Surfaces. 14 April Surfaces 14 April /29

Transcription

1 Surfaces 14 April 2014 Surfaces 14 April /29

2 Last Week Last week, when we discussed graph theory, we saw that the maximum colors any map might need depends on the surface on which the map is drawn. If we draw a map on a piece of paper, or on a sphere, we need at most 4 colors. However, we need up to 7 if we draw a map on a torus (a doughnut). We also saw Euler s formula, which relates the number of vertices, edges, and faces of a planar graph, or a solid. It turns out that we saw a special case of a result which has to do with all surfaces. This week we will look at examples of surfaces, how to build examples, and what can we do to surfaces without changing its basic properties. Surfaces 14 April /29

3 Two Simple Surfaces Two surfaces with which we are familiar are the plane and the sphere. Surfaces 14 April /29

4 The Torus We ve also seen the torus, which is a pretty familiar shape. The second picture shows how to build the torus with little rectangles. We ll think about this more on Wednesday. Surfaces 14 April /29

5 The Möbius Strip Escher s Ants Surfaces 14 April /29

6 Escher s Möbius Birds Surfaces 14 April /29

7 Building a Möbius Strip To build a Möbius strip, take a strip of paper. Twist the strip and then glue or tape the ends together. Surfaces 14 April /29

8 The Möbius strip has the unexpected property that it has only one side. If you start walking on blue, eventually you will end up on white. So, while the strip of paper we use to build the Möbius strip has two sides, the result has only one. Surfaces 14 April /29

9 Topology To construct some other surfaces and to understand what we get, we need to discuss some ideas involving how we can change a shape. Specifically, we ll discuss the mathematical area of topology. Very roughly, topology is concerned with shapes, and the properties of shapes that do not change when smoothly deforming them. Such deformations include stretching and shrinking but not tearing or gluing. However, we will see that tearing and gluing are useful in producing new shapes. Surfaces 14 April /29

10 To start to make some sense of this, consider the following two shapes: If we had the line segment made from play dough, we could bend it to make the semicircle. These shapes are called (topologically) equivalent. To a topologist, they are the same shape. Similarly, a cube and a sphere are equivalent. Here is a video showing how to turn a sphere into a cube. Surfaces 14 April /29

11 Mathematicians can t Tell the Difference Between a Coffee Cup and a Donut An example of a smooth deformation is in the following video. YouTube video of turning a coffee cup into a donut Surfaces 14 April /29

12 Two shapes which are not equivalent are the line segment and circle. We can t turn one into the other just by smooth deformations. We can make the line segment from the circle by cutting the circle. Alternatively, we can make the circle from the line segment by first smoothly deforming the line segment, then gluing the ends together. Surfaces 14 April /29

13 Building Surfaces From a Rectangle Some surfaces can be described by starting with a rectangle and gluing certain sides together. We can turn a rectangle into a cube, and then into a sphere. We can take a rectangle, smoothly deform it to the following shape, then bend it into a cube, by gluing edges. Surfaces 14 April /29

14 Building the Möbius Strip From a Rectangle We ve already seen how to build a Möbius strip from a rectangle. The following picture indicates what and how we glue. The arrows indicate that we glue the top left corner with the bottom right. We must glue so that the arrows go in the same direction. Surfaces 14 April /29

15 Clicker Question What kind of surface would we get if we glued the left and right sides together by putting the top corners together? A a sphere B a torus C a cylinder D a cone E something else Surfaces 14 April /29

16 Answer You d get a cylinder (without top or bottom). One way to see this is to take a piece of paper and put two edges together. Surfaces 14 April /29

17 Building the Torus From a Rectangle To build the torus, we glue the top and bottom sides together, and glue the left and right sides together. The arrows indicate that we keep the sides orientation unchanged. That means, for example, we glue the top end of the left side to the top end of the right side. Here is a video showing how to construct a torus, and here is a second video. Surfaces 14 April /29

18 Clicker Question What surface would we get if we glue the two sides of a rectangle induced in the picture: A A sphere B A cylinder C A cone D A torus E Something else Surfaces 14 April /29

19 Answer We d get a cone, at least after we deform the result some after doing the gluing. Surfaces 14 April /29

20 Exotic Surfaces There are surfaces which cannot exist in 3-dimensional space, but nonetheless exist mathematically (in 4-space). We ll see a couple of these, the projective plane and the Klein bottle. We ll construct these surfaces by taking a rectangle and gluing edges, but in a different way than we did for the Möbius strip and the torus. Surfaces 14 April /29

21 The Projective Plane The projective plane originated from artists in the Renaissance, who started to use projective geometry, or perspective geometry, for drawing. Previously artists did not have good methods for drawing buildings or landscapes and demonstrating depth perception. Without proper techniques, one can draw geometrically impossible pictures. Escher, who understood very well perspective drawing, new how to violate the rules. Surfaces 14 April /29

22 Escher s Waterfall Surfaces 14 April /29

23 Building the Projective Plane From a Rectangle The projective plane can be constructed from a rectangle by gluing according to the following picture. Gluing the left and right sides is the same as for the Möbius strip, but then we do extra gluing. If you try to do this by gluing the opposing edges of the Möbius strip, you ll find you can t do it without tearing. This is because the projective plane cannot exist in 3 dimensions. Surfaces 14 April /29

24 The Klein Bottle The Klein bottle has the unusual property that it has neither inside nor outside. It isn t really a bottle in the usual sense! Like the projective plane, it does not exist in 3 dimensions. In spite of this, people draw pictures to represent it. These pictures aren t really of the Klein bottle, but they attempt to give some idea of what is the shape. Surfaces 14 April /29

25 Here are some additional pictures of the Klein bottle. Surfaces 14 April /29

26 Building a Klein Bottle From a Rectangle The Klein bottle can be formed from gluing sides of a rectangle. Surfaces 14 April /29

27 Construction of a Klein Bottle The following video gives a representation of how to build a Klein bottle from a rectangle. It isn t possible for the Klein bottle to live in 3 dimensions, so sadly we cannot show the actual Klein bottle in a video. It can live in 4 dimensions. However, this doesn t help us picture what it looks like. Surfaces 14 April /29

28 Next Time On Wednesday we will continue to talk about surfaces, and see some properties of surfaces which do not change when we do a smooth deformation. We will see that these properties allow us to classify and distinguish surfaces. Surfaces 14 April /29