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: Is a triangle
projectively related to another triangle?
Is an circle projectively related to an ellipse?
Is a triangle projectively
related to 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.
The following two letters are topologically
equivalent.
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.
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.
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.
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 
VE 

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 VE is not the same.
However, as the table shows, there are
curves that have the same number VE which are also not equivalent.
These can be distinguished by other criterion, such as connectivity when a point is removed.
The number VE 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 VE without being equivalent.
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.
Euler's Formula for the plane
or the sphere:
Theorem: For any connected graph in the plane,
The number 2 is called the "Euler characteristic
for the plane." ( and the sphere).
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!
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!
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 VE+R=
0
Summary: We have learned about
(and proved) Euler's Formula
for the plane or the sphere: