Euler Characteristic

Size: px
Start display at page:

Download "Euler Characteristic"

Transcription

1 Euler Characteristic

2 Face Classification set_view(gl_render); set_scene(gl_render); glgetdoublev(gl_modelview_matrix, modelview_matrix1); glgetdoublev(gl_projection_matrix, projection_matrix1); glgetintegerv(gl_viewport, viewport1); gluproject((gldouble) poly->tlist[i]->center.entry[0], (GLdouble) poly->tlist[i]- >center.entry[1], (GLdouble)poly->tlist[i]->center.entry[2], modelview_matrix1, projection_matrix1, viewport1, &face_norm_start.entry[0], &face_norm_start.entry[1], &face_norm_start.entry[2]);

3 Topics Today Platonic solids Corner structure

4 Topics Today Platonic solids Corner structure

5 Platonic Solids shiftingsands.com.au/platonicsolids.html

6 Platonic Solids shiftingsands.com.au/platonicsolids.html

7 Platonic Solids davidf.faricy.net/polyhedra/platonic_solids.html

8 Platonic Solids

9 Are We Missing Anything?

10 Are We Missing Anything? All regular polyhedron must be convex.

11 Are We Missing Anything? All regular polyhedron must be convex. When n=3?

12 Are We Missing Anything? All regular polyhedron must be convex. When n=3? m=3: tetrahedron

13 Are We Missing Anything? All regular polyhedron must be convex. When n=3? m=3: tetrahedron m=4: octahedron

14 Are We Missing Anything? All regular polyhedron must be convex. When n=3? m=3: tetrahedron m=4: octahedron m=5: icosahedron

15 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4?

16 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4? m=3, Hexahedron (cube)

17 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4? When n=5?

18 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4? When n=5? m=3: dodecahedron

19 Are We Missing Anything? For example, is it possible to have m=3 and n=3 but f<>4?

20 Euler Characteristics L=V-E+F=2 Why?

21 Elementary Collapse on Edges

22 Elementary Collapse for Edges

23 Elementary Collapse for Edges

24 Elementary Collapse for Edges

25 Elementary Collapse for Edges

26 Elementary Collapse for Edges

27 Elementary Collapse for Edges

28 Elementary Collapse for Edges

29 Elementary Collapse for Edges

30 Elementary Collapse for Edges

31 Elementary Collapse for Edges

32 Elementary Collapse for Edges

33 Elementary Collapse for Edges V=E for closed a simple planar curve.

34 Elementary Collapse for Edges V=E for closed a simple planar curve. What about 3D surfaces?

35 Elementary Collapse for Edges V=E for closed a simple planar curve. What about 3D surfaces? Need to consider merging faces.

36 Elementary Collapse for Faces

37 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C

38 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C

39 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C

40 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C

41 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C

42 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C

43 Proof of Euler s Theorem on a Cube H G H G E E F C A B A B C

44 Proof of Euler s Theorem on a Cube G G E E F C A B A B C

45 Proof of Euler s Theorem on a Cube G G F C A B A B C

46 Proof of Euler s Theorem on a Cube F C A B A B

47 Proof of Euler s Theorem on a Cube F B B

48 Proof of Euler s Theorem on a Cube F

49 Proof of Euler s Theorem on a Cube F F

50 Another Look H G H G E E F D C A B A B C

51 Another Look H G H G E F E F D C D C A B A B

52 Dual of a Hexahedron H G E F A D B C

53 Dual of a Hexahedron H G E F A D C

54 Dual of a Hexahedron H G E F A D C

55 Dual of a Hexahedron H G E F A D C

56 Dual of a Hexahedron

57 Dual Shape What is the dual of Octahedron Icosahedron Dodecahedron Tetrahedron Does the dual operation change the Euler characteristic? What operations will change it?

58 Are We Missing Anything? For example, is it possible to have m=3 and n=3 but f<>4?

59 Are We Missing Anything? For example, is it possible to have m=3 and n=3 but f<>4? No, we are not.

60 Proof v-e+f=2 n=number of edges in the polyong m=number of faces (edges) meeting at a vertex

61 Proof v-e+f=2 n=number of edges in the polygon m=number of faces (edges) meeting at a vertex We have 2e=nf

62 Proof v-e+f=2 n=number of edges in the polygon m=number of faces (edges) meeting at a vertex We have 2e=nf mv=nf

63 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2

64 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2 3f/3-3f/2+f=2

65 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2 3f/3-3f/2+f=2 f/2=2

66 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2 3f/3-3f/2+f=2 f/2=2 f=4

67 Any questions?

Platonic Solids and the Euler Characteristic

Platonic Solids and the Euler Characteristic Platonic Solids and the Euler Characteristic Keith Jones Sanford Society, SUNY Oneonta September 2013 What is a Platonic Solid? A Platonic Solid is a 3-dimensional object with flat faces and straight edges

More information

Five Platonic Solids: Three Proofs

Five Platonic Solids: Three Proofs Five Platonic Solids: Three Proofs Vincent J. Matsko IMSA, Dodecahedron Day Workshop 18 November 2011 Convex Polygons convex polygons nonconvex polygons Euler s Formula If V denotes the number of vertices

More information

Example: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices:

Example: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices: 11.1: Space Figures and Cross Sections Polyhedron: solid that is bounded by polygons Faces: polygons that enclose a polyhedron Edge: line segment that faces meet and form Vertex: point or corner where

More information

Week 7 Convex Hulls in 3D

Week 7 Convex Hulls in 3D 1 Week 7 Convex Hulls in 3D 2 Polyhedra A polyhedron is the natural generalization of a 2D polygon to 3D 3 Closed Polyhedral Surface A closed polyhedral surface is a finite set of interior disjoint polygons

More information

Euler Characteristic

Euler Characteristic Euler Characteristic Rebecca Robinson May 15, 2007 Euler Characteristic Rebecca Robinson 1 PLANAR GRAPHS 1 Planar graphs v = 5, e = 4, f = 1 v e + f = 2 v = 6, e = 7, f = 3 v = 4, e = 6, f = 4 v e + f

More information

Ma/CS 6b Class 9: Euler s Formula

Ma/CS 6b Class 9: Euler s Formula Ma/CS 6b Class 9: Euler s Formula By Adam Sheffer Recall: Plane Graphs A plane graph is a drawing of a graph in the plane such that the edges are noncrossing curves. 1 Recall: Planar Graphs The drawing

More information

Lecture 19: Introduction To Topology

Lecture 19: Introduction To Topology Chris Tralie, Duke University 3/24/2016 Announcements Group Assignment 2 Due Wednesday 3/30 First project milestone Friday 4/8/2016 Welcome to unit 3! Table of Contents The Euler Characteristic Spherical

More information

Math 311. Polyhedra Name: A Candel CSUN Math

Math 311. Polyhedra Name: A Candel CSUN Math 1. A polygon may be described as a finite region of the plane enclosed by a finite number of segments, arranged in such a way that (a) exactly two segments meets at every vertex, and (b) it is possible

More information

1. CONVEX POLYGONS. Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D.

1. CONVEX POLYGONS. Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. 1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third

More information

Question. Why is the third shape not convex?

Question. Why is the third shape not convex? 1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third

More information

Classifying 3D Shapes

Classifying 3D Shapes Classifying 3D Shapes Middle School Texas Essential Knowledge and Skills (TEKS) Math 5.4B Algebraic reasoning The student applies mathematical process standards to develop concepts of expressions and equations.

More information

One simple example is that of a cube. Each face is a square (=regular quadrilateral) and each vertex is connected to exactly three squares.

One simple example is that of a cube. Each face is a square (=regular quadrilateral) and each vertex is connected to exactly three squares. Berkeley Math Circle Intermediate I, 1/23, 1/20, 2/6 Presenter: Elysée Wilson-Egolf Topic: Polygons, Polyhedra, Polytope Series Part 1 Polygon Angle Formula Let s start simple. How do we find the sum of

More information

Chapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings

Chapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings Chapter 12 and 11.1 Planar graphs, regular polyhedra, and graph colorings Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 12: Planar Graphs Math 184A / Fall 2017 1 / 45 12.1 12.2. Planar graphs Definition

More information

We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.

We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the

More information

Grade VIII. Mathematics Geometry Notes. #GrowWithGreen

Grade VIII. Mathematics Geometry Notes. #GrowWithGreen Grade VIII Mathematics Geometry Notes #GrowWithGreen Polygons can be classified according to their number of sides (or vertices). The sum of all the interior angles of an n -sided polygon is given by,

More information

Lesson/Unit Plan Name: Platonic Solids Using geometric nets to explore Platonic solids and discovering Euler s formula.

Lesson/Unit Plan Name: Platonic Solids Using geometric nets to explore Platonic solids and discovering Euler s formula. Grade Level/Course: Grade 6 Lesson/Unit Plan Name: Platonic Solids Using geometric nets to explore Platonic solids and discovering Euler s formula. Rationale/Lesson Abstract: An activity where the students

More information

Section 9.4. Volume and Surface Area. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 9.4. Volume and Surface Area. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 9.4 Volume and Surface Area What You Will Learn Volume Surface Area 9.4-2 Volume Volume is the measure of the capacity of a three-dimensional figure. It is the amount of material you can put inside

More information

INTRODUCTION TO GRAPH THEORY. 1. Definitions

INTRODUCTION TO GRAPH THEORY. 1. Definitions INTRODUCTION TO GRAPH THEORY D. JAKOBSON 1. Definitions A graph G consists of vertices {v 1, v 2,..., v n } and edges {e 1, e 2,..., e m } connecting pairs of vertices. An edge e = (uv) is incident with

More information

Euler's formula and Platonic solids

Euler's formula and Platonic solids University of Washington Euler's formula and Platonic solids Name: David Clark, Kelsey Kyllo, Kurt Maugerle, Yue Yuan Zhang Course Number: Math 445 Professor: Julia Pevtsova Date: 2013/06/03 Table of Contents:

More information

Convex Hulls (3D) O Rourke, Chapter 4

Convex Hulls (3D) O Rourke, Chapter 4 Convex Hulls (3D) O Rourke, Chapter 4 Outline Polyhedra Polytopes Euler Characteristic (Oriented) Mesh Representation Polyhedra Definition: A polyhedron is a solid region in 3D space whose boundary is

More information

Explore Solids

Explore Solids 1212.1 Explore Solids Surface Area and Volume of Solids 12.2 Surface Area of Prisms and Cylinders 12.3 Surface Area of Pyramids and Cones 12.4 Volume of Prisms and Cylinders 12.5 Volume of Pyramids and

More information

11.4 Three-Dimensional Figures

11.4 Three-Dimensional Figures 11. Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called

More information

Week 9: Planar and non-planar graphs. 7 and 9 November, 2018

Week 9: Planar and non-planar graphs. 7 and 9 November, 2018 (1/27) MA284 : Discrete Mathematics Week 9: Planar and non-planar graphs http://www.maths.nuigalway.ie/ niall/ma284/ 7 and 9 November, 2018 1 Planar graphs and Euler s formula 2 Non-planar graphs K 5 K

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

3D shapes introduction

3D shapes introduction 3D shapes introduction 2D shapes have 2 dimensions width and height. They re flat. height 3D shapes have 3 dimensions height, width and depth. Sometimes we call them solids. When we draw them, we often

More information

Major Facilities for Mathematical Thinking and Understanding. (2) Vision, spatial sense and kinesthetic (motion) sense.

Major Facilities for Mathematical Thinking and Understanding. (2) Vision, spatial sense and kinesthetic (motion) sense. Major Facilities for Mathematical Thinking and Understanding. (2) Vision, spatial sense and kinesthetic (motion) sense. Left brain Right brain Hear what you see. See what you hear. Mobius Strip http://www.metacafe.com/watch/331665/

More information

Intermediate Math Circles Fall 2018 Patterns & Counting

Intermediate Math Circles Fall 2018 Patterns & Counting Intermediate Math Circles Fall 2018 Patterns & Counting Michael Miniou The Centre for Education in Mathematics and Computing Faculty of Mathematics University of Waterloo December 5, 2018 Michael Miniou

More information

Jordan Curves. A curve is a subset of IR 2 of the form

Jordan Curves. A curve is a subset of IR 2 of the form Jordan Curves A curve is a subset of IR 2 of the form α = {γ(x) : x [0,1]}, where γ : [0,1] IR 2 is a continuous mapping from the closed interval [0,1] to the plane. γ(0) and γ(1) are called the endpoints

More information

CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics CS 2336 Discrete Mathematics Lecture 15 Graphs: Planar Graphs 1 Outline What is a Planar Graph? Euler Planar Formula Platonic Solids Five Color Theorem Kuratowski s Theorem 2 What is a Planar Graph? Definition

More information

THE PLATONIC SOLIDS BOOK DAN RADIN

THE PLATONIC SOLIDS BOOK DAN RADIN THE PLATONIC SOLIDS BOOK DAN RADIN Copyright 2008 by Daniel R. Radin All rights reserved. Published by CreateSpace Publishing 3-D renderings were created on a thirteen-year-old Macintosh computer using

More information

Unit I: Euler's Formula (and applications).

Unit I: Euler's Formula (and applications). Unit I: Euler's Formula (and applications). We define a roadmap to be a nonempty finite collection of possibly curvedlil1e segments in a piane, each with exactly two endpoints, such that if any pair of

More information

Edge Unfoldings of Platonic Solids Never Overlap

Edge Unfoldings of Platonic Solids Never Overlap Edge Unfoldings of Platonic Solids Never Overlap Takashi Horiyama (Saitama Univ.) joint work with Wataru Shoji 1 Unfolding Simple polygon unfolded by cutting along the surface of a polyhedron Two kinds

More information

Rectangular prism. The two bases of a prism. bases

Rectangular prism. The two bases of a prism. bases Page 1 of 8 9.1 Solid Figures Goal Identify and name solid figures. Key Words solid polyhedron base face edge The three-dimensional shapes on this page are examples of solid figures, or solids. When a

More information

Polyhedra. Kavitha d/o Krishnan

Polyhedra. Kavitha d/o Krishnan Polyhedra Kavitha d/o Krishnan Supervisor: Associate Professor Helmer Aslaksen Department of Mathematics National University of Singapore Semester I 2001/2002 Abstract Introduction The report focuses on

More information

Key Concept Euler s Formula

Key Concept Euler s Formula 11-1 Space Figures and Cross Sections Objectives To recognize polyhedrons and their parts To visualize cross sections of space figures Common Core State Standards G-GMD.B.4 Identify the shapes of two-dimensional

More information

1 The Platonic Solids

1 The Platonic Solids 1 The We take the celebration of Dodecahedron Day as an opportunity embark on a discussion of perhaps the best-known and most celebrated of all polyhedra the Platonic solids. Before doing so, however,

More information

Multiply using the grid method.

Multiply using the grid method. Multiply using the grid method. Learning Objective Read and plot coordinates in all quadrants DEFINITION Grid A pattern of horizontal and vertical lines, usually forming squares. DEFINITION Coordinate

More information

Polygons and Convexity

Polygons and Convexity Geometry Week 4 Sec 2.5 to ch. 2 test Polygons and Convexity section 2.5 convex set has the property that any two of its points determine a segment contained in the set concave set a set that is not convex

More information

Today we will be exploring three-dimensional objects, those that possess length, width, and depth.

Today we will be exploring three-dimensional objects, those that possess length, width, and depth. Lesson 22 Lesson 22, page 1 of 13 Glencoe Geometry Chapter 11.1 3-D figures & Polyhedra Today we will be exploring three-dimensional objects, those that possess length, width, and depth. In Euclidean,

More information

Planar Graphs, Solids, and Surfaces. Planar Graphs 1/28

Planar Graphs, Solids, and Surfaces. Planar Graphs 1/28 Planar Graphs, Solids, and Surfaces Planar Graphs 1/28 Last time we discussed the Four Color Theorem, which says that any map can be colored with at most 4 colors and not have two regions that share a

More information

SMMG September 16 th, 2006 featuring Dr. Jessica Purcell Geometry out of the Paper: An Introduction to Manifolds

SMMG September 16 th, 2006 featuring Dr. Jessica Purcell Geometry out of the Paper: An Introduction to Manifolds 1. Explore a Cylinder SMMG September 16 th, 2006 featuring Dr. Jessica Purcell Geometry out of the Paper: An Introduction to Manifolds Take a strip of paper. Bring the two ends of the strip together to

More information

The Volume of a Platonic Solid

The Volume of a Platonic Solid University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-007 The Volume of a Platonic Solid Cindy Steinkruger

More information

REGULAR TILINGS. Hints: There are only three regular tilings.

REGULAR TILINGS. Hints: There are only three regular tilings. REGULAR TILINGS Description: A regular tiling is a tiling of the plane consisting of multiple copies of a single regular polygon, meeting edge to edge. How many can you construct? Comments: While these

More information

Map-colouring with Polydron

Map-colouring with Polydron Map-colouring with Polydron The 4 Colour Map Theorem says that you never need more than 4 colours to colour a map so that regions with the same colour don t touch. You have to count the region round the

More information

1 Appendix to notes 2, on Hyperbolic geometry:

1 Appendix to notes 2, on Hyperbolic geometry: 1230, notes 3 1 Appendix to notes 2, on Hyperbolic geometry: The axioms of hyperbolic geometry are axioms 1-4 of Euclid, plus an alternative to axiom 5: Axiom 5-h: Given a line l and a point p not on l,

More information

A Physical Proof for Five and Only Five Regular Solids

A Physical Proof for Five and Only Five Regular Solids A Physical Proof for Five and Only Five Regular Solids Robert McDermott Center for High Performance Computing University of Utah Salt Lake City, Utah, 84112, USA E-mail: mcdermott@chpc.utah.edu Abstract

More information

Jitterbug Defined Polyhedra: The Shape and Dynamics of Space

Jitterbug Defined Polyhedra: The Shape and Dynamics of Space Jitterbug Defined Polyhedra: The Shape and Dynamics of Space by Robert W. Gray rwgray@rwgrayprojects.com Oct. 25-26, 2001 This presentation may be found at: http://www.rwgrayprojects.com/oswegooct2001/presentation/prsentationweb.html

More information

Triangles and Squares David Eppstein, ICS Theory Group, April 20, 2001

Triangles and Squares David Eppstein, ICS Theory Group, April 20, 2001 Triangles and Squares David Eppstein, ICS Theory Group, April 20, 2001 Which unit-side-length convex polygons can be formed by packing together unit squares and unit equilateral triangles? For instance

More information

Week 9: Planar and non-planar graphs. 1st and 3rd of November, 2017

Week 9: Planar and non-planar graphs. 1st and 3rd of November, 2017 (1/26) MA284 : Discrete Mathematics Week 9: Planar and non-planar graphs http://www.maths.nuigalway.ie/~niall/ma284/ 1st and 3rd of November, 2017 1 Recall... planar graphs and Euler s formula 2 Non-planar

More information

Answer Key: Three-Dimensional Cross Sections

Answer Key: Three-Dimensional Cross Sections Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection

More information

Euclid forgot to require that the vertices should be the same, so his definition includes the deltahedra.

Euclid forgot to require that the vertices should be the same, so his definition includes the deltahedra. 2 1. What is a Platonic solid? What is a deltahedron? Give at least one example of a deltahedron that is t a Platonic solid. What is the error Euclid made when he defined a Platonic solid? Solution: A

More information

Jordan Curves. A curve is a subset of IR 2 of the form

Jordan Curves. A curve is a subset of IR 2 of the form Jordan Curves A curve is a subset of IR 2 of the form α = {γ(x) : x [0, 1]}, where γ : [0, 1] IR 2 is a continuous mapping from the closed interval [0, 1] to the plane. γ(0) and γ(1) are called the endpoints

More information

Platonic Polyhedra and How to Construct Them

Platonic Polyhedra and How to Construct Them Platonic Polyhedra and How to Construct Them Tarun Biswas June 17, 2016 The platonic polyhedra (or platonic solids) are convex regular polyhedra that have identical regular polygons as faces They are characterized

More information

(1) Page #2 26 Even. (2) Page 596 #1 14. (3) Page #15 25 ; FF #26 and 28. (4) Page 603 #1 18. (5) Page #19 26

(1) Page #2 26 Even. (2) Page 596 #1 14. (3) Page #15 25 ; FF #26 and 28. (4) Page 603 #1 18. (5) Page #19 26 Geometry/Trigonometry Unit 10: Surface Area and Volume of Solids Notes Name: Date: Period: # (1) Page 590 591 #2 26 Even (2) Page 596 #1 14 (3) Page 596 597 #15 25 ; FF #26 and 28 (4) Page 603 #1 18 (5)

More information

LESSON. Bigger and Bigger. Years 5 to 9. Enlarging Figures to Construct Polyhedra Nets

LESSON. Bigger and Bigger. Years 5 to 9. Enlarging Figures to Construct Polyhedra Nets LESSON 4 Bigger and Bigger Years 5 to 9 Enlarging Figures to Construct Polyhedra Nets This lesson involves students using their MATHOMAT to enlarge regular polygons to produce nets of selected polyhedra,

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

Computer Graphics using OpenGL, 3 rd Edition F. S. Hill, Jr. and S. Kelley

Computer Graphics using OpenGL, 3 rd Edition F. S. Hill, Jr. and S. Kelley Computer Graphics using OpenGL, 3 rd Edition F. S. Hill, Jr. and S. Kelley Chapter 6.1-3 Modeling Shapes with Polygonal Meshes S. M. Lea University of North Carolina at Greensboro 2007, Prentice Hall 3D

More information

Platonic Solids. Jennie Sköld. January 21, Karlstad University. Symmetries: Groups Algebras and Tensor Calculus FYAD08

Platonic Solids. Jennie Sköld. January 21, Karlstad University. Symmetries: Groups Algebras and Tensor Calculus FYAD08 Platonic Solids Jennie Sköld January 21, 2015 Symmetries: Groups Algebras and Tensor Calculus FYAD08 Karlstad University 1 Contents 1 What are Platonic Solids? 3 2 Symmetries in 3-Space 5 2.1 Isometries

More information

Planar Graphs and Surfaces. Graphs 2 1/58

Planar Graphs and Surfaces. Graphs 2 1/58 Planar Graphs and Surfaces Graphs 2 1/58 Last time we discussed the Four Color Theorem, which says that any map can be colored with at most 4 colors and not have two regions that share a border having

More information

A Study of the Rigidity of Regular Polytopes

A Study of the Rigidity of Regular Polytopes A Study of the Rigidity of Regular Polytopes A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Helene

More information

Zipper Unfoldings of Polyhedral Complexes

Zipper Unfoldings of Polyhedral Complexes Zipper Unfoldings of Polyhedral Complexes Erik D. Demaine Martin L. Demaine Anna Lubiw Arlo Shallit Jonah L. Shallit Abstract We explore which polyhedra and polyhedral complexes can be formed by folding

More information

of Nebraska - Lincoln

of Nebraska - Lincoln University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of

More information

Sonobe Origami for enriching understanding of geometric concepts in three dimensions. DONNA A. DIETZ American University Washington, D.C.

Sonobe Origami for enriching understanding of geometric concepts in three dimensions. DONNA A. DIETZ American University Washington, D.C. Sonobe Origami for enriching understanding of geometric concepts in three dimensions DONNA A. DIETZ American University Washington, D.C. Donna Dietz, American University Sonobe Origami for enriching understanding

More information

CARDSTOCK MODELING Math Manipulative Kit. Student Activity Book

CARDSTOCK MODELING Math Manipulative Kit. Student Activity Book CARDSTOCK MODELING Math Manipulative Kit Student Activity Book TABLE OF CONTENTS Activity Sheet for L.E. #1 - Getting Started...3-4 Activity Sheet for L.E. #2 - Squares and Cubes (Hexahedrons)...5-8 Activity

More information

7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

More information

Non-flat tilings with flat tiles

Non-flat tilings with flat tiles Non-flat tilings with flat tiles Rinus Roelofs Sculptor Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: rinus@rinusroelofs.nl www.rinusroelofs.nl Abstract In general a tiling is considered to be a

More information

Principles and Standards for School Mathematics. Content Standards. Process Standards. Emphasis across the Grades. Principles

Principles and Standards for School Mathematics. Content Standards. Process Standards. Emphasis across the Grades. Principles 1 Navigating through Geometry Grades 3-5 Principles and Standards for School Mathematics Presented by Dr. Karol L. Yeatts Navigations Writer Navigating through Algebra Grades 3-5 Navigating through Number

More information

3.D. The Platonic solids

3.D. The Platonic solids 3.D. The Platonic solids The purpose of this addendum to the course notes is to provide more information about regular solid figures, which played an important role in Greek mathematics and philosophy.

More information

State if each pair of triangles is similar. If so, state how you know they are similar (AA, SAS, SSS) and complete the similarity statement.

State if each pair of triangles is similar. If so, state how you know they are similar (AA, SAS, SSS) and complete the similarity statement. Geometry 1-2 est #7 Review Name Date Period State if each pair of triangles is similar. If so, state how you know they are similar (AA, SAS, SSS) and complete the similarity statement. 1) Q R 2) V F H

More information

Zipper Unfoldings of Polyhedral Complexes. Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit

Zipper Unfoldings of Polyhedral Complexes. Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit Zipper Unfoldings of Polyhedral Complexes Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit 1 Unfolding Polyhedra Durer 1400 s Durer, 1498 snub cube 2 Unfolding Polyhedra Octahedron all

More information

Part Two Development of Single Solids - The Five Plutonic Solids

Part Two Development of Single Solids - The Five Plutonic Solids 1 Part Two Development of Single Solids - The Five Plutonic Solids We will now proceed to learn different topics of descriptive geometry using AutoCAD 2D tools and settings. Throughout this and subsequent

More information

Abstract Construction Projects and the Imagination

Abstract Construction Projects and the Imagination Abstract Construction Projects and the Imagination Hands-on projects for understanding abstract mathematical concepts through the use of polyhedral models and planar designs The 3-dimensional projects

More information

Ready To Go On? Skills Intervention 10-1 Solid Geometry

Ready To Go On? Skills Intervention 10-1 Solid Geometry 10A Find these vocabulary words in Lesson 10-1 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 10-1 Solid Geometry face edge vertex prism cylinder pyramid cone cube net cross

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

Section 3.4 Basic Results of Graph Theory

Section 3.4 Basic Results of Graph Theory 1 Basic Results of Graph Theory Section 3.4 Basic Results of Graph Theory Purpose of Section: To formally introduce the symmetric relation of a (undirected) graph. We introduce such topics as Euler Tours,

More information

Chapter 11 Part 2. Measurement of Figures and Solids

Chapter 11 Part 2. Measurement of Figures and Solids Chapter 11 Part 2 Measurement of Figures and Solids 11.5 Explore Solids Objective: Identify Solids Essential Question: When is a solid a polyhedron? Using properties of polyhedra A is a solid that is bounded

More information

Math 462: Review questions

Math 462: Review questions Math 462: Review questions Paul Hacking 4/22/10 (1) What is the angle between two interior diagonals of a cube joining opposite vertices? [Hint: It is probably quickest to use a description of the cube

More information

The Construction of Uniform Polyhedron with the aid of GeoGebra

The Construction of Uniform Polyhedron with the aid of GeoGebra The Construction of Uniform Polyhedron with the aid of GeoGebra JiangPing QiuFaWen 71692686@qq.com 3009827@qq.com gifted Department High-school northeast yucai school northeast yucai school 110179 110179

More information

Connected Holes. Rinus Roelofs Sculptor Lansinkweg AL Hengelo The Netherlands

Connected Holes. Rinus Roelofs Sculptor Lansinkweg AL Hengelo The Netherlands Connected Holes Rinus Roelofs Sculptor Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: rinus@rinusroelofs.nl www.rinusroelofs.nl Abstract It is possible to make interwoven structures by using two

More information

Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Combinatorics: The Fine Art of Counting Week Nine Menu raphic rub This week s colorful menu includes a number of graphic staples. Feel free to taste a small portion of each item or pick your favorite dish

More information

Tiling of Sphere by Congruent Pentagons

Tiling of Sphere by Congruent Pentagons Tiling of Sphere by Congruent Pentagons Min Yan September 9, 2017 webpage for further reading: http://www.math.ust.hk/ mamyan/research/urop.shtml We consider tilings of the sphere by congruent pentagons.

More information

Graphics and Visualization

Graphics and Visualization International University Bremen Spring Semester 2006 Recap Representing graphic objects by homogenous points and vectors Using affine transforms to modify objects Using projections to display objects

More information

Assignments are handed in on Tuesdays in even weeks. Deadlines are:

Assignments are handed in on Tuesdays in even weeks. Deadlines are: Tutorials at 2 3, 3 4 and 4 5 in M413b, on Tuesdays, in odd weeks. i.e. on the following dates. Tuesday the 28th January, 11th February, 25th February, 11th March, 25th March, 6th May. Assignments are

More information

Class Generated Review Sheet for Math 213 Final

Class Generated Review Sheet for Math 213 Final Class Generated Review Sheet for Math 213 Final Key Ideas 9.1 A line segment consists of two point on a plane and all the points in between them. Complementary: The sum of the two angles is 90 degrees

More information

Algorithms. Graphs. Algorithms

Algorithms. Graphs. Algorithms Algorithms Graphs Algorithms Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Algorithms 1 Graphs Vertices: Nodes, points, computers, users, items,...

More information

Mathematics Concepts 2 Exam 1 Version 4 21 September 2018

Mathematics Concepts 2 Exam 1 Version 4 21 September 2018 Mathematics Concepts 2 Exam 1 Version 4 21 September 2018 Name: Permissible Aides: The small ruler distributed by the proctor Prohibited: Class Notes Class Handouts Study Guides and Materials The Book

More information

Mathematics Concepts 2 Exam 1 Version 2 22 September 2017

Mathematics Concepts 2 Exam 1 Version 2 22 September 2017 Mathematics Concepts 2 Exam 1 Version 2 22 September 2017 Name: Permissible Aides: The small ruler distributed by the proctor Prohibited: Class Notes Class Handouts Study Guides and Materials The Book

More information

Algorithms: Graphs. Amotz Bar-Noy. Spring 2012 CUNY. Amotz Bar-Noy (CUNY) Graphs Spring / 95

Algorithms: Graphs. Amotz Bar-Noy. Spring 2012 CUNY. Amotz Bar-Noy (CUNY) Graphs Spring / 95 Algorithms: Graphs Amotz Bar-Noy CUNY Spring 2012 Amotz Bar-Noy (CUNY) Graphs Spring 2012 1 / 95 Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Amotz

More information

Math 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi

Math 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi Math 210 Manifold III, Spring 2018 Euler Characteristics of Surfaces Hirotaka Tamanoi 1. Euler Characteristic of Surfaces Leonhard Euler noticed that the number v of vertices, the number e of edges and

More information

D A S O D A. Identifying and Classifying 3-D Objects. Examples

D A S O D A. Identifying and Classifying 3-D Objects. Examples Identifying Classifying 3-D Objects Examples Have you noticed that many of the products we purchase come in packages or boxes? Take a look at the products below. A) Did you notice that all the sides or

More information

3D shapes types and properties

3D shapes types and properties 3D shapes types and properties 1 How do 3D shapes differ from 2D shapes? Imagine you re giving an explana on to a younger child. What would you say and/or draw? Remember the surfaces of a 3D shape are

More information

Math 489 Project 1: Explore a Math Problem L. Hogben 1 Possible Topics for Project 1: Explore a Math Problem draft 1/13/03

Math 489 Project 1: Explore a Math Problem L. Hogben 1 Possible Topics for Project 1: Explore a Math Problem draft 1/13/03 Math 489 Project 1: Explore a Math Problem L. Hogben 1 Possible Topics for Project 1: Explore a Math Problem draft 1/13/03 Number Base and Regularity We use base 10. The Babylonians used base 60. Discuss

More information

Curves, Surfaces and Recursive Subdivision

Curves, Surfaces and Recursive Subdivision Department of Computer Sciences Graphics Fall 25 (Lecture ) Curves, Surfaces and Recursive Subdivision Conics: Curves and Quadrics: Surfaces Implicit form arametric form Rational Bézier Forms Recursive

More information

The Game of Criss-Cross

The Game of Criss-Cross Chapter 5 The Game of Criss-Cross Euler Characteristic ( ) Overview. The regions on a map and the faces of a cube both illustrate a very natural sort of situation: they are each examples of regions that

More information

Operations on Maps. Mircea V. Diudea. Faculty of Chemistry and Chemical Engineering Babes-Bolyai

Operations on Maps. Mircea V. Diudea. Faculty of Chemistry and Chemical Engineering Babes-Bolyai Operations on Maps Mircea V. Diudea Faculty of Chemistry and Chemical Engineering Babes-Bolyai Bolyai University 400028 Cluj,, ROMANIA diudea@chem.ubbcluj.ro 1 Contents Cage Building by Map Operations

More information

Part 1. Twice the number of edges = 2 9 = 18. Thus the Handshaking lemma... The sum of the vertex degrees = twice the number of edges holds.

Part 1. Twice the number of edges = 2 9 = 18. Thus the Handshaking lemma... The sum of the vertex degrees = twice the number of edges holds. MT6 Examination 16 Q1 (a) Part 1 Part1 Solutions (b) egree Sequence (,,,,, ) (c) Sum of the vertex degrees = + + + + + = 18 Twice the number of edges = 9 = 18. Thus the Handshaking lemma... The sum of

More information

Flavor of Computational Geometry. Convex Hull in 2D. Shireen Y. Elhabian Aly A. Farag University of Louisville

Flavor of Computational Geometry. Convex Hull in 2D. Shireen Y. Elhabian Aly A. Farag University of Louisville Flavor of Computational Geometry Convex Hull in 2D Shireen Y. Elhabian Aly A. Farag University of Louisville February 2010 Agenda Introduction Definitions of Convexity and Convex Hulls Naïve Algorithms

More information

Edge-transitive tessellations with non-negative Euler characteristic

Edge-transitive tessellations with non-negative Euler characteristic October, 2009 p. Edge-transitive tessellations with non-negative Euler characteristic Alen Orbanić Daniel Pellicer Tomaž Pisanski Thomas Tucker Arjana Žitnik October, 2009 p. Maps 2-CELL EMBEDDING of a

More information

Computing the Symmetry Groups of the Platonic Solids with the Help of Maple

Computing the Symmetry Groups of the Platonic Solids with the Help of Maple Computing the Symmetry Groups of the Platonic Solids with the Help of Maple In this note we will determine the symmetry groups of the Platonic solids. We will use Maple to help us do this. The five Platonic

More information

Math 213 Student Note Outlines. Sections

Math 213 Student Note Outlines. Sections Math 213 Student Note Outlines Sections 9.2 9.3 11.1 11.2 11.3 9.2 KEY IDEAS, page 1 of 2 Polygon Vertex Angles Sum Regular Polygons: Vertex Angles Congruence Definition Regular Polygons Definition Tessellation

More information