Chapter 6: Graphs 6.2 The Euler Characteristic

1. Chapter 6: Graphs6.2 The Euler Characteristic

2. Draw A Graph! • Any connected graph you want, but don’t make it too simple or too crazy complicated • Only rule: No edges can cross (unless there’s a vertex where they’re crossing) OK: Not OK:

3. Now Count on Your Graph • Number of Vertices: V = ? • Number of Edges E = ? • Number of Regions (including the region outside your graph) R = ?

4. V-E+R: The Euler Characteristic ANY connected graph drawn on a flat plane with no edge crossings will have the same value for V-E+R: it will always be 2. (We’ll talk about why in a minute.)

5. V-E+R: The Euler Characteristic ANY connected graph drawn on a flat plane with no edge crossings will have the same value for V-E+R: it will always be 2. (We’ll talk about why in a minute.) The value of V-E+R for a surface is called its Euler Characteristic, so the Euler Characteristic for the plane is 2.

6. V-E+R: The Euler Characteristic The Euler Characteristic is different on different surfaces. More on this later. For now, we’re going to stick with graphs on a flat plane.

7. Why is V-E+R=2 on a flat plane? Start with simplest possible graph, count V-E+R: Now, to draw any connected graph at all, you can do it by just adding to this in 2 different ways, over and over.

8. Adding an Edge but no Vertex • How does this change V? E? R? • How does this change V-E+R?

9. Adding an Edge to a new Vertex • How does this change V? E? R? • How does this change V-E+R?

10. So… …we can draw ANY connected graph on a flat plane by starting with the basic one-edge, two-vertex graph and building it up step by step.

11. So… …we can draw ANY connected graph on a flat plane by starting with the basic one-edge, two-vertex graph and building it up step by step. …the starting graph has V-E+R=2, and each step keeps that unchanged.

12. So… …we can draw ANY connected graph on a flat plane by starting with the basic one-edge, two-vertex graph and building it up step by step. …the starting graph has V-E+R=2, and each step keeps that unchanged. …therefore, whatever graph we end up with still has V-E+R=2!

13. Other Surfaces: Spheres • Think of graph drawn on a balloon. Then flatten it out: • Same V, E, R, so same Euler Characteristic!

14. Other Surfaces: Torus But some surfaces have different Euler Characteristics, for example a torus (donut): The Euler Characteristic of a torus is 0, not 2.

15. Application to 3-D Solid Shapes We can think of “inflating” a polyhedron with colored edges and corners until it looks like a graph on a sphere: Th This comes from a cube.

16. Application to 3-D Solid Shapes • and for a polyhedron are the same thing as V, E, and R for a graph on a sphere, so we know that For any polyhedron with no holes in it,

17. Application to 3-D Solid Shapes This lets us finally see why there are only 5 regular polyhedra!

18. Application to 3-D Solid Shapes Remember that in any regular polyhedron, every face has the same # of edges, which we’ll call = # of edges / face Also every vertex has the same # of edges attached to it, so we’ll call that number = # of edges / vertex

19. Application to 3-D Solid Shapes Also remember that and So and The Euler Characteristic equation turns into

20. Application to 3-D Solid Shapes But we know is positive, so But also and must be 3 or bigger, and whole numbers

21. Application to 3-D Solid Shapes Turns out there are only 5 possibilities for and , and they lead to the 5 regular solids we know about already. So those are the only possible ones!