Curvature and Euler Characteristic

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Grade Vs Difficulty:
  EasyModerateChallengingPerplexing
1-2
 
 
3-4
 
 
5-6
 
 
7-8
 
 
9-10
 
 
11-12
 
 
13-14
 
 

There are many things that can be

done under this heading, and this lesson relates to many other lessons. It is related to the Solid
Geometry
lesson.It is also related to the Criss Cross activity, and the
other curvature activity.

The first video shows Chaim Goodman Strauss leading a math circle exploring curvature via paper strips. Cutting triangles out of paper, then cutting the corners apart and placing the corners together is a start of this activity that would be suitable for first graders. The first graders could also place paper strips on surfaces and make polygons with funky angles from paper strips. The calculations with spherical triangles would be better appreciated with people in the 5th grader or higher. Hence the difficulty
matrix reflects the paper strip activity for lower grades, and the rest for 5th and higher grades.

http://youtu.be/DrE2N1ygs_E

Hwang, Andrew D., "Paper Surface Geometry: Surveying a Locally Euclidean Universe"
American Mathematical Monthly, Volume 120, Number 6, June 2013 , pp. 487-499(13)
discusses similar concepts.

The following video shows Dave Auckly giving a lecture to a large audience on the topic:

http://youtu.be/d7ukq7jf9Tc

The first attachment is a paper suitable for undergraduates on the subject of the Atiyah-Singer index theorem. The second is more suitable for graduate students.

More to come: rotation defect, Pick's theorem, wall paper, spherical symmetry, crystal classes...