Euclid: congruence, similar... measurements, scale
Questions: Is a triangle congruent/similar to another triangle?
Is an circle congruent /similar to an ellipse?
Is a triangle congruent/similar to a square?

Projective: We will discuss this in greater detail later in the course. Projections preserve lines, points of intersection and contact (tangency).
Questions: Can a triangle project onto any other triangle?
Can a circle project onto to an ellipse?
Can a triangle project onto a square?

• Topology vocabulary:
  

  "Bounded" In the plane- A line segment is bounded. A line is not bounded. A circle is bounded.  A parabola is not bounded. "Bounded" means the object can be visualized in a box.

  
  "closed"  A line segment with endpoints is closed. A circle is closed.  A line is closed. A line segment missing one or both endpoints in not closed. A circle missing one point is not closed.

  
  

  
  "open"  A line segment missing both endpoints is open. A line is open.  A circle missing one point is open.
  

  
  
  "connected" A line, a line segment, a circle, a circle missing one point,  and a parabola are all connected.
  A line segment missing an interior point is not connected.
  It has two pieces.
  

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  The following two  letters   are closed, bounded, connected and topologically equivalent.
  

  I   C

  The following two  letters   are topologically equivalent to each other but not to the previous two letters. This can be seen by removing a single point from these letters which will not disconnect the curves, as it does with the previous letters.
  

  O  D
  

  
  The letter T missing the point where the top meets the vertical line segment is not connected. It has three pieces.

  
  The letter Y missing the point where the top meets the vertical line segment also is not connected. It has three pieces.
  T and Y are topologically  the same (equivalent)! this can be seen by bending the tops to the letters up or down and stretching and shrinking the lengths as well.
  

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  Questions: Is a triangle topologically related to another triangle?
  Yes. Stretch the sides and you'll also transform the angles.
  

  Is an circle topologically related to a line?  No. 
  The circle is bounded and still connected when you remove a point.
  The line is not bounded and is disconnected when you remove a point.
  
  Is a triangle topologically related to a square?a circle? yes.
  

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  Topology and measurements:
  

• Counting on a line.  Counting on a curve.
  Keep count on the number of vertices and segments for a graph.   Compare:

  	Vertices	Edges	V-E
  Line segment I	2	1	1
  circle O	1	1	0
  8	1	2	-1
  9	2	2	0
  B	2	3	-1

  
  Curves will not be topologically equivalent if the number V-E is not the same.
  

  However, as the table shows, there are curves that have the same number V-E which are also not equivalent.
  

  These can be distinguished by other criterion, such as connectivity when a point is removed.

  
  The number V-E is a characteristic of the curve that will be the same for any topologically equivalent curve.
  

  However, this number does not completely classify the curves topologically, since curves can have the same number V-E without being equivalent.

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  Counting in the plane or Counting on a surface.
  Vertices, edges, regions.

  	Vertices	Regions	Edges
  Line segment	2	1	1
  circle	1	2	1
  8	1	3	2
  9	2	2	2
  B	2	3	3
  	6	8	12

  Counting in the plane or Counting on a sphere.

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  Euler's Formula for the plane or the sphere:
  Theorem: For any connected graph in the plane,
  

  V+R = E + 2.
  
  OR  V -E+R =2  
  

  The number 2 is called the "Euler characteristic for the plane."  ( and the sphere).
  

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  Proof:  Deconstruct the graph.
  First remove edges to "connect regions" until there is only one region.  The relation between V+R and E+ 2 stays the same.  (One edge goes with one region.)
  
  Now remove one vertex and edge to "trim the tree".
  Again the relation between V+R and E+2 stays the same. ( One edge goes with each vertex.) 
  
  Finally we have just one vertex, no edges, and one region. 
  So the relationship is that V+R = E + 2!
  

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  For any connected graph on the sphere, V+R = E + 2

  Proof:
  Choose a point P on the sphere not on the graph.
  Place a plane touching the sphere opposite to P.
  Then project the graph onto the plane.
  
  The projected graph will have the same number of vertices, edges and regions.
  
  The counting in the plane shows that the formula is true for the projected graph, and thus for the original graph on the sphere!
  

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  What about graphs on the Torus?
  We have a connected graph on the torus with one vertex, two edges and only on region. But this information does not match for the euler formula for the plane or the sphere. SO.. the torus must be topologically different from the sphere or the plane!
  In fact for the torus we see that it is possible for V+R= E   or  V-E+R= 0

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  Summary: We have learned about (and proved) Euler's Formula for the plane or the sphere:
  

The number 2 is called the "Euler characteristic for the plane."  ( and the sphere).

This formula can help solve some problems on the plane and elsewhere.
We'll continue to examine more applications of this topological property of graphs (networks) in the plane.
Counting Activity.

The Utilities problem:
There are three houses being planned on a new development owned by Aaron(A), Beth(B), and Carol (C). The phone company(P), cable company(L), and electric company(E) are planning to put in underground lines to the three houses and to minimize interference between the services, they would like to set up separate lines from there connection boxes to the houses that do not cross on the map of the development.
Draw a design the will achieve this objective.

Discussion: Analysis. Suppose this were possible and consider the consequences.
If the graph were drawn it would have V = 6 vertices and ____ E=?___ edges.
Based on Euler's formula, the graph would therefore have to have _____R=?___ regions in the plane.


By design each region would have to have at least 4 edges. Thus if we count the edges by the regions we would have at least 4R edges counting each edge twice- because an edge is a boundary for exactly two regions. Thus the number of edges must be more than 2R.
Is this possible? What can we conclude?

Complete graphs. A complete graph is a graph in which each vertex of the graph is connected by an edge to every other vertex.

For example:

2 vertices: 1 edge - a line segment
3 vertices: 3 edges -  a triangle
4 vertices: 6  edges - a tetrahedron
Notice these all have planar representations.
The technical way this is decribed is to say these graphs can be imbedded in the plane.
What about 5 vertices?
Draw a complete graph using 5 vertices.

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Discussion: Analysis. Suppose this were possible and consider the consequences.
If the graph were drawn it would have V = 5 vertices and ____ E= ?___ edges.
Based on Euler's formula, the graph would therefore have to have _____R=7___ regions in the plane.
by design each region would have to have at least 3 edges. Thus if we count the edges by the regions we would have at least 3R=21 edges counting each edge twice- because an edge is a boundary for exactly two regions. Thus the number of edges must be more than 3/2R=10 1/2.
Is this possible? What can we conclude?

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Coloring problems: Suppose we have a map that consists of a graph that determines R regions (countries) with E edges (borders).  How many colors are needed to color the regions so that countries that share a border (not merely a vertex) have different colors.
The Junkyard on the color problem.