Geometry out of the Paper

Dr. Jessica Purcell

University of Texas at Austin

An Introduction to Manifolds

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

Dimension

“measurement in length, width, and thickness” --Dictionary.com

0-dimensionsA point.

1-dimensionA line.

2-dimensionsA plane.

2 coordinates: (x,y)

3-dimensionsSpace.

3 coordinates: (x,y,z)

4-dimensions? Space and time.

4 coordinates: (x, y, z, w)

5-dimensions? ? ? ?

5 coordinates: (x, y, z, w, t)

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

A sphere is a 2-dimensional manifold.

Sphere

x2 + y2 + z2 = 1

3-Sphere

3-dimensional manifold

x2 + y2 + z2 + w2 = 1

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.

Activity 1

Building manifolds.

Cylinder

Torus

Moebius Band

- M. C. Escher

Klein Bottle

http://www.gakushuin.ac.jp/~881791/kuroki/Klein.GIF

What’s the difference?

Cylinder

• Two boundaries

• Two sides

Moebius strip

• One boundary

• One side

What’s the difference? (II)

Torus

• 0 boundaries

• 2 sides

Klein bottle

• 0 boundaries

• 1 side

What’s the difference? (III)

Sphere

• 0 boundaries

• 2 sides

Torus

• 0 boundaries

• 2 sides

Break

Activity 2

Euler’s Formula

Example

Example

v=3; e=3; f=2.

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

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.

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.

Cauchy’s Proof Concluded

• You are left with triangles with 3 exterior edges only. This looks like my example.

Euler Characteristic for Moebius Strip

v=? e=? f=?

v – e + f = ?

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!