campbell Tessellations: Wallpapers, Escher & Soccer Balls Robert Campbell Tessellation Examples What Is What is a Tessellation? A Tessellation (or tiling) is a pattern made by copies of one or

More information

6.3 Poincare's Theorem

6.3 Poincare's Theorem Figure 6.5: The second cut. for some g 0. 6.3 Poincare's Theorem Theorem 6.3.1 (Poincare). Let D be a polygon diagram drawn in the hyperbolic plane such that the lengths of its edges and the interior angles

More information

SHAPE AND STRUCTURE. Shape and Structure. An explanation of Mathematical terminology

SHAPE AND STRUCTURE. Shape and Structure. An explanation of Mathematical terminology Shape and Structure An explanation of Mathematical terminology 2005 1 POINT A dot Dots join to make lines LINE A line is 1 dimensional (length) A line is a series of points touching each other and extending

More information

Helpful Hint When you are given a frieze pattern, you may assume that the pattern continues forever in both directions Notes: Tessellations

Helpful Hint When you are given a frieze pattern, you may assume that the pattern continues forever in both directions Notes: Tessellations A pattern has translation symmetry if it can be translated along a vector so that the image coincides with the preimage. A frieze pattern is a pattern that has translation symmetry along a line. Both of

More information

This is a tessellation.

This is a tessellation. This is a tessellation. What shapes do you see? Describe them. How are the shapes alike? How are the shapes different? POM Do the Tessellation P 1 What happens at the corners (vertices) of the shapes?

More information

Repeating Hyperbolic Pattern Algorithms Special Cases

Repeating Hyperbolic Pattern Algorithms Special Cases Repeating Hyperbolic Pattern Algorithms Special Cases Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

6.2 Classification of Closed Surfaces

6.2 Classification of Closed Surfaces Table 6.1: A polygon diagram 6.1.2 Second Proof: Compactifying Teichmuller Space 6.2 Classification of Closed Surfaces We saw that each surface has a triangulation. Compact surfaces have finite triangulations.

More information

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 16 & 17 2018 Non-Euclidean Geometry and the Globe (Euclidean) Geometry Review:

More information

Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 1 AND

Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 1 AND Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 1 AND Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 2 WHAT YOU WILL LEARN Transformational geometry,

More information

Middle School Geometry. Session 3

Middle School Geometry. Session 3 Middle School Geometry Session 3 Topic Transformational Geometry: Tessellations Activity Name Sums of the Measures of Angles of Triangles Do Congruent Triangles Tessellate? Do Congruent Quadrilaterals

More information

Visualization of Hyperbolic Tessellations

Visualization of Hyperbolic Tessellations Visualization of Hyperbolic Tessellations Jakob von Raumer April 9, 2013 KARLSRUHE INSTITUTE OF TECHNOLOGY KIT University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association

More information

MATH 113 Section 9.2: Symmetry Transformations

MATH 113 Section 9.2: Symmetry Transformations MATH 113 Section 9.2: Symmetry Transformations Prof. Jonathan Duncan Walla Walla University Winter Quarter, 2008 Outline 1 What is Symmetry 2 Types of Symmetry Reflective Symmetry Rotational Symmetry Translational

More information

08. Non-Euclidean Geometry 1. Euclidean Geometry

08. Non-Euclidean Geometry 1. Euclidean Geometry 08. Non-Euclidean Geometry 1. Euclidean Geometry The Elements. ~300 B.C. ~100 A.D. Earliest existing copy 1570 A.D. First English translation 1956 Dover Edition 13 books of propositions, based on 5 postulates.

More information

Elementary Planar Geometry

Elementary Planar Geometry Elementary Planar Geometry What is a geometric solid? It is the part of space occupied by a physical object. A geometric solid is separated from the surrounding space by a surface. A part of the surface

More information

Main Idea: classify polygons and determine which polygons can form a tessellation.

Main Idea: classify polygons and determine which polygons can form a tessellation. 10 8: Polygons and Tesselations Main Idea: classify polygons and determine which polygons can form a tessellation. Vocabulary: polygon A simple closed figure in a plane formed by three or more line segments

More information

Geometry Vocabulary Math Fundamentals Reference Sheet Page 1

Geometry Vocabulary Math Fundamentals Reference Sheet Page 1 Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and

More information

INTRODUCTION TO 3-MANIFOLDS

INTRODUCTION TO 3-MANIFOLDS INTRODUCTION TO 3-MANIFOLDS NIK AKSAMIT As we know, a topological n-manifold X is a Hausdorff space such that every point contained in it has a neighborhood (is contained in an open set) homeomorphic to

More information

Boardworks Ltd KS3 Mathematics. S1 Lines and Angles

Boardworks Ltd KS3 Mathematics. S1 Lines and Angles 1 KS3 Mathematics S1 Lines and Angles 2 Contents S1 Lines and angles S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1.3 Calculating angles S1.4 Angles in polygons 3 Lines In Mathematics,

More information

The National Strategies Secondary Mathematics exemplification: Y8, 9

The National Strategies Secondary Mathematics exemplification: Y8, 9 Mathematics exemplification: Y8, 9 183 As outcomes, Year 8 pupils should, for example: Understand a proof that the sum of the angles of a triangle is 180 and of a quadrilateral is 360, and that the exterior

More information

Course Number: Course Title: Geometry

Course Number: Course Title: Geometry Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line

More information

What makes geometry Euclidean or Non-Euclidean?

What makes geometry Euclidean or Non-Euclidean? What makes geometry Euclidean or Non-Euclidean? I-1. Each two distinct points determine a line I-2. Three noncollinear points determine a plane The 5 Axioms of I-3. If two points lie in a plane, then any

More information

Geometry and Gravitation

Geometry and Gravitation Chapter 15 Geometry and Gravitation 15.1 Introduction to Geometry Geometry is one of the oldest branches of mathematics, competing with number theory for historical primacy. Like all good science, its

More information

Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons

Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons Robert W. Fathauer Tessellations Tempe, AZ 85281, U.S.A. E-mail: tessella@futureone.com Abstract Two infinite families

More information

The Interplay Between Hyperbolic Symmetry and History

The Interplay Between Hyperbolic Symmetry and History The Interplay Between Hyperbolic Symmetry and History Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

PLC Papers Created For:

PLC Papers Created For: PLC Papers Created For: Year 10 Topic Practice Papers: Polygons Polygons 1 Grade 4 Look at the shapes below A B C Shape A, B and C are polygons Write down the mathematical name for each of the polygons

More information

Chapter 20 Tilings For All Practical Purposes: Effective Teaching Chapter Briefing Chapter Topics to the Point Tilings with Regular Polygons

Chapter 20 Tilings For All Practical Purposes: Effective Teaching Chapter Briefing Chapter Topics to the Point Tilings with Regular Polygons Chapter 20 Tilings For All Practical Purposes: Effective Teaching With this day and age of technology, most students are adept at using E-mail as a form of communication. Many institutions automatically

More information

A FAMILY OF THREE ELEMENT M.C. ESCHER PATTERNS

A FAMILY OF THREE ELEMENT M.C. ESCHER PATTERNS A FAMILY OF THREE ELEMENT M.C. ESCHER PATTERNS Douglas J. DUNHAM University of Minnesota Duluth, USA ABSTRACT: In 1952, the Dutch artist M.C. Escher created his striking Notebook Drawing 85. It is a repeating

More information

Geometry Vocabulary. acute angle-an angle measuring less than 90 degrees

Geometry Vocabulary. acute angle-an angle measuring less than 90 degrees Geometry Vocabulary acute angle-an angle measuring less than 90 degrees angle-the turn or bend between two intersecting lines, line segments, rays, or planes angle bisector-an angle bisector is a ray that

More information

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 16 & 17 2018 Non-Euclidean Geometry and the Globe (Euclidean) Geometry Review:

More information

Math 311. Trees Name: A Candel CSUN Math

Math 311. Trees Name: A Candel CSUN Math 1. A simple path in a graph is a path with no repeated edges. A simple circuit is a circuit without repeated edges. 2. Trees are special kinds of graphs. A tree is a connected graph with no simple circuits.

More information

DATE PERIOD. Lesson Reading Guide

DATE PERIOD. Lesson Reading Guide NAME DATE PERIOD Lesson Reading Guide Get Ready for the Lesson Read the introduction at the top of page 316 in your textbook. Write your answers below. 1. Predict the number of triangles and the sum of

More information

Worksheet 30: Wednesday April 22 Tessselations: Tiling The Plane

Worksheet 30: Wednesday April 22 Tessselations: Tiling The Plane Definition Worksheet 30: Wednesday April 22 Tessselations: Tiling The Plane A tiling of the plane or tesselation is a pattern that covers the plane with non-overlapping figures A periodic tiling is one

More information

Teaching diary. Francis Bonahon University of Southern California

Teaching diary. Francis Bonahon University of Southern California Teaching diary In the Fall 2010, I used the book Low-dimensional geometry: from euclidean surfaces to hyperbolic knots as the textbook in the class Math 434, Geometry and Transformations, at USC. Most

More information

3D Hyperbolic Tiling and Horosphere Cross Section

3D Hyperbolic Tiling and Horosphere Cross Section 3D Hyperbolic Tiling and Horosphere Cross Section Vladimir Bulatov, Shapeways Joint AMS/MAA meeting San Diego, January 10, 2018 Inversive Geometry Convenient container to work with 3 dimensional hyperbolic

More information

Describe Plane Shapes

Describe Plane Shapes Lesson 12.1 Describe Plane Shapes You can use math words to describe plane shapes. point an exact position or location line endpoints line segment ray a straight path that goes in two directions without

More information

Lesson 1: Creating T- Spline Forms. In Samples section of your Data Panel, browse to: Fusion 101 Training > 03 Sculpt > 03_Sculpting_Introduction.

Lesson 1: Creating T- Spline Forms. In Samples section of your Data Panel, browse to: Fusion 101 Training > 03 Sculpt > 03_Sculpting_Introduction. 3.1: Sculpting Sculpting in Fusion 360 allows for the intuitive freeform creation of organic solid bodies and surfaces by leveraging the T- Splines technology. In the Sculpt Workspace, you can rapidly

More information

Transformation, tessellation and symmetry line symmetry

Transformation, tessellation and symmetry line symmetry Transformation, tessellation and symmetry line symmetry Reflective or line symmetry describes mirror image, when one half of a shape or picture matches the other exactly. The middle line that divides the

More information

And Now From a New Angle Special Angles and Postulates LEARNING GOALS

And Now From a New Angle Special Angles and Postulates LEARNING GOALS And Now From a New Angle Special Angles and Postulates LEARNING GOALS KEY TERMS. In this lesson, you will: Calculate the complement and supplement of an angle. Classify adjacent angles, linear pairs, and

More information

An Algorithm to Generate Repeating Hyperbolic Patterns

An Algorithm to Generate Repeating Hyperbolic Patterns An Algorithm to Generate Repeating Hyperbolic Patterns Douglas Dunham Department of Computer Science University of innesota, Duluth Duluth, N 5581-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

"Unpacking the Standards" 4th Grade Student Friendly "I Can" Statements I Can Statements I can explain why, when and how I got my answer.

Unpacking the Standards 4th Grade Student Friendly I Can Statements I Can Statements I can explain why, when and how I got my answer. 0406.1.1 4th Grade I can explain why, when and how I got my answer. 0406.1.2 I can identify the range of an appropriate estimate. I can identify the range of over-estimates. I can identify the range of

More information

H.Geometry Chapter 7 Definition Sheet

H.Geometry Chapter 7 Definition Sheet Section 7.1 (Part 1) Definition of: - A mapping of points in a figure to points in a resulting figure - Manipulating an original figure to get a new figure - The original figure - The resulting figure

More information

An Interactive Java Program to Generate Hyperbolic Repeating Patterns Based on Regular Tessellations Including Hyperbolic Lines and Equidistant Curves

An Interactive Java Program to Generate Hyperbolic Repeating Patterns Based on Regular Tessellations Including Hyperbolic Lines and Equidistant Curves An Interactive Java Program to Generate Hyperbolic Repeating Patterns Based on Regular Tessellations Including Hyperbolic Lines and Equidistant Curves A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE

More information

What is a... Manifold?

What is a... Manifold? What is a... Manifold? Steve Hurder Manifolds happens all the time! We just have to know them when we see them. Manifolds have dimension, just like Euclidean space: 1-dimension is the line, 2-dimension

More information

Glossary of dictionary terms in the AP geometry units

Glossary of dictionary terms in the AP geometry units Glossary of dictionary terms in the AP geometry units affine linear equation: an equation in which both sides are sums of terms that are either a number times y or a number times x or just a number [SlL2-D5]

More information

Section 12.1 Translations and Rotations

Section 12.1 Translations and Rotations Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry. We look at two types of isometries in this section: translations and rotations. Translations A

More information

Fractal Tilings Based on Dissections of Polyhexes

Fractal Tilings Based on Dissections of Polyhexes To be published in the Proceedings of Bridges 2005 Fractal Tilings Based on Dissections of Polyhexes Robert W. Fathauer Tessellations Company 3913 E. Bronco Trail Phoenix, AZ 85044, USA E-mail: tessellations@cox.net

More information

YEAR 9 SPRING TERM PROJECT POLYGONS and SYMMETRY

YEAR 9 SPRING TERM PROJECT POLYGONS and SYMMETRY YEAR 9 SPRING TERM PROJECT POLYGONS and SYMMETRY Focus of the Project These investigations are all centred on the theme polygons and symmetry allowing students to develop their geometric thinking and reasoning

More information

Networks as Manifolds

Networks as Manifolds Networks as Manifolds Isabella Thiesen Freie Universitaet Berlin Abstract The aim of this project is to identify the manifolds corresponding to networks that are generated by simple substitution rules

More information

Curriki Geometry Glossary

Curriki Geometry Glossary Curriki Geometry Glossary The following terms are used throughout the Curriki Geometry projects and represent the core vocabulary and concepts that students should know to meet Common Core State Standards.

More information

Dgp _ lecture 2. Curves

Dgp _ lecture 2. Curves Dgp _ lecture 2 Curves Questions? This lecture will be asking questions about curves, their Relationship to surfaces, and how they are used and controlled. Topics of discussion will be: Free form Curves

More information

Worksheet 29: Friday November 20 Tessellations: Tiling The Plane

Worksheet 29: Friday November 20 Tessellations: Tiling The Plane Definition Worksheet 29: Friday November 20 Tessellations: Tiling The Plane A tiling of the plane or tesselation is a pattern that covers the plane with non-overlapping figures A periodic tiling is one

More information

Objective: Students will

Objective: Students will Please read the entire PowerPoint before beginning. Objective: Students will (1) Understand the concept of and the process of making tessellations. (2) Create tessellations using: Rotation, Translation,

More information

We can use square dot paper to draw each view (top, front, and sides) of the three dimensional objects:

We can use square dot paper to draw each view (top, front, and sides) of the three dimensional objects: Unit Eight Geometry Name: 8.1 Sketching Views of Objects When a photo of an object is not available, the object may be drawn on triangular dot paper. This is called isometric paper. Isometric means equal

More information

A simple problem that has a solution that is far deeper than expected!

A simple problem that has a solution that is far deeper than expected! The Water, Gas, Electricity Problem A simple problem that has a solution that is far deeper than expected! Consider the diagram below of three houses and three utilities: water, gas, and electricity. Each

More information

acute angle An angle with a measure less than that of a right angle. Houghton Mifflin Co. 2 Grade 5 Unit 6

acute angle An angle with a measure less than that of a right angle. Houghton Mifflin Co. 2 Grade 5 Unit 6 acute angle An angle with a measure less than that of a right angle. Houghton Mifflin Co. 2 Grade 5 Unit 6 angle An angle is formed by two rays with a common end point. Houghton Mifflin Co. 3 Grade 5 Unit

More information

EXPERIENCING GEOMETRY

EXPERIENCING GEOMETRY EXPERIENCING GEOMETRY EUCLIDEAN AND NON-EUCLIDEAN WITH HISTORY THIRD EDITION David W. Henderson Daina Taimina Cornell University, Ithaca, New York PEARSON Prentice Hall Upper Saddle River, New Jersey 07458

More information

TESSELLATION. For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art. M.C.

TESSELLATION. For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art. M.C. TESSELLATION For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art. M.C. Escher Activity 1: Guessing the lesson Doc. 1 Word Cloud 1) What do you

More information

Angle, symmetry and transformation

Angle, symmetry and transformation Terms Illustrations Definition Acute angle An angle greater than 0 and less than 90. Alternate angles Where two straight lines are cut by a third, as in the diagrams, the angles d and f (also c and e)

More information

Patterns on Triply Periodic Uniform Polyhedra

Patterns on Triply Periodic Uniform Polyhedra Patterns on Triply Periodic Uniform Polyhedra Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

Image Sampling and Quantisation

Image Sampling and Quantisation Image Sampling and Quantisation Introduction to Signal and Image Processing Prof. Dr. Philippe Cattin MIAC, University of Basel 1 of 46 22.02.2016 09:17 Contents Contents 1 Motivation 2 Sampling Introduction

More information

Hyperbolic Spirals and Spiral Patterns

Hyperbolic Spirals and Spiral Patterns ISAMA The International Society of the Arts, Mathematics, and Architecture BRIDGES Mathematical Connections in Art, Music, and Science Hyperbolic Spirals and Spiral Patterns Douglas Dunham Department of

More information

A Quick Introduction to Non-Euclidean Geometry. A Tiling of the Poincare Plane From Geometry: Plane and Fancy, David Singer, page 61.

A Quick Introduction to Non-Euclidean Geometry. A Tiling of the Poincare Plane From Geometry: Plane and Fancy, David Singer, page 61. A Quick Introduction to Non-Euclidean Geometry A Tiling of the Poincare Plane From Geometry: Plane and Fancy, David Singer, page 61. Dr. Robert Gardner Presented at Science Hill High School March 22, 2006

More information

Image Sampling & Quantisation

Image Sampling & Quantisation Image Sampling & Quantisation Biomedical Image Analysis Prof. Dr. Philippe Cattin MIAC, University of Basel Contents 1 Motivation 2 Sampling Introduction and Motivation Sampling Example Quantisation Example

More information

Slope, Distance, Midpoint

Slope, Distance, Midpoint Line segments in a coordinate plane can be analyzed by finding various characteristics of the line including slope, length, and midpoint. These values are useful in applications and coordinate proofs.

More information

Counting Vertices in Isohedral Tilings

Counting Vertices in Isohedral Tilings Counting Vertices in Isohedral Tilings John Choi Nicholas Pippenger, Advisor Arthur Benjamin, Reader May, 01 Department of Mathematics Copyright c 01 John Choi. The author grants Harvey Mudd College and

More information

LESSON SUMMARY. Properties of Shapes

LESSON SUMMARY. Properties of Shapes LESSON SUMMARY CXC CSEC MATHEMATICS UNIT Seven: Geometry Lesson 13 Properties of Shapes Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume 1 and 2. (Some helpful exercises and page numbers

More information

Grade 6 Math Circles February 19th/20th. Tessellations

Grade 6 Math Circles February 19th/20th. Tessellations Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles February 19th/20th Tessellations Introduction to Tessellations tessellation is a

More information

ACT Math and Science - Problem Drill 11: Plane Geometry

ACT Math and Science - Problem Drill 11: Plane Geometry ACT Math and Science - Problem Drill 11: Plane Geometry No. 1 of 10 1. Which geometric object has no dimensions, no length, width or thickness? (A) Angle (B) Line (C) Plane (D) Point (E) Polygon An angle

More information