GaussŐs intrinsic and extrinsic measurements were quite interesting. In 2D geometry, a circle is defined as all the segments of a given radius from a given center. But say some flatlanders wanted to draw a circle. Say, for example, these flatlanders lived on a sphere. Say the sphere had a radius of ten (and thus a diameter of twenty), and the flatlanders drew a circle with a radius of 19. Certainly the area of the circle would be not nearly what it was calculated to be by ideal, Euclidian geometry. In addition to this it would turn out that the points near the middle of the circle (on the midpoints of the radii) would be much further away from each other in three space and two space than the points on the perimeter of the circle. It would take the flatlander nearly twenty times as long to walk around the inner circle composed of the midpoints of the radii then to walk around the perimeter of the circle!
Does this then mean that if I draw a circle, here in three space, it is curved in the fourth dimension? Are there such discrepancies as the example given above in our universe. Such discrepancies I realize would increase if we were doing experiments involving such a large proportion of our universe.
The diagram at the bottom of page 197 is very helpful in explaining the relationship between Klien bottle construction and Mobius band construction. It is also the same method with which you explained torus and cylinder construction, so it helped me relate. I got a little lost on the figure eight pictures of the Klein bottle, for have never seen one that way, and I was just used to the idea of a Klein bottle being a tube that is wrapped around and stuck through itself so that the two openings end up together.