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Geometry out of the Paper
Dr. Jessica Purcell
University of Texas at Austin
An Introduction to Manifolds
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Based on map of St. Isidore of Seville
600-636 A.D.
“The earth is named from its roundness
(orbis) which is like a wheel. For the Ocean flows round it on all
sides and encircles itsboundaries.”
10th Century Map
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Dimension
“measurement in length, width, and thickness” --Dictionary.com
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0-dimensionsA point.
1-dimensionA line.
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2-dimensionsA plane.
2 coordinates: (x,y)
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3-dimensionsSpace.
3 coordinates: (x,y,z)
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4-dimensions? Space and time.
4 coordinates: (x, y, z, w)
5-dimensions? ? ? ?
5 coordinates: (x, y, z, w, t)
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n-dimensional manifold
Any point has a neighborhood that looks like a region in n-dimensional space.
A circle is a 1-dimensional manifold.
x2 + y2 = 1
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A sphere is a 2-dimensional manifold.
Sphere
x2 + y2 + z2 = 1
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3-Sphere
3-dimensional manifold
x2 + y2 + z2 + w2 = 1
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Manifold with boundaryEvery point has a neighborhood that either:
• looks like a region in n-dimensional space, or
• looks like a region in n-dimensional half space.
A disk is a 2-manifold with boundary a circle.
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Activity 1
Building manifolds.
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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.GIF
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What’s the difference?
Cylinder
• Two boundaries
• Two sides
Moebius strip
• One boundary
• One side
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What’s the difference? (II)
Torus
• 0 boundaries
• 2 sides
Klein bottle
• 0 boundaries
• 1 side
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What’s the difference? (III)
Sphere
• 0 boundaries
• 2 sides
Torus
• 0 boundaries
• 2 sides
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Break
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Activity 2
Euler’s Formula
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Example
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Example
v=3; e=3; f=2.
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Answer to #7: 2
Is the answer really always 2, or did it just happen that none of us drew the right picture to get something besides 2?
Mathematical proof: Show that no matter how many vertices, edges and faces we have, if we follow the rules, then
v – e + f = 2
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Cauchy’s Proof of Euler’s Formula
• If there is any face with more than 3 sides, draw a diagonal. Repeat. Eventually, everything is divided into triangles.
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Cauchy’s Proof Continued
• Repeat the following two steps:1. Remove triangles with one edge on the exterior.
2. Remove triangles with two edges on exterior.
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Cauchy’s Proof Concluded
• You are left with triangles with 3 exterior edges only. This looks like my example.
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Euler Characteristic for Moebius Strip
v=? e=? f=?
v – e + f = ?
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Concluding Remarks
• To a bug on an n-dimensional manifold, the world looks n-dimensional.
• It is hard to tell manifolds apart when you’re standing inside them.
• Different manifolds have many different interesting properties. Keep exploring!