(b)
(a)
(c)




















































3
2
2
3 2
5











4-1 (figure eight)
3-1 (trefoil)
5-1
5-2
6-1
6-2
6-3






















































gas water electricity
Hilbert Klein Poincare


























































































































































































































































































































































































































































































































































































































































area
strip
curved surface
zero curvature
the curvature
angle
curvature pi
negative
curvature
curvature
what is
this curvature?
- 2 pi


































































































































































































































































































































































































































































































































































































































































































mirror