History of Topology. Semester I, Graham Ellis NUI Galway, Ireland

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

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

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

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/

The Fundamental Group, Braids and Circles

The Fundamental Group, Braids and Circles Pete Goetz Mathematics Colloquium Humboldt State University April 17, 2014 Outline 1) Topology 2) Path Homotopy and The Fundamental Group 3) Covering Spaces and

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

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

Outreach Lecture: Polytopes

Outreach Lecture: Polytopes Darrell Allgaier (Grove City College), David Perkinson (Reed College), Sarah Ann Stewart (North Central College), John Thurber (Eastern Oregon University) Abstract This is an

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

Assignment 8; Due Friday, March 10

Assignment 8; Due Friday, March 10 The previous two exercise sets covered lots of material. We ll end the course with two short assignments. This one asks you to visualize an important family of three

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

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

Curvature Berkeley Math Circle January 08, 2013

Curvature Berkeley Math Circle January 08, 2013 Linda Green linda@marinmathcircle.org Parts of this handout are taken from Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman, and Bill

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

Topological Data Analysis - I. Afra Zomorodian Department of Computer Science Dartmouth College

Topological Data Analysis - I Afra Zomorodian Department of Computer Science Dartmouth College September 3, 2007 1 Acquisition Vision: Images (2D) GIS: Terrains (3D) Graphics: Surfaces (3D) Medicine: MRI

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

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

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

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

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

Research in Computational Differential Geomet

Research in Computational Differential Geometry November 5, 2014 Approximations Often we have a series of approximations which we think are getting close to looking like some shape. Approximations Often

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

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

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,

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

POLYTOPES. Grünbaum and Shephard [40] remarked that there were three developments which foreshadowed the modern theory of convex polytopes.

POLYTOPES MARGARET A. READDY 1. Lecture I: Introduction to Polytopes and Face Enumeration Grünbaum and Shephard [40] remarked that there were three developments which foreshadowed the modern theory of

Surfaces and Topology Transcript

Surfaces and Topology Transcript Date: Tuesday, 21 January 2014-1:00PM Location: Museum of London 21 January 2014 Surfaces and Topology Professor Raymond Flood Welcome to the fourth of my lectures this

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

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

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

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

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

GSAC TALK: THE WORD PROBLEM. 1. The 8-8 Tessellation. Consider the following infinite tessellation of the open unit disc. Actually, we want to think

GSAC TALK: THE WORD PROBLEM 1. The 8-8 Tessellation Consider the following infinite tessellation of the open unit disc. Actually, we want to think of the disc as a model for R 2 and we can do that using

THE POINCARÉ CONJECTURE The Mathematics of Smooth Behavior

THE POINCARÉ CONJECTURE The Mathematics of Smooth Behavior Freshman Seminar University of California, Irvine Bernard Russo University of California, Irvine Spring 2015 Bernard Russo (UCI) THE POINCARÉ

CS Object Representation. Aditi Majumder, CS 112 Slide 1

CS 112 - Object Representation Aditi Majumder, CS 112 Slide 1 What is Graphics? Modeling Computer representation of the 3D world Analysis For efficient rendering For catering the model to different applications..

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

1 Euler characteristics

Tutorials: MA342: Tutorial Problems 2014-15 Tuesday, 1-2pm, Venue = AC214 Wednesday, 2-3pm, Venue = AC201 Tutor: Adib Makroon 1 Euler characteristics 1. Draw a graph on a sphere S 2 PROBLEMS in such a

Euler Characteristic

Euler Characteristic Sudesh Kalyanswamy 1 Introduction Euler characteristic is a very important topological property which started out as nothing more than a simple formula involving polyhedra. Euler observed

3D convex hulls. Computational Geometry [csci 3250] Laura Toma Bowdoin College

3D convex hulls Computational Geometry [csci 3250] Laura Toma Bowdoin College Convex Hull in 3D The problem: Given a set P of points in 3D, compute their convex hull convex polyhedron 2D 3D polygon

CS3621 Midterm Solution (Fall 2005) 150 points

CS362 Midterm Solution Fall 25. Geometric Transformation CS362 Midterm Solution (Fall 25) 5 points (a) [5 points] Find the 2D transformation matrix for the reflection about the y-axis transformation (i.e.,

EULER S FORMULA AND THE FIVE COLOR THEOREM

EULER S FORMULA AND THE FIVE COLOR THEOREM MIN JAE SONG Abstract. In this paper, we will define the necessary concepts to formulate map coloring problems. Then, we will prove Euler s formula and apply

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

Surprises in high dimensions. Martin Lotz Galois Group, April 22, 2015

Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015 Ladd Ehlinger Jr. (dir). Flatland, 2007. Life in 2D Life in 2D Edwin A. Abbott. Flatland: A Romance of Many Dimensions, 1884. The novella

Three applications of Euler s formula. Chapter 10

Three applications of Euler s formula Chapter 10 A graph is planar if it can be drawn in the plane R without crossing edges (or, equivalently, on the -dimensional sphere S ). We talk of a plane graph if

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling Welcome to the lectures on computer graphics. We have

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

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

Area of Lattice Point Polygons. Andrejs Treibergs. Wednesday, January 16, 2019

Undergraduate Colloquium: Area of Lattice Point Polygons Andrejs Treibergs University of Utah Wednesday, January 16, 2019 2. Undergraduate Colloquium Lecture on Area of Lattice Point Polygons The URL for

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

MA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces

MA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces David L. Finn Today, we continue our discussion of subdivision surfaces, by first looking in more detail at the midpoint method and

Solutions to the Second Midterm Exam, Math 170, Section 002 Spring 2012

Solutions to the Second Midterm Exam, Math 170, Section 002 Spring 2012 Multiple choice questions. Question 1. Suppose we have a rectangle with one side of length 5 and a diagonal of length 13. What is

The Art Gallery Problem

The Art Gallery Problem Connor Bohlken Nebraska Wesleyan University cbohlken@nebrwesleyan.edu May 11, 2016 Abstract Given a polygonal art gallery of n vertices, how many security guards are necessary to

Lecture 7: Jan 31, Some definitions related to Simplical Complex. 7.2 Topological Equivalence and Homeomorphism

CS 6170 Computational Topology: Topological Data Analysis University of Utah Spring 2017 School of Computing Lecture 7: Jan 31, 2017 Lecturer: Prof. Bei Wang Scribe: Avani Sharma,

Introduction to Solid Modeling

Introduction to Solid Modeling Hongxin Zhang and Jieqing Feng 2007-01-15 State Key Lab of CAD&CG Zhejiang University Contents Solid Representations: An Introduction Wireframe Models Boundary Representations

COMBINATORIAL GEOMETRY

4 TH CLASS STARTER 9/23/14 3 RD CLASS STARTER 9/16/14 If you re allowed to make only a single, planar (flat) cut in order to slice a cube of cheese into two pieces, what shapes can the resulting faces

CS195H Homework 5. Due:March 12th, 2015

CS195H Homework 5 Due:March 12th, 2015 As usual, please work in pairs. Math Stuff For us, a surface is a finite collection of triangles (or other polygons, but let s stick with triangles for now) with

Math 170, Section 002 Spring 2012 Practice Exam 2 with Solutions

Math 170, Section 002 Spring 2012 Practice Exam 2 with Solutions Contents 1 Problems 2 2 Solution key 10 3 Solutions 11 1 1 Problems Question 1: A right triangle has hypothenuse of length 25 in and an

Perimeter Magic Polygons

Perimeter Magic Polygons In, Terrel Trotter, Jr., then a math teacher in Urbana Illinois, published an article called Magic Triangles of Order n. In, he published a follow up article called Perimeter Magic

4.2 Simplicial Homology Groups

4.2. SIMPLICIAL HOMOLOGY GROUPS 93 4.2 Simplicial Homology Groups 4.2.1 Simplicial Complexes Let p 0, p 1,... p k be k + 1 points in R n, with k n. We identify points in R n with the vectors that point

Geometry and Topology SHP Fall 16

Geometry and Topology SHP Fall 16 Henry Liu Last updated: December 10, 2016 Contents 0 About Mathematics 2 1 Surfaces 3 1.1 Examples of surfaces......................... 3 1.2 Equivalence of surfaces........................

COMBINATORIAL METHODS IN ALGEBRAIC TOPOLOGY

COMBINATORIAL METHODS IN ALGEBRAIC TOPOLOGY 1. Geometric and abstract simplicial complexes Let v 0, v 1,..., v k be points in R n. These points determine a hyperplane in R n, consisting of linear combinations

Intersecting Simple Surfaces. Dr. Scott Schaefer

Intersecting Simple Surfaces Dr. Scott Schaefer 1 Types of Surfaces Infinite Planes Polygons Convex Ray Shooting Winding Number Spheres Cylinders 2/66 Infinite Planes Defined by a unit normal n and a point

Geometry and Topology SHP Spring 17

Geometry and Topology SHP Spring 17 Henry Liu Last updated: April 29, 2017 Contents 0 About Mathematics 3 1 Surfaces 4 1.1 Examples of surfaces......................... 4 1.2 Equivalence of surfaces........................

Surfaces Beyond Classification

Chapter XII Surfaces Beyond Classification In most of the textbooks which present topological classification of compact surfaces the classification is the top result. However the topology of 2- manifolds

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.

INTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES

INTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES RACHEL CARANDANG Abstract. This paper provides an overview of the homology groups of a 2- dimensional complex. It then demonstrates a proof of the Invariance

The following is a summary, hand-waving certain things which actually should be proven.

1 Basics of Planar Graphs The following is a summary, hand-waving certain things which actually should be proven. 1.1 Plane Graphs A plane graph is a graph embedded in the plane such that no pair of lines

Conway s Magic Theorem Session Description This session will look at the mathematics behind Conway's Magic Theorem. This theorem can be used to prove that there are 17 plane symmetry (wallpaper) groups

Surfaces from a polygon. Olivier Bernardi Massachusetts Instiitute of Technology

Surfaces from a polygon Olivier Bernardi Massachusetts Instiitute of Technology IAP lecture series 2010 What is it about? We consider surfaces obtained by gluing polygons. What is it about? We consider

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:

4. Simplicial Complexes and Simplicial Homology

MATH41071/MATH61071 Algebraic topology Autumn Semester 2017 2018 4. Simplicial Complexes and Simplicial Homology Geometric simplicial complexes 4.1 Definition. A finite subset { v 0, v 1,..., v r } R n

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

Euler s Theorem. Brett Chenoweth. February 26, 2013

Euler s Theorem Brett Chenoweth February 26, 2013 1 Introduction This summer I have spent six weeks of my holidays working on a research project funded by the AMSI. The title of my project was Euler s

The orientability of small covers and coloring simple polytopes. Nishimura, Yasuzo; Nakayama, Hisashi. Osaka Journal of Mathematics. 42(1) P.243-P.

Title Author(s) The orientability of small covers and coloring simple polytopes Nishimura, Yasuzo; Nakayama, Hisashi Citation Osaka Journal of Mathematics. 42(1) P.243-P.256 Issue Date 2005-03 Text Version

Euler s Formula. Math 123. March 2006

Euler s Formula Math 123 March 2006 1 Purpose Although Euler s Formula is relatively simple to memorize, it is actually a manifestation of a very deep mathematical phenomenon. In this activity, we will

Geometry and Topology SHP Fall 17

Geometry and Topology SHP Fall 17 Henry Liu Last updated: December 16, 2017 Contents 0 About Mathematics 3 1 Surfaces 4 1.1 Examples of surfaces......................... 4 1.2 Equivalence of surfaces........................

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

Classification of Surfaces

Classification of Surfaces 16 April 2014 Classification of Surfaces 16 April 2014 1/29 Last Time On Monday we saw some examples of surfaces and how we can build some by starting with a rectangle and gluing

Hyperbolic structures and triangulations

CHAPTER Hyperbolic structures and triangulations In chapter 3, we learned that hyperbolic structures lead to developing maps and holonomy, and that the developing map is a covering map if and only if the

Bands: A Physical Data Structure to Represent Both Orientable and Non-Orientable 2-Manifold Meshes

Bands: A Physical Data Structure to Represent Both Orientable and Non-Orientable 2-Manifold Meshes Abstract This paper presents a physical data structure to represent both orientable and non-orientable

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

HW Graph Theory Name (andrewid) - X. 1: Draw K 7 on a torus with no edge crossings.

1: Draw K 7 on a torus with no edge crossings. A quick calculation reveals that an embedding of K 7 on the torus is a -cell embedding. At that point, it is hard to go wrong if you start drawing C 3 faces,

Hyperbolic Structures from Ideal Triangulations

Hyperbolic Structures from Ideal Triangulations Craig Hodgson University of Melbourne Geometric structures on 3-manifolds Thurston s idea: We would like to find geometric structures (or metrics) on 3-manifolds

66 III Complexes. R p (r) }.

66 III Complexes III.4 Alpha Complexes In this section, we use a radius constraint to introduce a family of subcomplexes of the Delaunay complex. These complexes are similar to the Čech complexes but differ

UNIT 7 Mensuration Activities

UNIT 7 Mensuration Activities Activities 7.1 Tangram 7.2 Closed Doodles 7.3 Map Colouring 7.4 Euler's Formula 7.5 Square-based Oblique Pyramid 7.6 Klein Cube (3 pages) 7.7 Transforming Polygons 7.8 Tubes

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

8.B. The result of Regiomontanus on tetrahedra

8.B. The result of Regiomontanus on tetrahedra We have already mentioned that Plato s theory that the five regular polyhedra represent the fundamental elements of nature, and in supplement (3.D) to the

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

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

PRIMES Circle: High School Math Enrichment at MIT

PRIMES Circle: High School Math Enrichment at MIT Department of Mathematics, MIT ivogt@mit.edu June 25th, 2015 PRIMES Umbrella organization: research and outreach Four programs: MIT PRIMES: research on

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

Senior Math Circles November 25, 2009 Graph Theory II

University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles November 5, 009 Graph Theory II Planar Graphs A graph that can be drawn in R without

Computer Aided Engineering Design Prof. Anupam Saxena Department of Mechanical Engineering Indian Institute of Technology, Kanpur.

(Refer Slide Time: 00:28) Computer Aided Engineering Design Prof. Anupam Saxena Department of Mechanical Engineering Indian Institute of Technology, Kanpur Lecture - 6 Hello, this is lecture number 6 of

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

Triangulations of hyperbolic 3-manifolds admitting strict angle structures

Triangulations of hyperbolic 3-manifolds admitting strict angle structures Craig D. Hodgson, J. Hyam Rubinstein and Henry Segerman segerman@unimelb.edu.au University of Melbourne January 4 th 2012 Ideal

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

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

Heegaard splittings and virtual fibers

Heegaard splittings and virtual fibers Joseph Maher maher@math.okstate.edu Oklahoma State University May 2008 Theorem: Let M be a closed hyperbolic 3-manifold, with a sequence of finite covers of bounded

The Beautiful Mathematical Explorations of Maryam Mirzakhani

The Beautiful Mathematical Explorations of Maryam Mirzakhani After her untimely death, Maryam Mirzakhani s life is best remembered through her work. By Moira Chas Jan Vondrák Maryam Mirzakhani in 2014.

Mgr. ubomíra Tomková GEOMETRY

GEOMETRY NAMING ANGLES: any angle less than 90º is an acute angle any angle equal to 90º is a right angle any angle between 90º and 80º is an obtuse angle any angle between 80º and 60º is a reflex angle