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 make a loop and tape them together. You now have a cylinder. a) Take a colored marker and begin coloring around an edge of the cylinder until you get back to where you started. Are all the edges colored now? If not, take a different color and repeat. In the end, how many edges we call them boundary components are there? b) Let s now count the number of sides: with a colored pen start drawing a line down the middle of a side and continue without lifting until you return to where you started. Is there a line on all sides of the cylinder now? If not, take a different color and repeat. In the end, how many sides does the cylinder have? 2. Use Your Imagination I Imagine taping two long cylinders together at all their boundaries. What kind of shape do you get? Discuss with your neighbors! 3. Explore a Möbius Band First you need to build your own! Get a new strip of paper and, before you tape the ends together, give one end of the strip a half twist. You then have a Möbius band. Let s answer the same questions as above: a) Take a colored marker and begin coloring around an edge of the Möbius band. Repeat as needed. How many boundary components are there? b) Use different colored pens to count the number of sides. How many are there? c) Imagine cutting your Möbius band all around down the center. What do you predict will happen? Now do the actual cutting. How many pieces do you get? How many boundary components do you have now? Is this new object another Möbius band? Why or why not?
d) Cut again along the center. How many pieces do you get now? How many boundary components are there on each? Are these Möbius bands or other manifolds that you have seen before? Discuss with your neighbors! August Ferdinand Möbius (1790-1868) 4. A new manifold from more twists Start with a new strip of paper (preferably one of the thinner ones) and let s build a new manifold. Now, before you tape the ends together, give one end of the strip three half twists. a) How many boundary components are there? b) How many sides? c) Is the object you just built a Möbius band? Why or why not? d) Cut right down the center and answer the same questions: How many boundary components? How many sides? Try to describe the new object. 5. Use Your Imagination II Imagine taping two Möbius bands together along their boundaries. Would you get the same thing as when taping two cylinders? Discuss it with your neighbors!
6. Manifold Factory We saw that by twisting, taping and cutting, you can build new manifolds. Experiment a bit and come up with rules of how many boundary components/sides you will get depending on how many twists you make or how often you cut etc. 7. Connect the Dots I Take a letter sized piece of paper. At random, draw a bunch of dots across the page don t go too crazy cause later we ll ask you to count them! Next, connect your dots according to the following rules: - lines between dots may not cross - every dot on your page must be connected to every other dot through a sequence of lines. You have now separated your page into various regions. Using mathematical language, we will now denote the dots as vertices, the lines as edges and the regions as faces. a) As a first exercise, count the number of vertices, edges and faces in your picture. (Don't forget to count the outside as a region too.) Then take the sum of the number of vertices and faces and compare with the number of edges. What do you observe? Compare your answer with that of your neighbor. Is it the same? Is it different? Switch your paper with your neighbor and check his/her answer! b) Let s investigate this in simple pictures: o Below, draw a dot and then a line segment from there with another dot at the end of the segment. Now count the number of vertices, edges and faces.
o Now, draw another line segment that starts at one of the dots and goes in any direction. Add a dot at the end and count again. o Finally, join the two end dots by an edge to make a triangle. Count again. c) Now let s go the other way! Start with the following picture and remove all external edges and vertices one by one until you are left with only the inside triangle. In each step, do the counting.
d) It s time for you to make a conjecture! For any connected graph in the plane, the number v e + f =. This amazing fact was first observed about 350 years ago by René Descartes, however, it took another 200 years until Leonhard Euler came up with a rigorous proof. In his honor we now call it Euler s Formula. Leonhard Euler (1707-1783) 8. Use Your Imagination III. One of your faces above was not like the others. To make the outside face into a real face, imagine gluing together all the edges and corners of the paper, so there are no more boundaries in the outside face. You get a manifold we have seen before, with no boundaries. What is this manifold?
9. Connect the Dots II. The Euler formula you discovered in part 7 is known as the Euler characteristic of the sphere. (Why the sphere? See part 8.) The other manifolds you have created today also have an Euler characteristic. a) Draw vertices, edges, and faces on a strip of paper. Note for each face to be a real face, with edges for sides, you will have to have an edge along each piece of the boundary. b) Count the number of edges, vertices, and faces. Compute v e + f =. This is the Euler characteristic of a disk. c) Now imagine taping the boundaries of the strip of paper together, as when you create a cylinder, Möbius band, torus, or Klein bottle. Notice some of your edges and vertices will be taped together and any such edges and vertices should be counted only once! Thus, you will get new numbers for v and e. Try to compute v e + f for the following: Euler characteristic of a Möbius band: Euler characteristic of a Torus: Euler characteristic of a Klein bottle:
10.Take Home Challenges - Try to follow in Euler s footsteps and prove that your conjecture in number 7 is correct! That is, show that for any connected graph in the plane the number of vertices plus the number of faces is always two more than the number of edges, no matter what picture you draw. (Hints: Use different triangulations and describe why they give the same Euler characteristic. In different situations, how does adding a vertex or an edge possibly change the number of faces?) - Calculate the Euler Characteristic (that is the number v-e+f) for each of the five regular solids, the tetrahedron, cube, octahedron, dodecahedron and icosahedron!