Chapter 6: Graphs 6.2 The Euler Characteristic. Draw A Graph! Any connected graph you want, but...
-
Upload
kathryn-richard -
Category
Documents
-
view
224 -
download
4
Transcript of Chapter 6: Graphs 6.2 The Euler Characteristic. Draw A Graph! Any connected graph you want, but...
Chapter 6: Graphs
6.2 The Euler Characteristic
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:
Now Count on Your Graph
• Number of Vertices: V = ?
• Number of Edges E = ?• Number of Regions (including the region
outside your graph)R = ?
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.)
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.
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.
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.
Adding an Edge but no Vertex
• How does this change V? E? R?
• How does this change V-E+R?
Adding an Edge to a new Vertex
• How does this change V? E? R?
• How does this change V-E+R?
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.
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.
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!
Other Surfaces: Spheres
• Think of graph drawn on a balloon. Then flatten it out:
• Same V, E, R, so same Euler Characteristic!
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.
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.
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,
Application to 3-D Solid Shapes
This lets us finally see why there are only 5 regular polyhedra!
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
Application to 3-D Solid Shapes
Also remember that and
So and
The Euler Characteristic equation turns into
Application to 3-D Solid Shapes
But we know is positive, so
But also and must be 3 or bigger, and whole numbers
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!