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 the same color. Graphs 2 2/58

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 the same color. It turns out that this result has something to do with the topology of the plane. If we draw maps on other surfaces, we may need a different number of colors. Graphs 2 2/58

For example, if we draw a map on a torus (a doughnut), we may need up to 7 colors, as the following pictures indicate. http://faculty.smcm.edu/sgoldstine/torus7.html Graphs 2 3/58

For example, if we draw a map on a torus (a doughnut), we may need up to 7 colors, as the following pictures indicate. http://faculty.smcm.edu/sgoldstine/torus7.html Here is an animation of a map on the torus which needs 7 colors. The map on the torus shown above has 7 regions and each region is adjacent to each of the other 6 regions. This is why it needs 7 colors. Graphs 2 3/58

While it may seem that determining the minimum number of colors to color a map drawn on a torus would be much harder than for maps drawn on a plane, it is just the opposite. Graphs 2 4/58

While it may seem that determining the minimum number of colors to color a map drawn on a torus would be much harder than for maps drawn on a plane, it is just the opposite. The Four Color Theorem for maps on a plane took a huge amount of effort, 61 pages in Scientific American, and lots of computer calculation, the result for the torus can be written down in a few pages, and was proved much earlier. Graphs 2 4/58

Planar Graphs The graphs that arise from a map turn out to have the following property: They can be drawn in such a way that no two edges cross. These are called planar graphs. Graphs 2 5/58

Planar Graphs The graphs that arise from a map turn out to have the following property: They can be drawn in such a way that no two edges cross. These are called planar graphs. The pentagon graph we saw the previous class is not a planar graph. This is why it does not represent the graph of a map and why it does not violate the Four Color Theorem. Graphs 2 5/58

Let s consider the following graph with 4 vertices and 6 edges: Graphs 2 6/58

Clicker Question Can we redraw the graph so that the edges do not cross? A Yes B No Graphs 2 7/58

Answer Yes we can redraw it by having one of the diagonal edges drawn outside of the square. Thus, it is a planar graph, even though the original drawing does not indicate so. Graphs 2 8/58

Faces of a Planar Graph Graphs 2 9/58

Faces of a Planar Graph For solids, such as those above, there is a reasonable meaning of vertex, edge, and face. For example, the cube has 8 vertices (or corners), 12 edges, and 6 square faces. Graphs 2 9/58

The cube can be drawn as a graph in multiple ways. Here are two such. One is a pretty typical way to draw a cube, and the other is less so. Graphs 2 10/58

The cube can be drawn as a graph in multiple ways. Here are two such. One is a pretty typical way to draw a cube, and the other is less so. The second graph is a planar graph. One way to view the six faces of the cube in it are to consider the 5 regions inside the graph along with the outside. Graphs 2 10/58

In general, with a planar graph, we can see regions inside the graph, and we say the number of faces of the graph is the number of regions inside plus the outside. In this way a graph that represents a solid will have the same number of faces as the solid. Graphs 2 11/58

Clicker Question How many faces does the Octahedron have? Graphs 2 12/58

Answer There are 8 faces. Looking at the figure on the left, there are 4 faces on the top half, and 4 on the bottom half. Looking at the graph on the right, there are 7 regions inside the graph, and 1 outside, making 8 faces. Graphs 2 13/58

Is there a relationship between the Numbers of Vertices, Edges, and Faces? V E F Solid 4 6 4 Tetrahedron 8 12 6 Square 6 12 8 Octahedron 20 30 12 Dodecahedron 12 30 20 Icosahedron Stare at this table for a bit and see if you can find any relationship between V, E, F that holds for all of the solids. Graphs 2 14/58

Euler s Formula One of Euler s other contributions to graph theory was the following result about planar graphs: Graphs 2 15/58

Euler s Formula One of Euler s other contributions to graph theory was the following result about planar graphs: If V is the number of vertices, E the number of edges, and F the number of regions formed by a planar graph, then Graphs 2 15/58

Platonic Solids The solids in this picture are called Platonic solids, named after the Greek philosopher Plato. These solids are the most regular solids that are built from plane figures. While the Ancient Greeks were well aware of these five shapes, they didn t know if there were any others. Graphs 2 16/58

How Many Platonic Solids Are There? Euler s formula can help us determine all Platonic solids. The file Platonic Solids.pdf describes how one can see that there are exactly five such solids from the formula and some algebra. Graphs 2 17/58

Examples of Surfaces Two surfaces with which we are familiar are the plane and the sphere. Graphs 2 18/58

The Torus We ve also seen the torus, which is a pretty familiar shape. Graphs 2 19/58

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. Graphs 2 19/58

The Möbius Strip Escher s Ants Graphs 2 20/58

Escher s Möbius Birds Graphs 2 21/58

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. 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. This means the Möbits strip is not orientable. Graphs 2 22/58

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. Graphs 2 23/58

To start to make some sense of this, consider the following two shapes: Graphs 2 24/58

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. Graphs 2 24/58

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 Graphs 2 25/58

Two shapes which are not equivalent are the line segment and circle. Graphs 2 26/58

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. Graphs 2 26/58

Clicker Question Can the donut below be smoothly deformed into a sphere? A Yes B No C I m not sure Graphs 2 27/58

Answer A Yes, it can be, at least theoretically. We just need to squash the donut carefully, and maybe stretch the middle, to do so. This is easier to visualize in your mind than to actually do it. Graphs 2 28/58

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. Graphs 2 29/58

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. Graphs 2 30/58

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. Graphs 2 30/58

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 Graphs 2 31/58

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. Graphs 2 32/58

Building the Torus From a Rectangle Graphs 2 33/58

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. Graphs 2 33/58

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 Graphs 2 34/58

Answer We d get a cone, at least after we deform the result some after doing the gluing. Graphs 2 35/58

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. Graphs 2 36/58

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. Graphs 2 37/58

Escher s Waterfall Graphs 2 38/58

Building the Projective Plane From a Rectangle The projective plane can be constructed from a rectangle by gluing according to the following picture. Graphs 2 39/58

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. Graphs 2 39/58

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. Here are some additional pictures of the Klein bottle. Graphs 2 40/58

Building a Klein Bottle From a Rectangle The Klein bottle can be formed from gluing sides of a rectangle. Graphs 2 41/58

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 we cannot show the actual Klein bottle in a video. The Klein bottle does live in 4 dimensions. However, that doesn t help us picture what it looks like. Construction of a Klein Bottle Graphs 2 42/58

Building Surfaces From Little Panels We haven t, and won t, give a formal definition of surface. But, one way to think about them is that a surface, being a two-dimensional object, can be built by gluing together little square panels together. Graphs 2 43/58

The Euler Characteristic of a Surface If you build a surface with little panels, then you have effectively drawn a map on that surface. We can then talk about the vertices, edges, and faces of the map. If V is the number of vertices, E the number of edges, and F the number of faces, we define the Euler characteristic of the surface as V E + F We saw Euler s formula last time, which then says that the Euler characteristic of a sphere is 2. Essentially, Euler s formula is talking about a sphere. Graphs 2 44/58

There can be many ways to build a surface from little panels. It is not at all obvious that one gets the same Euler characteristic, regardless of how the surface is built. However, it can be proven that this is the case. The Euler characteristic is a property of surfaces that does not change when you smoothly deform the surface. That is, if you reshape the surface with stretching or shrinking, the Euler characteristic does not change. Cutting and gluing can effect the Euler characteristic. Graphs 2 45/58

The Euler Characteristic of a Sphere One way to see that the Euler characteristic of a sphere is 2 is to consider the following map. Graphs 2 46/58

The Euler Characteristic of a Sphere One way to see that the Euler characteristic of a sphere is 2 is to consider the following map. There are two vertices and two edges (the forward half circle and the back half circle). There are two faces of the map. Therefore, V = E = F = 2, so V E + F = 2. Graphs 2 46/58

We are using the fact that a hemisphere is topologically equivalent to a rectangle. That is, we can take a rectangle and smoothly deform it into a semicircle. Graphs 2 47/58

We are using the fact that a hemisphere is topologically equivalent to a rectangle. That is, we can take a rectangle and smoothly deform it into a semicircle. Here is a video that shows this. Graphs 2 47/58

Clicker Question What is the Euler characteristic of a Single Panel? To help answer this, what is the simplest map you can draw and cover the panel? How many vertices, edges, and faces does the map have? Graphs 2 48/58

Answer A single panel has 1 face. It has 4 vertices (the four corners) and 4 edges. Therefore, the Euler characteristic is V E + F = 4 4 + 1 = 1 Graphs 2 49/58

The Euler Characteristic of the Torus To calculate this we need to break the torus up into panels. We showed a picture of this earlier, but it is made up of lots of panels and counting them will be difficult. Graphs 2 50/58

Clicker Question How many faces are there? Graphs 2 51/58

Clicker Question How many faces are there? Answer: There are 32. There are 8 panels (4 blue, 4 yellow) in a ring around the hole. There are 4 such rings of panels, so 8 4 = 32 faces. Graphs 2 52/58

Clicker Question How many vertices are there? Graphs 2 53/58

Clicker Question How many vertices are there? Answer: There are 32. Looking at the circle on the top of the torus separating panels, we can count 8 vertices on the circle. There are four such circles, on the top, bottom, outside equator, and inside equator, so there are 8 4 = 32 total vertices. Graphs 2 54/58

How many edges are there? Graphs 2 55/58

How many edges are there? There are 64. Looking at the four circles we mentioned in the previous slide, each is made up of 8 edges, so there are 8 4 = 32 of those. But there are also edges making up smaller circles in another direction. There are 8 smaller circles, each made up of 4 edges, so there are 32 of these. Together they give 64 edges. Graphs 2 55/58

We then have V = F = 32 and E = 64. Then the Euler characteristic of the torus is V E + F = 32 64 + 32 = 0 Thus, the Euler characteristic is 0. Graphs 2 56/58

Classification of Surfaces It turns out that surfaces can be completely classified by knowing their Euler characteristic, whether or not they are orientable, and knowing about their boundary. We aren t going to go into any detail about this, but will summarize what has been discovered about this. Two surfaces without boundary are equivalent if they have the same Euler characteristic and are both Surfaces with are classified by their boundary, their orientability, and their Euler characteristic. Graphs 2 57/58

Have a good spring break! Graphs 2 58/58