Author:
Amanda Hager
Subject:
Mathematics
Material Type:
Lesson
Level:
Academic Lower Division
Provider:
University of Texas at Austin
Tags:
  • D2S2
  • Mindset
  • License:
    Creative Commons Attribution
    Language:
    English
    Media Formats:
    Text/HTML, Video

    Education Standards

    4.2 Four special surfaces

    4.2 Four special surfaces

    Overview

    TCCNS CourseMATH 1332: Contemporary Mathematics
    UT Austin CourseM 302: Introduction to Mathematics

    Introduction

    Suggested Resources and Preparation

    Materials and Technology

    • For the instructor: 1" wide strips of construction paper, scissors, tape to create Möbius bands and annuluses.

    Prerequisite Assumptions

    Students need to be familiar with the concepts of dimension and boundary as they are presented in Section 3.4.

    Overview and Student Objectives

    Lesson Length

    50 minutes

    Lesson Objectives

    Students will understand that:

    • Some 2D surfaces are orientable and some are non-orientable.
    • Identification diagrams are a technique we use to work with a object in space by flattening it out on paper (like a map of the globe).

    Students will be able to:

    • Describe how four different kinds of identification diagrams create the annulus, Möbius band, torus, and Klein bottle.
    • Perform experiments on Möbius bands made of paper by cutting or marking on them, documenting their results and making conjectures about mathematical properties. 
    • Determine whether or not an object has edges or is orientable using its identification diagram.

    Four special surfaces

    Annulus and Möbius Band

    Terms

    An annulus is a 2-D surface that is ring-shaped. You can make one by taking the short ends of a rectangle and attaching them without any twists. You can also think of it as a thin toilet-paper tube, or a cylinder with no top and no bottom.

     Blue annulustoilet tissue tube

    Möbius band is a 2-D surface that you get by taking the short ends of a rectangle and attaching them with a half-twist.

    green Möbius band resting on a table

    Image credit: David Benbennick.

    What are identification diagrams?

    Terms

    An identification diagram (sometimes called an edge identification diagram) is a 2-D picture that shows how to assemble a surface by attaching edges to each other. Different types of arrows are used to differentiate between pairs of edges. Diagrams can start with rectangles, hexagons, or any other flat shape.

     

    Torus and Klein Bottle

    Terms

    torus is a 2-D object that looks like a hollow donut.

    orange stacking ring toy resting on a wood table

    Klein bottle is a 2-D object that you can create from a rectangle by attaching one pair of opposite edges without twisting, then attaching the other pair of edges with a half-twist.

    gif of rectangle becoming a tube and then intersecting itself to be come a Klein bottle

     

    TL;DR

    ShapeNo. of dimensionsNo. of boundary piecesNo. of "sides"Identification Diagram
    Annulus222blue rectangle with two down-pointing arrows on left and right edges
    Möbius band211blue rectangle with one down-pointing arrow on left edge and one up-pointing arrow on right edge
    Torus202blue rectangle with two down-pointing arrows on left and right edges, and two right pointing double arrows on top and bottom edges
    Klein bottle201blue rectangle with one down-pointing arrow on left edge and one up-pointing arrow on right edge, and two right pointing double arrows on the top and bottom edges

     

    Quiz Questions

    Question 1

    4

    Question 2

    1, 1, 2, 2

    Question 3

    1, 2

    Question 4

    2

    Question 1

    What is a Möbius band?

    1. It is any strip of paper with the ends glued together.
    2. Any strip of paper with arrows on it that tell you how they attach.
    3. It is a heavy metal band from the 80s.
    4. It is a 2D object that you can make out of a rectangle by putting in a half twist and gluing two ends together.

    Question 2

    Fill in the blanks: A Möbius band has ____ side(s) and ____ edge(s).  An annulus has ____ sides(s) and ____ edge(s).

    Question 3

    Fill in the blanks: If you cut a Möbius band one third of the way from the edge and keep cutting, you get ____ piece(s).  If you cut an annulus one third of the way from the edge and keep cutting as long as possible, you get ____ piece(s).

    Question 4

    Your sister holds a strip of paper.  She gives one end a half twist, then she gives the other end a half twist in the same direction, then she tapes the ends together.  Does she get a Möbius strip?

    1. Yes
    2. No
    3. Impossible to determine.
    4. It depends.

    Homework Questions

    Question 4.2.1

    You can trace an edge with a finger, or you can trace the surface of the paper. If there's one edge, or if your finger is on the other side of the paper after one trip around, then it's a Möbius band.

    Question 4.2.2

    You do not. The resulting object has two sides and two edges, so it can't be a Möbius band.

    Question 4.2.3

    You get two copies of the same object but they're interlinked. These objects are not quite annuluses but they're not Möbius bands either. But they are orientable, meaning they have two sides and two edges.

    Question 4.2.4

    No. If you hold the strips together and try to keep them that way as you build a Möbius band, you find that you have to choose to assemble one or the other. You can't tape both.

    If we are putting the ends of the strips together without twisting, however, we see that the same-colored edges meet up. Two pieces of tape later and we have a pair of annuluses.

    Question 4.2.5

    This diagram is similar to the one for the Klein bottle. Students should draw a rectangle with two edges identified in opposite directions, and two edges left alone. If the left and right edges are identified, then draw a horizontal thin rectangle down the center of the object. That rectangle will be a Möbius band.

    Question 4.2.6

    blue rectangle with up-pointing double arrow on left, down-pointing double arrow on right, and right pointing arrows on top and bottom. Vertical orange rectangle inside of the blue rectangle is highlighted. To the right, a diagram of a Klein bottle is shown with an orange annulus on its gooseneck highlighted

    Question 4.2.7

    blue rectangle with up-pointing double arrow on left, down-pointing double arrow on right, and right pointing arrows on top and bottom. Horizontal orange rectangle inside of the blue rectangle is highlighted.

    Question 4.2.8

    Reflective writing prompt. Responses vary.

    Question 4.2.1

    Give two different techniques for looking at a loop of paper and determining if it is a Möbius band.

    Question 4.2.2

    Take a strip of paper, give one end two half-twists, and tape the ends. Do you get a Möbius band? Why or why not? (Hint: trace with your finger and count how many edges it has).

    Question 4.2.3

    Take a strip of paper, put two half-twists in it, and tape the ends together (you can use the same strip from problem 2). Then cut the paper along its midline all the way around. What do you get? Can you explain why you get that?

    Question 4.2.4

    Hold two differently-colored strips of paper together, say white and blue, then give them a half-twist. Are you able to connect white to white and blue to blue? Is the answer different if you try to keep the pieces of paper together and create annuluses with them?

    Question 4.2.5

    Show on an identification diagram that when you cut a Möbius band all the way around 1/3 of the way from the edge, that the middle section of the Möbius band is still a Möbius band.

    Question 4.2.6

    Prove that you can make two cuts in a Klein bottle and cut out an annulus. Draw a Klein bottle and show the cuts on the Klein bottle, then show an identification diagram for a Klein bottle and show the two cuts there.

    Question 4.2.7

    Prove that you can make one cut in a Klein bottle and cut out a Möbius band. (Hint: it should probably be perpendicular to the cuts in Question 4.2.6.

    Question 4.2.8

    Reflect on today, the day you are completing this assignment. Write down three things you did, said, or learned that made you feel proud. Why are you proud that you did, said, or learned these things?