Euler Numbers

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Euler Numbers
An Introduction to 2 Dimensional Manifolds

• Dr. Paul Sundheim
• University of Wisconsin at Waukesha
• with help from many sources on the web

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2-dimensions
A plane.
2 coordinates (x,y)
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Sphere
x2 y2 z2 1
A sphere is a 2-dimensional manifold.
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Cylinder
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Torus
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Moebius Band
- M. C. Escher
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Klein Bottle
http//www.gakushuin.ac.jp/881791/kuroki/Klein.GI
F
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Whats the difference?

• Cylinder
• Two boundaries
• Two sides

• Moebius strip
• One boundary
• One side

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Whats the difference? (II)

• Torus
• 0 boundaries
• 2 sides

• Klein bottle
• 0 boundaries
• 1 side

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Whats the difference? (III)

• Torus
• 0 boundaries
• 2 sides

• Sphere
• 0 boundaries
• 2 sides

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Manifold with boundary

• Every point has a neighborhood that either
• looks like a region in 2-dimensional space, or
• looks like a region in 2-dimensional half
  space.

A disk is a 2-manifold with boundary a circle.
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Terminology
Vertex
Edge
Face
3 vertices, 3 edges and 1 face.
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Triangulate Your Surface
Make sure to mark the vertices of the triangle
clearly. The edges do not have to be straight.
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Rules of Triangulation
1. Two triangles can meet in only one edge.
Like this
The vertices have to match up.
Or this
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Rules of Triangulation
1. Two triangles can meet in only one edge. 2.
Every point of the surface must be a point in at
least one triangle.
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Group Work for Triangulating the Sphere

• One person draws a triangle
• They pass the sphere and the pen to the next
  person
• They continue the triangulation by drawing one
  triangle
• They pass it to the next person
• etc.

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Calculate the Euler Number for Your Manifold
1. How many vertices are there? This number is
v 2. How many edges are there? This number is
e 3. How many faces are there? This number is f.
4. Now put the numbers in the formula v e
f ?
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Calculate the Euler Number for Your Manifold
v 4 e 6 f 4 v e f 2
1. How many vertices are there? This number is
v 2. How many edges are there? This number is
e 3. How many faces are there? This number is f.
4. Now put the numbers in the formula v e
f ?
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Time to Calculate Euler Numbers!

• Eulers number is denoted ?. So ? 2
• ? ?
• ? ?
• ? ?

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Eulers Numbers

• Eulers number is denoted ?.
• ? 2
• ? 0
• ? 0
• ? 2 2g where g is the number of holes

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Eulers Number
What is the Euler number for the following
manifold?
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Eulers Number
What is the Euler number for the following
manifold?
? 2 2g 2 2(2) 2 4 -2 so ? v e
f -2
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